WEBVTT 00:00:00.000 --> 00:00:09.760 align:middle line:84% [SQUEAKING] [RUSTLING] [CLICKING] 00:00:09.760 --> 00:00:12.800 align:middle line:84% HAZHIR RAHMANDAD: Good to see you all. 00:00:12.800 --> 00:00:15.600 align:middle line:84% We have the predator prey model today. 00:00:15.600 --> 00:00:20.740 align:middle line:84% And actually, a lot of you submitted models. 00:00:20.740 --> 00:00:22.200 align:middle line:90% So that's great. 00:00:22.200 --> 00:00:23.980 align:middle line:90% We can jump in. 00:00:23.980 --> 00:00:26.600 align:middle line:90% 00:00:26.600 --> 00:00:33.640 align:middle line:84% Actually, so predator prey model, very, very classic-- 00:00:33.640 --> 00:00:37.280 align:middle line:84% of course, we had a little bit of an extended version 00:00:37.280 --> 00:00:42.120 align:middle line:84% to think about carrying capacity for the prey-- think of a deer 00:00:42.120 --> 00:00:45.200 align:middle line:90% and wolf model with vegetation. 00:00:45.200 --> 00:00:50.320 align:middle line:84% And we wanted to focus on a simple but robust model 00:00:50.320 --> 00:00:54.600 align:middle line:84% and, to the extent that you can, think about model analysis 00:00:54.600 --> 00:00:56.760 align:middle line:90% as well. 00:00:56.760 --> 00:00:59.620 align:middle line:84% So that was the problem definition. 00:00:59.620 --> 00:01:02.020 align:middle line:84% And let me turn off the light on the presentation. 00:01:02.020 --> 00:01:05.400 align:middle line:84% That probably helps with the recording. 00:01:05.400 --> 00:01:12.760 align:middle line:84% So maybe we can get started with your reflections 00:01:12.760 --> 00:01:15.320 align:middle line:90% on things that were challenging. 00:01:15.320 --> 00:01:19.560 align:middle line:84% That's usually helpful for focusing the conversation 00:01:19.560 --> 00:01:22.280 align:middle line:84% on what might be more useful feedback as we 00:01:22.280 --> 00:01:25.080 align:middle line:84% go through the models because we have several models. 00:01:25.080 --> 00:01:28.380 align:middle line:84% And we may not have time to cover everything. 00:01:28.380 --> 00:01:33.580 align:middle line:84% So whoever likes to share, go ahead and share 00:01:33.580 --> 00:01:37.800 align:middle line:84% what were some of your major challenges 00:01:37.800 --> 00:01:41.720 align:middle line:84% as you were going through the model. 00:01:41.720 --> 00:01:44.200 align:middle line:90% STUDENT: [INAUDIBLE] 00:01:44.200 --> 00:01:50.300 align:middle line:90% 00:01:50.300 --> 00:01:51.800 align:middle line:84% HAZHIR RAHMANDAD: So can you tell me 00:01:51.800 --> 00:01:53.880 align:middle line:84% more what was challenging with it? 00:01:53.880 --> 00:01:55.760 align:middle line:90% STUDENT: Yeah. 00:01:55.760 --> 00:02:02.580 align:middle line:84% So because there are a lot of behaviors that [INAUDIBLE] 00:02:02.580 --> 00:02:03.830 align:middle line:90% HAZHIR RAHMANDAD: That's good. 00:02:03.830 --> 00:02:06.410 align:middle line:84% So what are the behaviors you were trying to get? 00:02:06.410 --> 00:02:10.370 align:middle line:84% STUDENT: I wanted [INAUDIBLE] actually 00:02:10.370 --> 00:02:15.730 align:middle line:84% be a little bit lower when deer were less dense and higher when 00:02:15.730 --> 00:02:17.190 align:middle line:90% they're more dense. 00:02:17.190 --> 00:02:27.210 align:middle line:84% And they're needs to be a maximum and minimum consumption 00:02:27.210 --> 00:02:28.810 align:middle line:84% level that wolves can actually live 00:02:28.810 --> 00:02:32.530 align:middle line:90% on that has to do with that. 00:02:32.530 --> 00:02:36.370 align:middle line:84% And honestly, the difficulty [INAUDIBLE] 00:02:36.370 --> 00:02:42.890 align:middle line:90% 00:02:42.890 --> 00:02:44.170 align:middle line:90% HAZHIR RAHMANDAD: Good. 00:02:44.170 --> 00:02:48.030 align:middle line:84% I think this might have been one of the more common challenges, 00:02:48.030 --> 00:02:50.570 align:middle line:84% not only for predation, but also for the other side-- that 00:02:50.570 --> 00:02:56.450 align:middle line:84% is, consumption of vegetation has a challenge to it. 00:02:56.450 --> 00:02:57.370 align:middle line:90% Good. 00:02:57.370 --> 00:03:02.970 align:middle line:84% Other challenges that anybody faced in modeling? 00:03:02.970 --> 00:03:04.490 align:middle line:84% STUDENT: The biggest challenge was 00:03:04.490 --> 00:03:09.033 align:middle line:84% I was trying to conceptualize the model based on [INAUDIBLE] 00:03:09.033 --> 00:03:09.950 align:middle line:90% HAZHIR RAHMANDAD: Yes. 00:03:09.950 --> 00:03:13.490 align:middle line:84% STUDENT: But then I have this memory [INAUDIBLE] 00:03:13.490 --> 00:03:14.605 align:middle line:90% which it didn't go well. 00:03:14.605 --> 00:03:16.230 align:middle line:84% HAZHIR RAHMANDAD: OK, very interesting. 00:03:16.230 --> 00:03:17.570 align:middle line:90% So tell us more. 00:03:17.570 --> 00:03:27.550 align:middle line:84% So it's interference from the memory of classical models. 00:03:27.550 --> 00:03:29.890 align:middle line:84% STUDENT: [INAUDIBLE] And I'm sorry to say 00:03:29.890 --> 00:03:33.930 align:middle line:84% that I did the whole model with a new conceptualization, a more 00:03:33.930 --> 00:03:35.570 align:middle line:90% realistic conceptualization. 00:03:35.570 --> 00:03:40.050 align:middle line:84% But then I went back to the mathematical model [INAUDIBLE] 00:03:40.050 --> 00:03:41.170 align:middle line:90% what I remembered. 00:03:41.170 --> 00:03:43.290 align:middle line:90% HAZHIR RAHMANDAD: Yes, yes. 00:03:43.290 --> 00:03:43.790 align:middle line:90% Yeah? 00:03:43.790 --> 00:03:51.650 align:middle line:84% STUDENT: So the other challenge was operationalization, extreme 00:03:51.650 --> 00:03:52.810 align:middle line:90% conditions. 00:03:52.810 --> 00:03:57.010 align:middle line:84% If we don't have any deer, we shouldn't have any wolves. 00:03:57.010 --> 00:04:00.650 align:middle line:84% And that was also one of the [INAUDIBLE] 00:04:00.650 --> 00:04:03.684 align:middle line:84% wolves did not grow on their own. 00:04:03.684 --> 00:04:04.726 align:middle line:90% HAZHIR RAHMANDAD: Mm-hmm. 00:04:04.726 --> 00:04:13.210 align:middle line:90% 00:04:13.210 --> 00:04:18.390 align:middle line:90% Yeah, good, good. 00:04:18.390 --> 00:04:20.610 align:middle line:90% Other challenges? 00:04:20.610 --> 00:04:22.170 align:middle line:90% Jason? 00:04:22.170 --> 00:04:30.810 align:middle line:84% STUDENT: [INAUDIBLE] I found it hard to get away 00:04:30.810 --> 00:04:34.650 align:middle line:84% from the very basic story of thinking 00:04:34.650 --> 00:04:40.210 align:middle line:84% in terms of an agent-based model of deer and wolves interacting. 00:04:40.210 --> 00:04:44.810 align:middle line:84% And at the end, what I settled on 00:04:44.810 --> 00:04:47.850 align:middle line:84% was something that [INAUDIBLE] you 00:04:47.850 --> 00:04:52.108 align:middle line:84% don't need to model those interactions directly. 00:04:52.108 --> 00:04:53.900 align:middle line:84% HAZHIR RAHMANDAD: So how does that thinking 00:04:53.900 --> 00:04:56.960 align:middle line:84% about those interactions make it more complex for you? 00:04:56.960 --> 00:05:06.220 align:middle line:84% STUDENT: It distracted from what is the important features 00:05:06.220 --> 00:05:13.140 align:middle line:84% [INAUDIBLE] At the end, the rates that we wanted to compare 00:05:13.140 --> 00:05:18.980 align:middle line:84% and put together don't require these more 00:05:18.980 --> 00:05:22.377 align:middle line:84% microscopic interactions to think about. 00:05:22.377 --> 00:05:23.252 align:middle line:90% HAZHIR RAHMANDAD: OK. 00:05:23.252 --> 00:05:25.860 align:middle line:84% So you were starting with microscopic interaction 00:05:25.860 --> 00:05:27.200 align:middle line:90% between deer and wolves. 00:05:27.200 --> 00:05:30.680 align:middle line:84% And you couldn't get to any good solution 00:05:30.680 --> 00:05:32.120 align:middle line:90% for the more aggregate problem. 00:05:32.120 --> 00:05:35.082 align:middle line:84% Could you get to a good solution for the individual-based model? 00:05:35.082 --> 00:05:37.040 align:middle line:84% If you were building an individual-based model, 00:05:37.040 --> 00:05:39.948 align:middle line:84% do you think it would have been easier? 00:05:39.948 --> 00:05:40.740 align:middle line:90% That's interesting. 00:05:40.740 --> 00:05:41.782 align:middle line:90% We can come back to that. 00:05:41.782 --> 00:05:42.400 align:middle line:90% I'm not sure. 00:05:42.400 --> 00:05:44.275 align:middle line:84% I think, actually, the individual-based model 00:05:44.275 --> 00:05:47.360 align:middle line:84% in this case is more complex than the aggregate one. 00:05:47.360 --> 00:05:49.740 align:middle line:90% But we can see. 00:05:49.740 --> 00:05:51.120 align:middle line:90% Any other reactions? 00:05:51.120 --> 00:05:54.740 align:middle line:84% STUDENT: Well, I just forgot the time step size was important. 00:05:54.740 --> 00:05:57.500 align:middle line:84% HAZHIR RAHMANDAD: So how does the time step play a role? 00:05:57.500 --> 00:05:59.540 align:middle line:84% STUDENT: I just couldn't get it to calibrate 00:05:59.540 --> 00:06:02.120 align:middle line:90% when I set it equal to 1. 00:06:02.120 --> 00:06:03.865 align:middle line:90% And I needed to set it to 0.25. 00:06:03.865 --> 00:06:05.240 align:middle line:84% HAZHIR RAHMANDAD: OK, good point. 00:06:05.240 --> 00:06:10.100 align:middle line:84% So that's a very general rule, lesson that time step 00:06:10.100 --> 00:06:13.740 align:middle line:84% is actually not a construct that has a conceptual meaning 00:06:13.740 --> 00:06:14.960 align:middle line:90% in the real world for us. 00:06:14.960 --> 00:06:17.060 align:middle line:84% Real world is a continuous world. 00:06:17.060 --> 00:06:20.803 align:middle line:84% So if the behavior of your model is sensitive to time step-- that 00:06:20.803 --> 00:06:22.220 align:middle line:84% is, you change it and the behavior 00:06:22.220 --> 00:06:25.040 align:middle line:84% changes-- it means that you have too big a time step. 00:06:25.040 --> 00:06:27.380 align:middle line:84% Just bring it down further, divide it by 2, 00:06:27.380 --> 00:06:30.300 align:middle line:84% and divide it by 2 until there is no change in the behavior. 00:06:30.300 --> 00:06:34.980 align:middle line:84% That's a very simple heuristic that usually works. 00:06:34.980 --> 00:06:36.282 align:middle line:90% Oh, Anne, go ahead. 00:06:36.282 --> 00:06:37.740 align:middle line:84% And somehow, we are not seeing you. 00:06:37.740 --> 00:06:40.780 align:middle line:84% I don't know what's the problem here. 00:06:40.780 --> 00:06:42.440 align:middle line:84% STUDENT: Are you sharing the screen? 00:06:42.440 --> 00:06:44.107 align:middle line:84% I don't think you're sharing the screen. 00:06:44.107 --> 00:06:48.190 align:middle line:84% HAZHIR RAHMANDAD: Ah, that might also be a problem. 00:06:48.190 --> 00:06:50.310 align:middle line:90% OK, so let's continue. 00:06:50.310 --> 00:06:52.850 align:middle line:84% Basically, we were discussing common challenges. 00:06:52.850 --> 00:06:55.430 align:middle line:84% Anything else that anybody wants to share on 00:06:55.430 --> 00:07:00.990 align:middle line:84% what was challenging in this week's modeling work for them? 00:07:00.990 --> 00:07:03.030 align:middle line:90% And then we dive deeper. 00:07:03.030 --> 00:07:07.150 align:middle line:84% STUDENT: I'm just debating whether [INAUDIBLE]. 00:07:07.150 --> 00:07:14.430 align:middle line:90% 00:07:14.430 --> 00:07:17.530 align:middle line:84% HAZHIR RAHMANDAD: So that's a generally an important question, 00:07:17.530 --> 00:07:21.470 align:middle line:84% whether you go with a single flow or separate birth and death 00:07:21.470 --> 00:07:21.970 align:middle line:90% rate. 00:07:21.970 --> 00:07:23.990 align:middle line:84% And in fact, separate birth and death rate 00:07:23.990 --> 00:07:26.310 align:middle line:84% from natural causes versus natural death 00:07:26.310 --> 00:07:27.970 align:middle line:90% rate from predation. 00:07:27.970 --> 00:07:30.750 align:middle line:90% So that's another good question. 00:07:30.750 --> 00:07:34.470 align:middle line:84% OK, let's look at some of the models that you have built 00:07:34.470 --> 00:07:39.570 align:middle line:84% and see how you have handled some of these complexities. 00:07:39.570 --> 00:07:42.850 align:middle line:84% So that would require opening a bunch of models. 00:07:42.850 --> 00:07:45.230 align:middle line:84% I'll just open the version B for those of you 00:07:45.230 --> 00:07:51.390 align:middle line:84% who have submitted two because we can look at them. 00:07:51.390 --> 00:07:54.890 align:middle line:84% OK, so this is a good enough number to get us started with. 00:07:54.890 --> 00:07:59.710 align:middle line:90% So this one is for Sahin. 00:07:59.710 --> 00:08:04.290 align:middle line:90% So let's take a look. 00:08:04.290 --> 00:08:07.170 align:middle line:84% So stocks, we have the prey population, 00:08:07.170 --> 00:08:09.930 align:middle line:84% predator population, and vegetation. 00:08:09.930 --> 00:08:14.910 align:middle line:84% So I think everybody is pretty much on the same page 00:08:14.910 --> 00:08:15.690 align:middle line:90% about that. 00:08:15.690 --> 00:08:19.870 align:middle line:84% Some of you have broken it down by two age groups. 00:08:19.870 --> 00:08:23.510 align:middle line:84% The majority have not, from a quick look. 00:08:23.510 --> 00:08:29.690 align:middle line:90% So let's look at one of them. 00:08:29.690 --> 00:08:34.049 align:middle line:84% And we can see how the ideas are working for the rest. 00:08:34.049 --> 00:08:38.309 align:middle line:90% So this is the vegetation side. 00:08:38.309 --> 00:08:42.630 align:middle line:84% And there is a regeneration rate for vegetation. 00:08:42.630 --> 00:08:47.270 align:middle line:84% The regeneration rate, I think, is fairly common 00:08:47.270 --> 00:08:53.790 align:middle line:84% and fairly standard to think of it as a fractional regeneration 00:08:53.790 --> 00:08:57.510 align:middle line:90% rate, as a first cut. 00:08:57.510 --> 00:09:00.330 align:middle line:84% So who has used a similar formulation? 00:09:00.330 --> 00:09:05.470 align:middle line:84% Basically, the stock multiplied by a fractional growth rate. 00:09:05.470 --> 00:09:07.830 align:middle line:84% Anybody has a formulation that is different 00:09:07.830 --> 00:09:13.230 align:middle line:84% and wants to share that or think about that? 00:09:13.230 --> 00:09:15.270 align:middle line:90% Let's look at your model then. 00:09:15.270 --> 00:09:17.890 align:middle line:84% STUDENT: I mean, it's not that much more complicated. 00:09:17.890 --> 00:09:20.043 align:middle line:84% It basically just creates the idea [INAUDIBLE] 00:09:20.043 --> 00:09:21.210 align:middle line:90% HAZHIR RAHMANDAD: Very good. 00:09:21.210 --> 00:09:26.150 align:middle line:84% So that's actually an important consideration, if we want to-- 00:09:26.150 --> 00:09:28.230 align:middle line:90% so let's see. 00:09:28.230 --> 00:09:31.150 align:middle line:84% I think I had your model somewhere. 00:09:31.150 --> 00:09:35.197 align:middle line:84% It's just hard to know which model we are looking at. 00:09:35.197 --> 00:09:36.030 align:middle line:90% STUDENT: [INAUDIBLE] 00:09:36.030 --> 00:09:36.988 align:middle line:90% HAZHIR RAHMANDAD: Yeah. 00:09:36.988 --> 00:09:40.070 align:middle line:90% STUDENT: [INAUDIBLE] 00:09:40.070 --> 00:09:46.040 align:middle line:84% HAZHIR RAHMANDAD: So here, you have max vegetation level 00:09:46.040 --> 00:09:48.240 align:middle line:90% impacting regrowth rates. 00:09:48.240 --> 00:09:49.420 align:middle line:90% Let's take a look at this. 00:09:49.420 --> 00:09:52.960 align:middle line:84% So max vegetation minus vegetation divided by time step 00:09:52.960 --> 00:09:53.460 align:middle line:90% here. 00:09:53.460 --> 00:09:57.903 align:middle line:90% So you are basically-- 00:09:57.903 --> 00:09:59.820 align:middle line:84% this is actually a very different formulation. 00:09:59.820 --> 00:10:01.460 align:middle line:90% It's not problematic. 00:10:01.460 --> 00:10:03.420 align:middle line:84% So there are two different logics 00:10:03.420 --> 00:10:04.920 align:middle line:84% that you can think about when you're 00:10:04.920 --> 00:10:07.300 align:middle line:84% thinking about regeneration of vegetation. 00:10:07.300 --> 00:10:10.720 align:middle line:84% One is that you need seeds that come from that 00:10:10.720 --> 00:10:13.280 align:middle line:90% vegetation to regenerate. 00:10:13.280 --> 00:10:17.800 align:middle line:84% The other is that, as vegetation gets denser, 00:10:17.800 --> 00:10:22.300 align:middle line:84% it competes with itself for food, for water, for sunlight. 00:10:22.300 --> 00:10:24.800 align:middle line:84% And that brings down its regrowth rate. 00:10:24.800 --> 00:10:28.160 align:middle line:84% And therefore, it cannot go above some level. 00:10:28.160 --> 00:10:33.000 align:middle line:84% So vegetation alone model without deer and nothing 00:10:33.000 --> 00:10:36.000 align:middle line:84% should not grow exponentially forever. 00:10:36.000 --> 00:10:41.420 align:middle line:84% So this is what Jake's model accounts for more strongly. 00:10:41.420 --> 00:10:44.760 align:middle line:84% So this model says, there is a total vegetation level 00:10:44.760 --> 00:10:47.480 align:middle line:84% that is feasible to have in the area. 00:10:47.480 --> 00:10:50.100 align:middle line:84% Now, here, we don't have the area explicitly. 00:10:50.100 --> 00:10:53.720 align:middle line:84% You could have a max density multiplied by the area 00:10:53.720 --> 00:10:58.200 align:middle line:84% to give you the max vegetation level. 00:10:58.200 --> 00:11:02.615 align:middle line:84% And then you look at that versus the current level of vegetation. 00:11:02.615 --> 00:11:04.240 align:middle line:84% And you would say, vegetation is trying 00:11:04.240 --> 00:11:06.300 align:middle line:90% to grow back to that level. 00:11:06.300 --> 00:11:10.920 align:middle line:84% So this formulation does that alternative conceptualization. 00:11:10.920 --> 00:11:16.000 align:middle line:84% I would say this is more robust for the extreme condition 00:11:16.000 --> 00:11:19.000 align:middle line:84% that we discussed-- that is, the exponential growth in vegetation 00:11:19.000 --> 00:11:21.280 align:middle line:90% forever is not going to work. 00:11:21.280 --> 00:11:23.340 align:middle line:90% It does have a downside. 00:11:23.340 --> 00:11:27.140 align:middle line:84% And that is, if you are running out of vegetation, 00:11:27.140 --> 00:11:30.320 align:middle line:84% the regeneration rate-- and therefore, the growth rate-- 00:11:30.320 --> 00:11:33.640 align:middle line:90% is actually going to collapse. 00:11:33.640 --> 00:11:37.640 align:middle line:84% And that is a challenge in many of the actual ecosystems. 00:11:37.640 --> 00:11:40.780 align:middle line:84% When you have too much of the prey, 00:11:40.780 --> 00:11:44.120 align:middle line:84% basically, deer in this case, eating the vegetation 00:11:44.120 --> 00:11:46.560 align:middle line:84% to the bone, then they don't regenerate. 00:11:46.560 --> 00:11:48.100 align:middle line:90% And that becomes a bottleneck. 00:11:48.100 --> 00:11:51.880 align:middle line:84% Whereas, in this model, the increase in the vegetation 00:11:51.880 --> 00:11:55.980 align:middle line:84% will be always very robust when you don't have much vegetation. 00:11:55.980 --> 00:11:59.000 align:middle line:84% So this is actually a non-trivial formulation 00:11:59.000 --> 00:12:05.080 align:middle line:84% to get both of these factors play a role in your overall rate 00:12:05.080 --> 00:12:06.182 align:middle line:90% of vegetation. 00:12:06.182 --> 00:12:07.640 align:middle line:84% STUDENT: Do you have any preference 00:12:07.640 --> 00:12:13.440 align:middle line:84% between adding max vegetation like this versus just having 00:12:13.440 --> 00:12:19.320 align:middle line:84% two variables that [INAUDIBLE] what you calculate 00:12:19.320 --> 00:12:22.740 align:middle line:90% your regeneration [INAUDIBLE] 00:12:22.740 --> 00:12:29.000 align:middle line:90% 00:12:29.000 --> 00:12:32.100 align:middle line:84% HAZHIR RAHMANDAD: So I'm not quite following the two options. 00:12:32.100 --> 00:12:36.537 align:middle line:84% So one is max regeneration rate or max total vegetation-- 00:12:36.537 --> 00:12:38.870 align:middle line:84% STUDENT: No, either you take it into account like this-- 00:12:38.870 --> 00:12:39.787 align:middle line:90% HAZHIR RAHMANDAD: Yes. 00:12:39.787 --> 00:12:49.250 align:middle line:84% STUDENT: [INAUDIBLE] there's one maximum amount of-- 00:12:49.250 --> 00:12:50.723 align:middle line:90% maximum amount of-- 00:12:50.723 --> 00:12:52.390 align:middle line:84% HAZHIR RAHMANDAD: Shipment in that case. 00:12:52.390 --> 00:12:54.640 align:middle line:84% STUDENT: Or like paperwork and processing just because 00:12:54.640 --> 00:12:58.108 align:middle line:90% of how many [INAUDIBLE] 00:12:58.108 --> 00:12:59.650 align:middle line:84% HAZHIR RAHMANDAD: So that formulation 00:12:59.650 --> 00:13:02.938 align:middle line:84% plays a more important role in the outflows. 00:13:02.938 --> 00:13:04.230 align:middle line:90% So we will get to the outflows. 00:13:04.230 --> 00:13:06.530 align:middle line:84% I think, for the inflow here, we need 00:13:06.530 --> 00:13:09.170 align:middle line:84% a formulation that actually brings both of those ideas 00:13:09.170 --> 00:13:11.250 align:middle line:90% together. 00:13:11.250 --> 00:13:13.290 align:middle line:84% STUDENT: So we have been trying this. 00:13:13.290 --> 00:13:16.290 align:middle line:84% And I think it has a little bit of downfall 00:13:16.290 --> 00:13:20.890 align:middle line:84% because, if the max vegetation is very high-- 00:13:20.890 --> 00:13:22.050 align:middle line:90% imagine it's like here. 00:13:22.050 --> 00:13:24.050 align:middle line:90% And you have two deer only. 00:13:24.050 --> 00:13:26.730 align:middle line:84% And suddenly, the inflow at the very first beginning 00:13:26.730 --> 00:13:32.090 align:middle line:84% is very large and they grow very rapidly, whereas, if we like 00:13:32.090 --> 00:13:34.110 align:middle line:84% multiply it by the fractional growth-- 00:13:34.110 --> 00:13:34.590 align:middle line:90% HAZHIR RAHMANDAD: Wait, wait. 00:13:34.590 --> 00:13:35.510 align:middle line:90% This is not for deer. 00:13:35.510 --> 00:13:37.290 align:middle line:90% This is for vegetation. 00:13:37.290 --> 00:13:39.390 align:middle line:84% STUDENT: I was just giving a sensible example. 00:13:39.390 --> 00:13:44.970 align:middle line:84% But in general, in any species, if the max level is very high, 00:13:44.970 --> 00:13:47.810 align:middle line:84% so suddenly the inflow is very large compared 00:13:47.810 --> 00:13:52.733 align:middle line:84% to the exponential growth that it controls [INAUDIBLE] 00:13:52.733 --> 00:13:53.650 align:middle line:90% HAZHIR RAHMANDAD: Yes. 00:13:53.650 --> 00:13:57.250 align:middle line:84% And it also partly reflects whether you are using 00:13:57.250 --> 00:14:00.010 align:middle line:90% a single flow or double flows. 00:14:00.010 --> 00:14:04.010 align:middle line:84% That is, in a two-flow model, you could actually 00:14:04.010 --> 00:14:07.050 align:middle line:84% have this regeneration rate being exponential, 00:14:07.050 --> 00:14:13.090 align:middle line:84% being a direct multiplier of the stock. 00:14:13.090 --> 00:14:18.410 align:middle line:84% But then you could have an outflow that grows significantly 00:14:18.410 --> 00:14:20.910 align:middle line:90% as you have more density. 00:14:20.910 --> 00:14:22.430 align:middle line:90% And at some point, it saturates. 00:14:22.430 --> 00:14:24.290 align:middle line:90% It doesn't allow it to go up. 00:14:24.290 --> 00:14:30.130 align:middle line:84% So that would be one way to capture the non-linearity rather 00:14:30.130 --> 00:14:32.980 align:middle line:84% than putting everything into the inflow. 00:14:32.980 --> 00:14:36.460 align:middle line:84% Now, there is some physics behind it either way 00:14:36.460 --> 00:14:37.760 align:middle line:90% that you need to capture. 00:14:37.760 --> 00:14:40.380 align:middle line:84% So one way of thinking about the physics of it 00:14:40.380 --> 00:14:43.800 align:middle line:84% is that, OK, the seeds are spread, but they don't grow. 00:14:43.800 --> 00:14:45.980 align:middle line:84% They don't even start to grow because they 00:14:45.980 --> 00:14:48.280 align:middle line:84% don't have the sunlight and the water that they need. 00:14:48.280 --> 00:14:51.480 align:middle line:84% And so as such, the inflow is disrupted. 00:14:51.480 --> 00:14:53.460 align:middle line:84% So I need to capture it in the inflow. 00:14:53.460 --> 00:14:55.420 align:middle line:84% The other conceptualization is saying, well, 00:14:55.420 --> 00:14:56.520 align:middle line:90% the seeds are spread. 00:14:56.520 --> 00:14:59.320 align:middle line:84% They start to grow, but they die out very soon. 00:14:59.320 --> 00:15:01.120 align:middle line:90% So the outflow is much higher. 00:15:01.120 --> 00:15:02.620 align:middle line:84% You could go with either of those 00:15:02.620 --> 00:15:06.220 align:middle line:90% and still have a story to tell. 00:15:06.220 --> 00:15:06.860 align:middle line:90% Alexander? 00:15:06.860 --> 00:15:08.600 align:middle line:84% STUDENT: So trying to capture both, 00:15:08.600 --> 00:15:11.740 align:middle line:84% I used the logistic function in the inflow. 00:15:11.740 --> 00:15:16.775 align:middle line:84% So it has the property where, if there's little 00:15:16.775 --> 00:15:18.400 align:middle line:84% then you'll have that reinforcing loop. 00:15:18.400 --> 00:15:19.752 align:middle line:90% But then it doesn't go above. 00:15:19.752 --> 00:15:21.960 align:middle line:84% HAZHIR RAHMANDAD: Yeah, that's generally a good idea. 00:15:21.960 --> 00:15:24.720 align:middle line:84% But let's actually check out your model. 00:15:24.720 --> 00:15:26.658 align:middle line:90% I don't remember. 00:15:26.658 --> 00:15:28.200 align:middle line:84% STUDENT: I didn't include vegetation. 00:15:28.200 --> 00:15:31.028 align:middle line:84% I just did part A. But I did it for deer. 00:15:31.028 --> 00:15:32.820 align:middle line:84% HAZHIR RAHMANDAD: OK, so we can take a look 00:15:32.820 --> 00:15:38.940 align:middle line:84% at the model you have for deer and see how that is playing 00:15:38.940 --> 00:15:42.140 align:middle line:90% a role in the formulation. 00:15:42.140 --> 00:15:44.660 align:middle line:84% STUDENT: So I called it the natural change 00:15:44.660 --> 00:15:45.880 align:middle line:90% in prey population. 00:15:45.880 --> 00:15:48.540 align:middle line:84% So that's supposed to capture-- it's not only births, 00:15:48.540 --> 00:15:50.280 align:middle line:84% but it's also supposed to be deaths. 00:15:50.280 --> 00:15:54.600 align:middle line:84% HAZHIR RAHMANDAD: So this is really a single flow version, 00:15:54.600 --> 00:15:56.100 align:middle line:90% net flow version. 00:15:56.100 --> 00:16:01.040 align:middle line:84% Now, net flow, this model is simple, but it is not trivial. 00:16:01.040 --> 00:16:03.660 align:middle line:84% And you will see that many of you 00:16:03.660 --> 00:16:06.940 align:middle line:84% had a challenge getting the inflows and outflows robust. 00:16:06.940 --> 00:16:10.500 align:middle line:84% So when you combine inflow and outflow, it is very risky. 00:16:10.500 --> 00:16:13.700 align:middle line:84% It becomes even harder because you 00:16:13.700 --> 00:16:16.140 align:middle line:84% need to keep in mind what is happening 00:16:16.140 --> 00:16:17.860 align:middle line:90% both for the inflow and outflow. 00:16:17.860 --> 00:16:23.200 align:middle line:84% So this is prey multiplied by their natural birth rate. 00:16:23.200 --> 00:16:27.660 align:middle line:84% This is the part that is the positive growth rate. 00:16:27.660 --> 00:16:33.580 align:middle line:84% And then here, you have this whole thing 00:16:33.580 --> 00:16:35.740 align:middle line:90% multiplied by a factor. 00:16:35.740 --> 00:16:40.020 align:middle line:84% This factor is net positive as long 00:16:40.020 --> 00:16:44.060 align:middle line:84% as prey is below its carrying capacity. 00:16:44.060 --> 00:16:47.200 align:middle line:84% And once prey exceeds its carrying capacity, 00:16:47.200 --> 00:16:49.860 align:middle line:90% that ratio becomes above 1. 00:16:49.860 --> 00:16:52.540 align:middle line:84% 1 minus that ratio becomes negative. 00:16:52.540 --> 00:16:56.400 align:middle line:90% Now, you have a net loss rate. 00:16:56.400 --> 00:17:01.780 align:middle line:84% So this is how Alexander has formulated it. 00:17:01.780 --> 00:17:03.180 align:middle line:90% And it works well. 00:17:03.180 --> 00:17:07.260 align:middle line:84% I think this does OK in creating a population that 00:17:07.260 --> 00:17:13.420 align:middle line:84% grows exponentially, gets to a saturation point, 00:17:13.420 --> 00:17:17.740 align:middle line:84% and stays there for a single species 00:17:17.740 --> 00:17:20.500 align:middle line:84% without influence from anything else. 00:17:20.500 --> 00:17:26.180 align:middle line:84% So this is a meaningful, decent formulation 00:17:26.180 --> 00:17:31.590 align:middle line:84% if you are doing a single flow version of it. 00:17:31.590 --> 00:17:33.950 align:middle line:84% My worry with this formulation is 00:17:33.950 --> 00:17:37.870 align:middle line:84% that the mechanics of birth and death are missing. 00:17:37.870 --> 00:17:41.070 align:middle line:84% So you don't as easily tell a story. 00:17:41.070 --> 00:17:43.350 align:middle line:84% It is more a mathematical function 00:17:43.350 --> 00:17:47.630 align:middle line:84% that works OK and gives you what you want and is perhaps 00:17:47.630 --> 00:17:49.530 align:middle line:90% equivalent to another model. 00:17:49.530 --> 00:17:54.710 align:middle line:84% That is, if you had the model with natural birth 00:17:54.710 --> 00:17:57.910 align:middle line:84% rate multiplied by prey as inflow, 00:17:57.910 --> 00:18:00.150 align:middle line:84% this model would then have an outflow, which 00:18:00.150 --> 00:18:02.790 align:middle line:84% is prey to the power of 2 divided 00:18:02.790 --> 00:18:06.790 align:middle line:84% by carrying capacity multiplied by natural birth rate. 00:18:06.790 --> 00:18:10.490 align:middle line:84% Now, that is not a very intuitive outflow 00:18:10.490 --> 00:18:13.390 align:middle line:84% because where does the prey to the power of 2 come from? 00:18:13.390 --> 00:18:17.430 align:middle line:84% It comes from a meaningful idea that too much density leads 00:18:17.430 --> 00:18:19.170 align:middle line:90% to more killing of the prey. 00:18:19.170 --> 00:18:23.310 align:middle line:84% So as such, you can tell a story. 00:18:23.310 --> 00:18:26.310 align:middle line:84% And mathematically, it works very nicely because then it 00:18:26.310 --> 00:18:28.050 align:middle line:90% saturates at carrying capacity. 00:18:28.050 --> 00:18:30.545 align:middle line:90% 00:18:30.545 --> 00:18:32.670 align:middle line:84% STUDENT: I spent like 45 minutes on this yesterday. 00:18:32.670 --> 00:18:34.950 align:middle line:84% If you did the math if I do the math, 00:18:34.950 --> 00:18:39.850 align:middle line:84% basically what it is is it's kind of odd, 00:18:39.850 --> 00:18:44.030 align:middle line:84% but basically the outflow is essentially the prey population 00:18:44.030 --> 00:18:47.490 align:middle line:84% divided by some reference lifespan-- 00:18:47.490 --> 00:18:50.750 align:middle line:84% so what the lifespan of prey is at the carrying capacity-- 00:18:50.750 --> 00:18:57.520 align:middle line:84% multiplied by-- I think it's the ratio of capacity to-- 00:18:57.520 --> 00:19:01.533 align:middle line:84% yeah, multiplied by the ratio of capacity to prey. 00:19:01.533 --> 00:19:03.950 align:middle line:84% HAZHIR RAHMANDAD: The prey should be in the numerator, not 00:19:03.950 --> 00:19:04.910 align:middle line:90% the denominator. 00:19:04.910 --> 00:19:08.110 align:middle line:84% STUDENT: As in like prey's in the numerator, and then 00:19:08.110 --> 00:19:09.630 align:middle line:90% the denominator, the lifespan. 00:19:09.630 --> 00:19:11.130 align:middle line:84% It's a first order draining process. 00:19:11.130 --> 00:19:12.255 align:middle line:90% HAZHIR RAHMANDAD: Yes, yes. 00:19:12.255 --> 00:19:13.590 align:middle line:90% It will be a first order. 00:19:13.590 --> 00:19:17.790 align:middle line:84% So basically, the natural birth rate of prey 00:19:17.790 --> 00:19:21.630 align:middle line:84% will be 1 over the lifespan in that version, 00:19:21.630 --> 00:19:24.210 align:middle line:84% in the outflow conceptualization. 00:19:24.210 --> 00:19:25.230 align:middle line:90% You're right. 00:19:25.230 --> 00:19:26.030 align:middle line:90% OK. 00:19:26.030 --> 00:19:27.770 align:middle line:90% So this is one way to do it. 00:19:27.770 --> 00:19:30.950 align:middle line:84% And it would work OK in this case. 00:19:30.950 --> 00:19:32.870 align:middle line:90% And I just forget. 00:19:32.870 --> 00:19:36.070 align:middle line:90% 00:19:36.070 --> 00:19:43.870 align:middle line:84% Any other formulations for vegetation growth rate? 00:19:43.870 --> 00:19:46.590 align:middle line:84% STUDENT: [INAUDIBLE] I have one mistake. 00:19:46.590 --> 00:19:49.010 align:middle line:84% But I only did it for the inflow. 00:19:49.010 --> 00:19:53.270 align:middle line:84% So I realized that's wrong [INAUDIBLE] 00:19:53.270 --> 00:19:56.350 align:middle line:84% HAZHIR RAHMANDAD: OK, so let's take a look at your model 00:19:56.350 --> 00:19:59.610 align:middle line:90% to see what we had. 00:19:59.610 --> 00:20:07.390 align:middle line:90% 00:20:07.390 --> 00:20:09.790 align:middle line:84% So this is the carrying capacity. 00:20:09.790 --> 00:20:13.590 align:middle line:84% And your vegetation growth has a max vegetation 00:20:13.590 --> 00:20:17.533 align:middle line:90% and vegetation growth rate. 00:20:17.533 --> 00:20:19.950 align:middle line:84% STUDENT: It shouldn't be growth rate times max vegetation, 00:20:19.950 --> 00:20:21.243 align:middle line:90% but just max vegetation. 00:20:21.243 --> 00:20:22.160 align:middle line:90% HAZHIR RAHMANDAD: Yes. 00:20:22.160 --> 00:20:25.720 align:middle line:84% So if you remove this max vegetation multiplication, 00:20:25.720 --> 00:20:29.900 align:middle line:84% because that one doesn't actually work, 00:20:29.900 --> 00:20:36.040 align:middle line:84% then you have vegetation growth rate-- 00:20:36.040 --> 00:20:43.120 align:middle line:84% so this has units problems as well or maybe not. 00:20:43.120 --> 00:20:45.520 align:middle line:84% So if the growth rate is one per year 00:20:45.520 --> 00:20:48.120 align:middle line:84% and you have max vegetation, then the units 00:20:48.120 --> 00:20:54.180 align:middle line:84% will be fine because the second part is dimensionless. 00:20:54.180 --> 00:20:58.680 align:middle line:90% 00:20:58.680 --> 00:21:00.763 align:middle line:84% STUDENT: Yeah, that should be just the vegetation. 00:21:00.763 --> 00:21:02.805 align:middle line:84% HAZHIR RAHMANDAD: So it should be the vegetation, 00:21:02.805 --> 00:21:03.817 align:middle line:90% if you want to-- yeah. 00:21:03.817 --> 00:21:05.400 align:middle line:84% And if you make it vegetation, then it 00:21:05.400 --> 00:21:09.480 align:middle line:84% becomes very similar to the formulation that Alexander had. 00:21:09.480 --> 00:21:11.260 align:middle line:84% STUDENT: But then it's wrong still. 00:21:11.260 --> 00:21:13.340 align:middle line:84% I mean, based on what you're saying, 00:21:13.340 --> 00:21:16.680 align:middle line:84% it's still wrong because then it can go negative. 00:21:16.680 --> 00:21:19.300 align:middle line:84% HAZHIR RAHMANDAD: So this can go negative. 00:21:19.300 --> 00:21:20.780 align:middle line:90% So if you don't have-- 00:21:20.780 --> 00:21:23.840 align:middle line:90% 00:21:23.840 --> 00:21:26.040 align:middle line:84% I mean, I don't know where-- oh, this is the stock, 00:21:26.040 --> 00:21:28.160 align:middle line:90% your carrying capacity. 00:21:28.160 --> 00:21:31.860 align:middle line:84% So it is not going to go negative, 00:21:31.860 --> 00:21:33.660 align:middle line:84% if you don't have any other inflows. 00:21:33.660 --> 00:21:36.560 align:middle line:84% So if this is the only inflow to the stock, 00:21:36.560 --> 00:21:41.260 align:middle line:84% the stock will grow until it reaches the max vegetation. 00:21:41.260 --> 00:21:46.440 align:middle line:84% But as soon as it reaches max vegetation, this will shut off. 00:21:46.440 --> 00:21:47.680 align:middle line:90% It becomes 0. 00:21:47.680 --> 00:21:51.400 align:middle line:84% And therefore, it cannot go above that max vegetation. 00:21:51.400 --> 00:21:53.940 align:middle line:84% In fact, even if it goes above max vegetation, 00:21:53.940 --> 00:21:55.940 align:middle line:84% this becomes an outflow for the vegetation. 00:21:55.940 --> 00:22:00.140 align:middle line:84% It kills vegetation and, as such, brings it back to the max. 00:22:00.140 --> 00:22:02.560 align:middle line:84% So it is not problematic, in the sense 00:22:02.560 --> 00:22:06.740 align:middle line:84% of going negative, if you think of it as a two-way flow, 00:22:06.740 --> 00:22:08.120 align:middle line:90% not as a one-way flow. 00:22:08.120 --> 00:22:10.680 align:middle line:84% STUDENT: Yeah, that's what I meant, that it can only work 00:22:10.680 --> 00:22:14.303 align:middle line:90% if it's a net flow [INAUDIBLE] 00:22:14.303 --> 00:22:15.220 align:middle line:90% HAZHIR RAHMANDAD: Yes. 00:22:15.220 --> 00:22:18.330 align:middle line:84% But as an inflow, it is still doing an OK job 00:22:18.330 --> 00:22:21.530 align:middle line:84% because here you have the consumption 00:22:21.530 --> 00:22:23.290 align:middle line:90% of vegetation by deer. 00:22:23.290 --> 00:22:27.490 align:middle line:84% You don't have the natural death of vegetation 00:22:27.490 --> 00:22:31.230 align:middle line:84% as the grass gets dry and dies after a while. 00:22:31.230 --> 00:22:36.590 align:middle line:84% So that part is actually already built into here. 00:22:36.590 --> 00:22:39.250 align:middle line:84% So it is a two-way flow in your model as well. 00:22:39.250 --> 00:22:40.290 align:middle line:90% OK. 00:22:40.290 --> 00:22:43.150 align:middle line:90% So you got an idea. 00:22:43.150 --> 00:22:47.530 align:middle line:84% So basically, yes, exponential growth type 00:22:47.530 --> 00:22:52.730 align:middle line:84% with only vegetation multiplied by a natural growth rate 00:22:52.730 --> 00:22:54.850 align:middle line:90% is a possibility. 00:22:54.850 --> 00:22:59.410 align:middle line:84% But the possibility has the downside of becoming too strong. 00:22:59.410 --> 00:23:04.530 align:middle line:84% So you do want to think about the impact of carrying capacity 00:23:04.530 --> 00:23:08.130 align:middle line:90% of the land for vegetation. 00:23:08.130 --> 00:23:11.690 align:middle line:84% One way to do it is similar to what Alexander did, 00:23:11.690 --> 00:23:15.050 align:middle line:84% which made me think of it as a two-way flow. 00:23:15.050 --> 00:23:24.450 align:middle line:84% Another is to have, basically, a consumption or a natural death 00:23:24.450 --> 00:23:29.330 align:middle line:84% rate of vegetation that will grow as vegetation 00:23:29.330 --> 00:23:31.210 align:middle line:90% becomes denser. 00:23:31.210 --> 00:23:32.750 align:middle line:84% You need to have one of those two. 00:23:32.750 --> 00:23:35.030 align:middle line:84% And because we have the same story with the deer, 00:23:35.030 --> 00:23:36.770 align:middle line:84% we can go through the formulations 00:23:36.770 --> 00:23:38.970 align:middle line:84% in the deer population and, in fact, 00:23:38.970 --> 00:23:44.170 align:middle line:84% with the wolves, that will give more info on how you 00:23:44.170 --> 00:23:46.170 align:middle line:90% have been thinking about it. 00:23:46.170 --> 00:23:51.030 align:middle line:90% So let's go back to-- 00:23:51.030 --> 00:23:53.970 align:middle line:90% 00:23:53.970 --> 00:23:57.170 align:middle line:84% oh, we can go to any of the models for-- 00:23:57.170 --> 00:24:00.310 align:middle line:90% let's just pick another one. 00:24:00.310 --> 00:24:03.450 align:middle line:90% So this one this is Alisa. 00:24:03.450 --> 00:24:13.830 align:middle line:84% OK, so Ali has just simple names-- n0, n1, n2-- 00:24:13.830 --> 00:24:17.570 align:middle line:90% which I suspect this is-- 00:24:17.570 --> 00:24:19.250 align:middle line:90% this is the wolf, predator. 00:24:19.250 --> 00:24:24.050 align:middle line:84% And then you have the deer as n1 and n2 being the grass. 00:24:24.050 --> 00:24:27.170 align:middle line:84% So here, you have the death rate that 00:24:27.170 --> 00:24:30.570 align:middle line:84% is coming from the-- the birth rate that is 00:24:30.570 --> 00:24:32.670 align:middle line:90% coming from just regeneration. 00:24:32.670 --> 00:24:36.170 align:middle line:84% And then you have the death rate that is-- 00:24:36.170 --> 00:24:38.290 align:middle line:90% let's look at this equation. 00:24:38.290 --> 00:24:41.650 align:middle line:84% So I'm a little bit-- so just looking at this graph, 00:24:41.650 --> 00:24:45.770 align:middle line:84% I can't follow what the logic of that equation is because there 00:24:45.770 --> 00:24:48.930 align:middle line:90% are four, five different-- 00:24:48.930 --> 00:24:50.850 align:middle line:84% six arrows going into that equation. 00:24:50.850 --> 00:24:53.310 align:middle line:90% So the logic will be complex. 00:24:53.310 --> 00:24:56.210 align:middle line:84% You are combining six different things into a single flow. 00:24:56.210 --> 00:24:57.510 align:middle line:90% Not a good idea. 00:24:57.510 --> 00:24:58.630 align:middle line:90% Break down the logic. 00:24:58.630 --> 00:25:00.830 align:middle line:84% It's clearer for anybody to look at it, 00:25:00.830 --> 00:25:04.010 align:middle line:84% but also will be clearer for you because you can then 00:25:04.010 --> 00:25:08.690 align:middle line:84% open up each of those pathways and see how it is impacting 00:25:08.690 --> 00:25:10.430 align:middle line:90% your overall flow variable. 00:25:10.430 --> 00:25:13.780 align:middle line:84% So that's just a simple process thing. 00:25:13.780 --> 00:25:16.520 align:middle line:84% Don't drop in 100 things together. 00:25:16.520 --> 00:25:20.260 align:middle line:84% Now, we have n2-- that is the grass-- 00:25:20.260 --> 00:25:26.380 align:middle line:84% multiplied by its natural fractional death rate 00:25:26.380 --> 00:25:32.020 align:middle line:90% plus n2 divided by normal n. 00:25:32.020 --> 00:25:33.320 align:middle line:90% What is normal n? 00:25:33.320 --> 00:25:34.905 align:middle line:90% STUDENT: It's a unit correction. 00:25:34.905 --> 00:25:35.780 align:middle line:90% HAZHIR RAHMANDAD: OK. 00:25:35.780 --> 00:25:39.020 align:middle line:84% So unit corrections are risky things to use. 00:25:39.020 --> 00:25:43.680 align:middle line:84% That is, if you're just adding something to fix your units, 00:25:43.680 --> 00:25:45.220 align:middle line:90% it means you're in trouble. 00:25:45.220 --> 00:25:47.900 align:middle line:90% Your logic is not working well. 00:25:47.900 --> 00:25:51.380 align:middle line:84% Almost always, there is a way to fix the units based 00:25:51.380 --> 00:25:56.560 align:middle line:84% on the mechanisms rather than adding just ad hoc parameters. 00:25:56.560 --> 00:25:59.200 align:middle line:84% So think of those as fault factors you shouldn't use. 00:25:59.200 --> 00:26:01.700 align:middle line:84% There are situations-- like, if you're raising something 00:26:01.700 --> 00:26:04.180 align:middle line:84% to a power in Vensim, it gives you an error no matter 00:26:04.180 --> 00:26:09.800 align:middle line:84% what, if you have a unit for that element, 00:26:09.800 --> 00:26:15.200 align:middle line:84% and then you want to make it dimensionless to not get it. 00:26:15.200 --> 00:26:17.980 align:middle line:84% So there are very rare cases you need fudge factors, 00:26:17.980 --> 00:26:20.020 align:middle line:90% but don't use them. 00:26:20.020 --> 00:26:22.860 align:middle line:90% But this is ultimately some-- 00:26:22.860 --> 00:26:24.760 align:middle line:90% maybe even a value of 1. 00:26:24.760 --> 00:26:30.220 align:middle line:84% But you have the grass multiplied by grass carrying 00:26:30.220 --> 00:26:32.980 align:middle line:90% capacity. 00:26:32.980 --> 00:26:34.860 align:middle line:84% And then this thing is multiplied 00:26:34.860 --> 00:26:39.540 align:middle line:90% by grass itself, ultimately. 00:26:39.540 --> 00:26:41.760 align:middle line:90% So I'm not following that logic. 00:26:41.760 --> 00:26:42.760 align:middle line:90% Can you tell what it is? 00:26:42.760 --> 00:26:44.635 align:middle line:84% STUDENT: The first thing that you pointed out 00:26:44.635 --> 00:26:45.920 align:middle line:90% is natural death rate. 00:26:45.920 --> 00:26:49.140 align:middle line:84% Second one is kind of like a carrying capacity 00:26:49.140 --> 00:26:51.060 align:middle line:90% if they reach a limit. 00:26:51.060 --> 00:26:54.457 align:middle line:90% So it slows-- 00:26:54.457 --> 00:26:56.040 align:middle line:84% HAZHIR RAHMANDAD: It slows it further? 00:26:56.040 --> 00:26:58.540 align:middle line:84% STUDENT: Yes, it slows it further, and then kills grass. 00:26:58.540 --> 00:27:00.290 align:middle line:84% HAZHIR RAHMANDAD: But why is it multiplied 00:27:00.290 --> 00:27:03.740 align:middle line:90% by carrying capacity n2? 00:27:03.740 --> 00:27:07.653 align:middle line:84% STUDENT: I was thinking of a factor that-- 00:27:07.653 --> 00:27:09.820 align:middle line:84% HAZHIR RAHMANDAD: Yes, but don't think about factors 00:27:09.820 --> 00:27:11.520 align:middle line:90% in this abstract way. 00:27:11.520 --> 00:27:14.480 align:middle line:84% Factors can be any number, any mathematical number. 00:27:14.480 --> 00:27:16.860 align:middle line:84% And then you have infinite possibilities. 00:27:16.860 --> 00:27:20.280 align:middle line:84% And you can't narrow down into something meaningful. 00:27:20.280 --> 00:27:23.200 align:middle line:84% So actually, carrying capacity is a meaningful concept. 00:27:23.200 --> 00:27:27.700 align:middle line:84% So if you have n2 divided by carrying capacity of n2-- that 00:27:27.700 --> 00:27:32.500 align:middle line:84% is, the total grass that possible-- that ratio 00:27:32.500 --> 00:27:36.780 align:middle line:84% becomes basically how close you are to carrying capacity. 00:27:36.780 --> 00:27:43.100 align:middle line:84% And then you are saying, as we get closer to that ratio of 1, 00:27:43.100 --> 00:27:48.180 align:middle line:84% then the natural death rate goes up and, in this case, linearly. 00:27:48.180 --> 00:27:51.020 align:middle line:90% So it can become double-- 00:27:51.020 --> 00:27:54.660 align:middle line:90% well, not double. 00:27:54.660 --> 00:27:59.615 align:middle line:84% Then it has to be multiplied by some fractional-- 00:27:59.615 --> 00:28:00.240 align:middle line:90% STUDENT: Right. 00:28:00.240 --> 00:28:03.300 align:middle line:84% [INAUDIBLE] same as the fractional death rate. 00:28:03.300 --> 00:28:05.327 align:middle line:84% That's why I did multiplication communication. 00:28:05.327 --> 00:28:06.202 align:middle line:90% HAZHIR RAHMANDAD: OK. 00:28:06.202 --> 00:28:09.830 align:middle line:84% And then you have ny divided by normal n multiplied 00:28:09.830 --> 00:28:12.170 align:middle line:90% by impact of predator n1. 00:28:12.170 --> 00:28:15.310 align:middle line:84% So basically, you are looking here 00:28:15.310 --> 00:28:21.470 align:middle line:90% at the deer eating the grass. 00:28:21.470 --> 00:28:24.950 align:middle line:84% But you're multiplying that by n2. 00:28:24.950 --> 00:28:26.590 align:middle line:90% Why is that? 00:28:26.590 --> 00:28:29.150 align:middle line:90% STUDENT: So here's the thing. 00:28:29.150 --> 00:28:34.283 align:middle line:84% If we don't have any grass, we shouldn't have any outflow. 00:28:34.283 --> 00:28:35.450 align:middle line:90% HAZHIR RAHMANDAD: Very good. 00:28:35.450 --> 00:28:38.990 align:middle line:84% So everybody has had a version of this thinking, 00:28:38.990 --> 00:28:41.290 align:middle line:84% I think, most of you have been thinking. 00:28:41.290 --> 00:28:43.670 align:middle line:84% And then the way you have implemented-- many of you-- 00:28:43.670 --> 00:28:45.850 align:middle line:84% I had a very quick look at some of the models-- 00:28:45.850 --> 00:28:49.590 align:middle line:84% was this idea that you multiply by the grass 00:28:49.590 --> 00:28:53.070 align:middle line:84% so that you capture the fact that, if you don't have grass, 00:28:53.070 --> 00:28:57.910 align:middle line:84% the consumption of grass should go down, 00:28:57.910 --> 00:29:02.050 align:middle line:84% which goes up to some point to address that need. 00:29:02.050 --> 00:29:03.750 align:middle line:84% But what kind of a problem does it 00:29:03.750 --> 00:29:12.310 align:middle line:84% create, if you just multiply the grass, in this case, by the deer 00:29:12.310 --> 00:29:15.510 align:middle line:84% by some constant as the outflow, which is 00:29:15.510 --> 00:29:17.090 align:middle line:90% what this formulation is doing? 00:29:17.090 --> 00:29:17.830 align:middle line:90% Katherine? 00:29:17.830 --> 00:29:21.210 align:middle line:84% STUDENT: It doesn't account for the appetite being constrained. 00:29:21.210 --> 00:29:23.950 align:middle line:84% So if you have infinite grass, then the deer 00:29:23.950 --> 00:29:26.915 align:middle line:84% are going to be eating so much grass [INAUDIBLE] 00:29:26.915 --> 00:29:28.290 align:middle line:84% HAZHIR RAHMANDAD: Yes, very good. 00:29:28.290 --> 00:29:30.973 align:middle line:84% So that's a very big robustness problem. 00:29:30.973 --> 00:29:32.890 align:middle line:84% And it's not really just a robustness problem. 00:29:32.890 --> 00:29:35.850 align:middle line:84% It is really the physics is gotten wrong. 00:29:35.850 --> 00:29:38.470 align:middle line:84% If you have infinite grass and one deer, 00:29:38.470 --> 00:29:41.870 align:middle line:84% that single deer will be consuming infinite grass. 00:29:41.870 --> 00:29:43.670 align:middle line:90% That is not going to happen. 00:29:43.670 --> 00:29:48.910 align:middle line:84% So there is a serious constraint on how much 00:29:48.910 --> 00:29:51.710 align:middle line:84% grass can be consumed by the deer 00:29:51.710 --> 00:29:57.710 align:middle line:84% that this type of multiplicative constraint does not address. 00:29:57.710 --> 00:30:00.070 align:middle line:90% Does that make sense? 00:30:00.070 --> 00:30:03.420 align:middle line:84% So let's actually look at that in a few other formulations, 00:30:03.420 --> 00:30:05.800 align:middle line:84% because this is really at the heart of the tougher 00:30:05.800 --> 00:30:09.560 align:middle line:90% formulations in this model. 00:30:09.560 --> 00:30:12.500 align:middle line:84% So let's look at some other random models. 00:30:12.500 --> 00:30:16.200 align:middle line:90% 00:30:16.200 --> 00:30:17.100 align:middle line:90% Whose is this one? 00:30:17.100 --> 00:30:18.620 align:middle line:90% This is Oriana's. 00:30:18.620 --> 00:30:23.520 align:middle line:90% 00:30:23.520 --> 00:30:25.760 align:middle line:90% So this is the vegetation. 00:30:25.760 --> 00:30:27.966 align:middle line:90% And you have-- 00:30:27.966 --> 00:30:30.400 align:middle line:90% STUDENT: [INAUDIBLE] 00:30:30.400 --> 00:30:33.460 align:middle line:84% HAZHIR RAHMANDAD: Yeah, we could go and look at the prey one. 00:30:33.460 --> 00:30:35.940 align:middle line:84% So you can see that the logic is actually very similar. 00:30:35.940 --> 00:30:40.580 align:middle line:84% So you can look at either of the prey or vegetation. 00:30:40.580 --> 00:30:46.540 align:middle line:84% And the underlying structure shouldn't be that different. 00:30:46.540 --> 00:30:51.462 align:middle line:84% So here, the version you have is prey-- 00:30:51.462 --> 00:30:52.920 align:middle line:84% STUDENT: This is like the prey that 00:30:52.920 --> 00:30:54.520 align:middle line:90% died because of the predators. 00:30:54.520 --> 00:30:56.260 align:middle line:90% HAZHIR RAHMANDAD: Yes, yes. 00:30:56.260 --> 00:31:00.520 align:middle line:84% And then the prey predator death rate is-- 00:31:00.520 --> 00:31:01.400 align:middle line:90% yes? 00:31:01.400 --> 00:31:03.782 align:middle line:90% STUDENT: [INAUDIBLE] 00:31:03.782 --> 00:31:06.240 align:middle line:90% 00:31:06.240 --> 00:31:12.920 align:middle line:84% HAZHIR RAHMANDAD: It depends on the predator and prey required 00:31:12.920 --> 00:31:15.040 align:middle line:90% per prey per day-- 00:31:15.040 --> 00:31:16.420 align:middle line:90% predator per day. 00:31:16.420 --> 00:31:21.680 align:middle line:84% So this is the constraint that Kathy mentioned that, OK, you 00:31:21.680 --> 00:31:25.320 align:middle line:84% want to have some amount of consumption based 00:31:25.320 --> 00:31:27.220 align:middle line:90% on how many predators you have. 00:31:27.220 --> 00:31:29.020 align:middle line:90% So this is a good indicator. 00:31:29.020 --> 00:31:35.360 align:middle line:84% This is the desired consumption of prey by the predator. 00:31:35.360 --> 00:31:36.580 align:middle line:90% So far so good. 00:31:36.580 --> 00:31:39.520 align:middle line:84% But in the next part when you multiply that 00:31:39.520 --> 00:31:43.640 align:middle line:84% by the prey population, you get two problems. 00:31:43.640 --> 00:31:47.420 align:middle line:84% First of all, you get the same equation that Ali had. 00:31:47.420 --> 00:31:50.320 align:middle line:84% Basically, it is the equivalent equation. 00:31:50.320 --> 00:31:52.600 align:middle line:84% The second problem is a unit problem. 00:31:52.600 --> 00:31:56.640 align:middle line:90% So what is the unit problem? 00:31:56.640 --> 00:32:01.960 align:middle line:84% This should be in deer per wolf per day or per month 00:32:01.960 --> 00:32:03.280 align:middle line:90% or whatever is the unit. 00:32:03.280 --> 00:32:07.560 align:middle line:84% Multiplied by the wolf, it becomes deer per month. 00:32:07.560 --> 00:32:10.480 align:middle line:84% Now, it is in the right unit for outflow. 00:32:10.480 --> 00:32:13.040 align:middle line:90% You multiply that by the deer. 00:32:13.040 --> 00:32:16.060 align:middle line:84% You would not get the right unit as well. 00:32:16.060 --> 00:32:17.540 align:middle line:90% So that's the challenge. 00:32:17.540 --> 00:32:20.840 align:middle line:84% And it probably shows up in the units as well. 00:32:20.840 --> 00:32:25.680 align:middle line:84% OK, let's look at some more formulations. 00:32:25.680 --> 00:32:31.520 align:middle line:84% I think-- let's see if we have any of the folks online 00:32:31.520 --> 00:32:34.480 align:middle line:90% their models we can check out. 00:32:34.480 --> 00:32:37.960 align:middle line:90% Let's look at Gerardo's model. 00:32:37.960 --> 00:32:42.280 align:middle line:84% So Gerardo has two stocks for each of the-- 00:32:42.280 --> 00:32:47.520 align:middle line:84% or at least-- yes, for these two groups and also for vegetation. 00:32:47.520 --> 00:32:48.780 align:middle line:90% Two stocks are fine. 00:32:48.780 --> 00:32:50.320 align:middle line:90% Nothing wrong about it. 00:32:50.320 --> 00:32:54.760 align:middle line:84% It adds to your headache for keeping track of the fact 00:32:54.760 --> 00:33:00.830 align:middle line:84% that, OK, yes, the baby prey could be eaten as well. 00:33:00.830 --> 00:33:03.070 align:middle line:84% So do you want to capture that or not? 00:33:03.070 --> 00:33:06.450 align:middle line:90% They are actually easy prey. 00:33:06.450 --> 00:33:12.790 align:middle line:84% And so in this version, that is not captured. 00:33:12.790 --> 00:33:16.710 align:middle line:84% So for now, let's assume this is not a problem. 00:33:16.710 --> 00:33:20.250 align:middle line:84% Let's not worry about the baby prey and their fate. 00:33:20.250 --> 00:33:27.010 align:middle line:84% Assume they all grow to maturity. 00:33:27.010 --> 00:33:29.970 align:middle line:84% And if you make that assumption, this average maturity time 00:33:29.970 --> 00:33:31.530 align:middle line:90% works fine. 00:33:31.530 --> 00:33:35.650 align:middle line:90% Now, we have starving. 00:33:35.650 --> 00:33:43.810 align:middle line:84% We have death rate from average prey lifetime and eaten prey, 00:33:43.810 --> 00:33:51.250 align:middle line:90% which is from the predators. 00:33:51.250 --> 00:33:54.030 align:middle line:84% So this is an interesting question. 00:33:54.030 --> 00:33:57.490 align:middle line:84% Do we need all of the three pathways? 00:33:57.490 --> 00:34:00.770 align:middle line:90% Or do we need fewer? 00:34:00.770 --> 00:34:06.610 align:middle line:84% So I like to have two, separating the hunting 00:34:06.610 --> 00:34:11.409 align:middle line:84% and harvesting from natural death. 00:34:11.409 --> 00:34:16.530 align:middle line:84% The separation between natural deaths and starving death 00:34:16.530 --> 00:34:20.929 align:middle line:90% is harder to pin down. 00:34:20.929 --> 00:34:27.010 align:middle line:84% It feels to me they are too close to each other. 00:34:27.010 --> 00:34:30.270 align:middle line:84% But sometimes, it may be helpful. 00:34:30.270 --> 00:34:33.449 align:middle line:90% 00:34:33.449 --> 00:34:33.949 align:middle line:90% OK. 00:34:33.949 --> 00:34:35.610 align:middle line:84% For now, let's focus on the eaten 00:34:35.610 --> 00:34:37.889 align:middle line:84% prey because those two-- that adds 00:34:37.889 --> 00:34:41.050 align:middle line:90% another layer of complexity. 00:34:41.050 --> 00:34:47.250 align:middle line:84% So eaten prey in Gerardo's equation 00:34:47.250 --> 00:34:50.530 align:middle line:84% is based on desired prey-- desired 00:34:50.530 --> 00:34:53.389 align:middle line:90% prey are coming from predators. 00:34:53.389 --> 00:34:56.370 align:middle line:84% So this is the same equation that Oriana also 00:34:56.370 --> 00:35:00.670 align:middle line:84% had, predators multiplied by their average consumption. 00:35:00.670 --> 00:35:01.590 align:middle line:90% This is the desired. 00:35:01.590 --> 00:35:02.350 align:middle line:90% That's good. 00:35:02.350 --> 00:35:05.250 align:middle line:90% This is what we want to have. 00:35:05.250 --> 00:35:09.310 align:middle line:90% Then let's go to the equation. 00:35:09.310 --> 00:35:11.690 align:middle line:90% This looks more complex now. 00:35:11.690 --> 00:35:17.890 align:middle line:84% So this is max of 0 and minimum of desired prey 00:35:17.890 --> 00:35:22.130 align:middle line:84% versus prey divided by feasible time to eat the prey. 00:35:22.130 --> 00:35:27.690 align:middle line:90% So is the logic clear? 00:35:27.690 --> 00:35:31.890 align:middle line:90% 00:35:31.890 --> 00:35:34.770 align:middle line:84% Who wants to take a stab at explaining this logic, what's 00:35:34.770 --> 00:35:35.590 align:middle line:90% happening here? 00:35:35.590 --> 00:35:43.570 align:middle line:90% 00:35:43.570 --> 00:35:46.170 align:middle line:90% STUDENT: [INAUDIBLE] 00:35:46.170 --> 00:35:48.610 align:middle line:90% 00:35:48.610 --> 00:35:50.940 align:middle line:84% HAZHIR RAHMANDAD: So basically, the max of 0 00:35:50.940 --> 00:35:54.100 align:middle line:84% is saying it shouldn't go negative. 00:35:54.100 --> 00:35:57.440 align:middle line:84% Now, this should not go negative anyway. 00:35:57.440 --> 00:36:01.880 align:middle line:84% So this is another kind of a stylistic idea to keep in mind. 00:36:01.880 --> 00:36:05.080 align:middle line:84% That is, the second part of the equation should be robust. 00:36:05.080 --> 00:36:07.700 align:middle line:84% It should not go negative no matter what. 00:36:07.700 --> 00:36:10.620 align:middle line:84% Putting this max 0 does two things 00:36:10.620 --> 00:36:12.040 align:middle line:90% that are not necessarily good. 00:36:12.040 --> 00:36:15.325 align:middle line:84% One is that it makes your equation a bit too complex. 00:36:15.325 --> 00:36:16.700 align:middle line:84% So when I'm looking at it, I have 00:36:16.700 --> 00:36:18.840 align:middle line:84% to process that and make sense of it. 00:36:18.840 --> 00:36:21.040 align:middle line:84% The second thing is that, for whatever reason, 00:36:21.040 --> 00:36:24.180 align:middle line:84% if your other equations lead to a negative value 00:36:24.180 --> 00:36:26.760 align:middle line:84% because they were not correctly formulated, 00:36:26.760 --> 00:36:28.620 align:middle line:90% this hides that error. 00:36:28.620 --> 00:36:31.260 align:middle line:84% So you actually don't want to add these max 00:36:31.260 --> 00:36:34.060 align:middle line:90% 0 thingies everywhere. 00:36:34.060 --> 00:36:36.820 align:middle line:84% As long as you have taught through why 00:36:36.820 --> 00:36:40.140 align:middle line:84% the equation shouldn't go negative, let 00:36:40.140 --> 00:36:41.720 align:middle line:90% the equation do its work. 00:36:41.720 --> 00:36:44.780 align:middle line:84% Don't add another layer of protection. 00:36:44.780 --> 00:36:49.120 align:middle line:84% OK, so simpler solution would be just the minimum. 00:36:49.120 --> 00:36:53.340 align:middle line:84% So now, this minimum, it is either 00:36:53.340 --> 00:36:58.180 align:middle line:84% what the prey wants to eat or what the-- 00:36:58.180 --> 00:37:01.500 align:middle line:84% sorry, what the predators want to eat 00:37:01.500 --> 00:37:03.560 align:middle line:90% or what the prey can offer. 00:37:03.560 --> 00:37:05.820 align:middle line:84% And that's what the prey can offer. 00:37:05.820 --> 00:37:09.660 align:middle line:84% Here, it's formulated very simply as number 00:37:09.660 --> 00:37:16.060 align:middle line:84% of prey divided by the minimum time to harvest or something-- 00:37:16.060 --> 00:37:18.820 align:middle line:84% that is, minimum time to find prey to eat 00:37:18.820 --> 00:37:20.448 align:middle line:90% or something like that. 00:37:20.448 --> 00:37:22.240 align:middle line:84% This is actually a pretty good formulation. 00:37:22.240 --> 00:37:24.020 align:middle line:90% This is a good starting point. 00:37:24.020 --> 00:37:30.420 align:middle line:84% This is perhaps the simplest formulation you can use. 00:37:30.420 --> 00:37:35.740 align:middle line:84% So if you want-- anybody who has a critique for this formulation? 00:37:35.740 --> 00:37:36.560 align:middle line:90% Alexander? 00:37:36.560 --> 00:37:41.320 align:middle line:84% STUDENT: So just the Min creates this sharp behavior. 00:37:41.320 --> 00:37:42.900 align:middle line:84% And so this is essentially the same 00:37:42.900 --> 00:37:45.520 align:middle line:90% as the shipment formulation. 00:37:45.520 --> 00:37:52.710 align:middle line:84% And so for that, you typically use a soft [INAUDIBLE] 00:37:52.710 --> 00:37:55.110 align:middle line:84% HAZHIR RAHMANDAD: So the critique that Alexander said 00:37:55.110 --> 00:37:58.090 align:middle line:84% is that this creates a very discontinuous behavior. 00:37:58.090 --> 00:38:02.630 align:middle line:84% That is, as long as pray divided by feasible time to eat prey 00:38:02.630 --> 00:38:08.670 align:middle line:84% is even slightly above desired prey rate, 00:38:08.670 --> 00:38:11.910 align:middle line:90% then you get the desired. 00:38:11.910 --> 00:38:17.150 align:middle line:84% Whereas in reality, in nature, preying on prey 00:38:17.150 --> 00:38:21.270 align:middle line:84% is kind of a very engaging task that 00:38:21.270 --> 00:38:24.810 align:middle line:84% very much depends on the density of the population. 00:38:24.810 --> 00:38:28.250 align:middle line:84% So it is not that it is hard to get it, 00:38:28.250 --> 00:38:33.110 align:middle line:84% and then it becomes completely abundant. 00:38:33.110 --> 00:38:36.370 align:middle line:84% So abundance emerges more continuously. 00:38:36.370 --> 00:38:40.190 align:middle line:84% So that softmax, which we have different ways of formulating 00:38:40.190 --> 00:38:42.310 align:middle line:90% it, could be useful. 00:38:42.310 --> 00:38:44.730 align:middle line:90% Is that clear, the core idea? 00:38:44.730 --> 00:38:48.230 align:middle line:84% But this is a better formulation than the other ones 00:38:48.230 --> 00:38:50.630 align:middle line:84% we have seen because it doesn't have the problem that, 00:38:50.630 --> 00:38:56.630 align:middle line:84% now, if we have infinite prey, the consumption of prey 00:38:56.630 --> 00:38:58.970 align:middle line:84% by a single predator goes to infinity. 00:38:58.970 --> 00:39:01.750 align:middle line:90% It would not happen. 00:39:01.750 --> 00:39:14.950 align:middle line:84% STUDENT: I was wondering if you do not [INAUDIBLE] Does 00:39:14.950 --> 00:39:16.990 align:middle line:84% that resolve some of your issues? 00:39:16.990 --> 00:39:20.150 align:middle line:84% I know, fundamentally, I guess the infectivity model isn't 00:39:20.150 --> 00:39:22.870 align:middle line:84% the same as prey eating, but you can 00:39:22.870 --> 00:39:25.150 align:middle line:84% imagine the prey going from alive 00:39:25.150 --> 00:39:26.690 align:middle line:90% to the stomachs of predators. 00:39:26.690 --> 00:39:29.670 align:middle line:84% It's like their contact frequency is how often they 00:39:29.670 --> 00:39:31.990 align:middle line:90% see each other [INAUDIBLE] 00:39:31.990 --> 00:39:33.353 align:middle line:90% 00:39:33.353 --> 00:39:34.270 align:middle line:90% HAZHIR RAHMANDAD: Yes. 00:39:34.270 --> 00:39:38.870 align:middle line:84% So the part that physically distracts 00:39:38.870 --> 00:39:42.150 align:middle line:84% from that conceptualization is that you 00:39:42.150 --> 00:39:45.570 align:middle line:84% see all these lions seeing all deer around them, 00:39:45.570 --> 00:39:47.130 align:middle line:84% but they are not hungry at the time. 00:39:47.130 --> 00:39:49.310 align:middle line:90% So they don't go and eat them. 00:39:49.310 --> 00:39:50.910 align:middle line:90% That's the part. 00:39:50.910 --> 00:39:53.730 align:middle line:84% Whereas the virus says, oh, if there is somebody, 00:39:53.730 --> 00:39:55.290 align:middle line:90% I'll catch them. 00:39:55.290 --> 00:39:57.470 align:middle line:90% It doesn't wait for you. 00:39:57.470 --> 00:39:59.050 align:middle line:90% It doesn't have any satiation. 00:39:59.050 --> 00:40:03.270 align:middle line:84% So the satiation is not captured in the Bass diffusion model. 00:40:03.270 --> 00:40:06.205 align:middle line:84% And that makes it not a good formulation. 00:40:06.205 --> 00:40:07.830 align:middle line:84% STUDENT: What about the fact that there 00:40:07.830 --> 00:40:08.913 align:middle line:90% are different populations? 00:40:08.913 --> 00:40:12.110 align:middle line:84% Because in Bass diffusion, it's one population just 00:40:12.110 --> 00:40:12.970 align:middle line:90% changing states. 00:40:12.970 --> 00:40:15.390 align:middle line:84% But these are different [? kind of things. ?] 00:40:15.390 --> 00:40:16.670 align:middle line:90% HAZHIR RAHMANDAD: Yeah, so-- 00:40:16.670 --> 00:40:19.710 align:middle line:90% STUDENT: [INAUDIBLE] 00:40:19.710 --> 00:40:25.470 align:middle line:84% HAZHIR RAHMANDAD: So if you had a kind of an animal that 00:40:25.470 --> 00:40:30.737 align:middle line:84% was just into killing their prey regardless of their hunger-- 00:40:30.737 --> 00:40:31.570 align:middle line:90% STUDENT: [INAUDIBLE] 00:40:31.570 --> 00:40:33.153 align:middle line:84% HAZHIR RAHMANDAD: Yeah, so like humans 00:40:33.153 --> 00:40:36.150 align:middle line:84% going and shooting animals without eating them, then 00:40:36.150 --> 00:40:40.520 align:middle line:84% you get closer to a multiplicative function 00:40:40.520 --> 00:40:43.860 align:middle line:84% that, as soon as you have an interaction, 00:40:43.860 --> 00:40:46.680 align:middle line:84% an interaction comes with the multiplication of the density 00:40:46.680 --> 00:40:48.560 align:middle line:90% of the two species. 00:40:48.560 --> 00:40:51.440 align:middle line:84% So in that sense, it gets closer to Bass diffusion 00:40:51.440 --> 00:40:54.380 align:middle line:84% or to that multiplying the two populations. 00:40:54.380 --> 00:40:58.920 align:middle line:84% Now, it is not a Bass diffusion model because of what you said. 00:40:58.920 --> 00:41:00.220 align:middle line:90% It's two different species. 00:41:00.220 --> 00:41:04.180 align:middle line:84% But the rate of decline for one is a function of both. 00:41:04.180 --> 00:41:05.760 align:middle line:90% That's the idea. 00:41:05.760 --> 00:41:09.252 align:middle line:84% STUDENT: So I didn't do this, but something-- 00:41:09.252 --> 00:41:10.460 align:middle line:90% I didn't do this formulation. 00:41:10.460 --> 00:41:12.127 align:middle line:84% But something I was thinking about a lot 00:41:12.127 --> 00:41:15.642 align:middle line:84% was something about the effectiveness of predators. 00:41:15.642 --> 00:41:18.100 align:middle line:84% Maybe they're really bad at hunting or something like that. 00:41:18.100 --> 00:41:20.060 align:middle line:84% How would this formulation capture that? 00:41:20.060 --> 00:41:21.382 align:middle line:90% Or how would you include that? 00:41:21.382 --> 00:41:22.840 align:middle line:84% HAZHIR RAHMANDAD: The effectiveness 00:41:22.840 --> 00:41:26.320 align:middle line:90% of the predators in hunting? 00:41:26.320 --> 00:41:27.080 align:middle line:90% STUDENT: Yeah. 00:41:27.080 --> 00:41:30.040 align:middle line:84% HAZHIR RAHMANDAD: So they have a desired consumption. 00:41:30.040 --> 00:41:31.920 align:middle line:84% But then there is a separate concept 00:41:31.920 --> 00:41:35.760 align:middle line:84% that you are saying is about their effectiveness that 00:41:35.760 --> 00:41:39.660 align:middle line:84% may bound how much they can consume. 00:41:39.660 --> 00:41:43.160 align:middle line:84% So that, basically, would come in this formulation 00:41:43.160 --> 00:41:44.580 align:middle line:90% from that softmax. 00:41:44.580 --> 00:41:47.840 align:middle line:84% The current one that Gerardo has wouldn't capture it. 00:41:47.840 --> 00:41:52.720 align:middle line:84% But a softmax formulation would say, OK, 00:41:52.720 --> 00:41:58.960 align:middle line:90% you can grow your consumption. 00:41:58.960 --> 00:42:03.740 align:middle line:84% But even if you have quite a bit of prey around, 00:42:03.740 --> 00:42:05.540 align:middle line:84% you don't get to your desired level. 00:42:05.540 --> 00:42:08.320 align:middle line:84% You need to have a huge amount of prey 00:42:08.320 --> 00:42:11.660 align:middle line:84% to be able to get to your desired level. 00:42:11.660 --> 00:42:15.600 align:middle line:84% Actually, maybe I should-- give me a second. 00:42:15.600 --> 00:42:19.320 align:middle line:90% Let's see if it works. 00:42:19.320 --> 00:42:32.760 align:middle line:84% Let's say here we have the max consumption 00:42:32.760 --> 00:42:41.260 align:middle line:84% possible from how many deer you have divided by desired. 00:42:41.260 --> 00:42:45.520 align:middle line:90% 00:42:45.520 --> 00:42:50.320 align:middle line:84% It's very hard to do this by hand, but that's what it is. 00:42:50.320 --> 00:42:56.560 align:middle line:84% And here is basically what fraction of the desired you 00:42:56.560 --> 00:43:01.480 align:middle line:90% get to consume. 00:43:01.480 --> 00:43:12.720 align:middle line:84% So what Gerard has is, at the point 1 and 1, 00:43:12.720 --> 00:43:16.360 align:middle line:84% we go the 45 degree line up to here. 00:43:16.360 --> 00:43:20.220 align:middle line:84% And from this point on, it's just a straight line. 00:43:20.220 --> 00:43:23.840 align:middle line:90% So this is Gerardo's equation. 00:43:23.840 --> 00:43:34.760 align:middle line:84% Now, Alexander's point was that you need a softer 00:43:34.760 --> 00:43:37.970 align:middle line:90% approach towards the end. 00:43:37.970 --> 00:43:43.710 align:middle line:84% So this becomes the alternative, that you have the softmax. 00:43:43.710 --> 00:43:46.050 align:middle line:84% And then the point that I was making, in response 00:43:46.050 --> 00:43:49.650 align:middle line:84% to Alexander's other comment about efficiency 00:43:49.650 --> 00:43:55.210 align:middle line:84% of predator in prey is that the slope of this curve 00:43:55.210 --> 00:43:56.030 align:middle line:90% could be different. 00:43:56.030 --> 00:43:57.390 align:middle line:90% So you could have this curve. 00:43:57.390 --> 00:44:00.410 align:middle line:84% Or you may have something that more looks like this. 00:44:00.410 --> 00:44:03.050 align:middle line:84% Oops, it goes down-- goes up like this. 00:44:03.050 --> 00:44:06.850 align:middle line:84% So the less efficient predators are 00:44:06.850 --> 00:44:11.910 align:middle line:84% suffering unless there is a lot more prey in the landscape. 00:44:11.910 --> 00:44:16.650 align:middle line:84% So that's one way to capture it, not the only way. 00:44:16.650 --> 00:44:18.330 align:middle line:90% OK. 00:44:18.330 --> 00:44:19.777 align:middle line:90% Do we have-- 00:44:19.777 --> 00:44:20.610 align:middle line:90% STUDENT: [INAUDIBLE] 00:44:20.610 --> 00:44:22.890 align:middle line:84% HAZHIR RAHMANDAD: What do you mean as an alternative 00:44:22.890 --> 00:44:24.310 align:middle line:90% to the mean formulation? 00:44:24.310 --> 00:44:25.050 align:middle line:90% I used a lot of-- 00:44:25.050 --> 00:44:26.710 align:middle line:84% I don't know a better alternative. 00:44:26.710 --> 00:44:28.650 align:middle line:84% Practically, we use a lookup function. 00:44:28.650 --> 00:44:29.230 align:middle line:90% OK, great. 00:44:29.230 --> 00:44:31.730 align:middle line:84% So let's actually-- then this is very much aligned with what 00:44:31.730 --> 00:44:33.430 align:middle line:90% we are right now discussing. 00:44:33.430 --> 00:44:37.370 align:middle line:84% So how can you solve this problem? 00:44:37.370 --> 00:44:44.290 align:middle line:84% There are two general ways-- analytical functions or table 00:44:44.290 --> 00:44:45.570 align:middle line:90% functions. 00:44:45.570 --> 00:44:47.790 align:middle line:84% Table function is pretty straightforward. 00:44:47.790 --> 00:44:50.250 align:middle line:84% You have this max divided by desired 00:44:50.250 --> 00:44:53.010 align:middle line:84% as the input to your table function. 00:44:53.010 --> 00:44:54.950 align:middle line:84% And you get the shape that you want. 00:44:54.950 --> 00:44:56.890 align:middle line:90% You plug it in. 00:44:56.890 --> 00:44:58.970 align:middle line:90% That's nice and clean. 00:44:58.970 --> 00:45:01.510 align:middle line:84% System dynamicists are comfortable with it. 00:45:01.510 --> 00:45:05.010 align:middle line:84% Outside of system dynamics, people would say, what's that? 00:45:05.010 --> 00:45:06.630 align:middle line:84% That's too many degrees of freedom. 00:45:06.630 --> 00:45:07.390 align:middle line:90% That's confusing. 00:45:07.390 --> 00:45:08.730 align:middle line:90% I can't follow it. 00:45:08.730 --> 00:45:10.490 align:middle line:90% Give me an equation. 00:45:10.490 --> 00:45:15.210 align:middle line:84% So then you need to have at least an analytical alternative 00:45:15.210 --> 00:45:18.350 align:middle line:90% for other types of audiences. 00:45:18.350 --> 00:45:21.170 align:middle line:84% So anybody has a feel for how you 00:45:21.170 --> 00:45:26.330 align:middle line:84% can get the flexibility for these different curves 00:45:26.330 --> 00:45:30.130 align:middle line:84% with an analytical function that does all of the things we want? 00:45:30.130 --> 00:45:31.920 align:middle line:84% Jason, what would be your suggestion? 00:45:31.920 --> 00:45:36.180 align:middle line:84% STUDENT: The one I landed on was input 00:45:36.180 --> 00:45:39.380 align:middle line:90% divided by 1 plus the input. 00:45:39.380 --> 00:45:42.100 align:middle line:84% To get flexibility, you could raise that whole quotient 00:45:42.100 --> 00:45:45.245 align:middle line:84% to some exponential power [INAUDIBLE] 00:45:45.245 --> 00:45:46.120 align:middle line:90% HAZHIR RAHMANDAD: OK. 00:45:46.120 --> 00:45:49.220 align:middle line:84% So let's call this x for simplicity. 00:45:49.220 --> 00:45:51.700 align:middle line:84% STUDENT: So it's x plus 1 over-- or x over-- 00:45:51.700 --> 00:45:55.100 align:middle line:84% HAZHIR RAHMANDAD: So your formulation is x-- 00:45:55.100 --> 00:45:57.420 align:middle line:90% STUDENT: Divided by 1 plus x. 00:45:57.420 --> 00:45:59.300 align:middle line:84% HAZHIR RAHMANDAD: Divided by $1 plus x. 00:45:59.300 --> 00:46:01.342 align:middle line:84% STUDENT: Right, because it gives you the property 00:46:01.342 --> 00:46:08.300 align:middle line:84% that when x is 0 you get the 0, 0 because the numerator is 0. 00:46:08.300 --> 00:46:09.540 align:middle line:90% HAZHIR RAHMANDAD: OK. 00:46:09.540 --> 00:46:11.900 align:middle line:84% STUDENT: And then it will approach-- 00:46:11.900 --> 00:46:13.500 align:middle line:84% HAZHIR RAHMANDAD: It would approach 1. 00:46:13.500 --> 00:46:15.500 align:middle line:84% STUDENT: It approaches 1 as x goes to infinity. 00:46:15.500 --> 00:46:17.700 align:middle line:90% HAZHIR RAHMANDAD: OK. 00:46:17.700 --> 00:46:20.420 align:middle line:84% STUDENT: The problem is it's kind of a slow approach to-- 00:46:20.420 --> 00:46:21.860 align:middle line:90% HAZHIR RAHMANDAD: Yes. 00:46:21.860 --> 00:46:24.060 align:middle line:84% STUDENT: So you could speed that up. 00:46:24.060 --> 00:46:25.900 align:middle line:90% HAZHIR RAHMANDAD: Yes. 00:46:25.900 --> 00:46:28.420 align:middle line:84% STUDENT: Speed that up by putting an exponent 00:46:28.420 --> 00:46:30.780 align:middle line:90% around the whole thing. 00:46:30.780 --> 00:46:33.397 align:middle line:84% HAZHIR RAHMANDAD: So you raise this to some larger power? 00:46:33.397 --> 00:46:33.980 align:middle line:90% STUDENT: Yeah. 00:46:33.980 --> 00:46:34.855 align:middle line:90% HAZHIR RAHMANDAD: OK. 00:46:34.855 --> 00:46:35.820 align:middle line:90% Are you following? 00:46:35.820 --> 00:46:38.400 align:middle line:84% This is exactly the type of thinking you want to do. 00:46:38.400 --> 00:46:42.220 align:middle line:84% That is, OK, I want something that starts from 0, goes to 1. 00:46:42.220 --> 00:46:44.520 align:middle line:90% x over 1 plus x does that. 00:46:44.520 --> 00:46:45.680 align:middle line:90% So that's nice. 00:46:45.680 --> 00:46:47.572 align:middle line:84% But then it doesn't approach fast enough. 00:46:47.572 --> 00:46:49.780 align:middle line:84% And it doesn't have any degrees of freedom right now. 00:46:49.780 --> 00:46:52.640 align:middle line:84% You don't have any parameter associated with it. 00:46:52.640 --> 00:46:56.040 align:middle line:84% So you can't get different saturation speeds. 00:46:56.040 --> 00:46:57.400 align:middle line:90% So this is not yet doing well. 00:46:57.400 --> 00:46:59.520 align:middle line:84% So now, Jason's next step was, oh, 00:46:59.520 --> 00:47:02.960 align:middle line:84% maybe we can do this, raise this to the power of alpha. 00:47:02.960 --> 00:47:06.020 align:middle line:90% 00:47:06.020 --> 00:47:11.580 align:middle line:84% And then that perhaps gives us something 00:47:11.580 --> 00:47:15.660 align:middle line:90% a little bit more flexible. 00:47:15.660 --> 00:47:20.380 align:middle line:84% Remember that, between 0 and 1, x values between 0 and 1 00:47:20.380 --> 00:47:24.080 align:middle line:84% is a very active region for this input. 00:47:24.080 --> 00:47:27.360 align:middle line:84% So you want a formulation that works in that region. 00:47:27.360 --> 00:47:33.060 align:middle line:84% So if you have, let's say, close to 1-- 00:47:33.060 --> 00:47:35.620 align:middle line:84% so if you get 1, in this case, this 00:47:35.620 --> 00:47:40.980 align:middle line:84% becomes 0.5 raised to the power of alpha. 00:47:40.980 --> 00:47:43.160 align:middle line:90% Well, it is important. 00:47:43.160 --> 00:47:50.120 align:middle line:84% So if it is 0.5, what do you want it to be? 00:47:50.120 --> 00:47:52.140 align:middle line:84% Do you want it to go above 0.5 or do 00:47:52.140 --> 00:47:54.000 align:middle line:90% you want it to go below 0.5? 00:47:54.000 --> 00:47:57.020 align:middle line:90% 00:47:57.020 --> 00:47:59.220 align:middle line:84% STUDENT: You want it to be below. 00:47:59.220 --> 00:48:01.440 align:middle line:84% HAZHIR RAHMANDAD: Yes, because if it is above 0.5, 00:48:01.440 --> 00:48:05.380 align:middle line:84% it is going to violate another constraint here, 00:48:05.380 --> 00:48:13.220 align:middle line:84% that you cannot exceed the y equals x line because the y 00:48:13.220 --> 00:48:17.220 align:middle line:84% equals x line-- that is, the output equals the input-- 00:48:17.220 --> 00:48:25.750 align:middle line:84% means your max divided by desired-- 00:48:25.750 --> 00:48:27.670 align:middle line:84% that is what you get out of this-- 00:48:27.670 --> 00:48:30.470 align:middle line:84% will need to be multiplied by the desired. 00:48:30.470 --> 00:48:39.470 align:middle line:84% So if it is y equals x line, then max divided by desired 00:48:39.470 --> 00:48:43.450 align:middle line:84% is going to be multiplied by the desired. 00:48:43.450 --> 00:48:45.070 align:middle line:90% This gives you the max. 00:48:45.070 --> 00:48:50.790 align:middle line:84% Now, any value above one above x equals y-- if this is y, 00:48:50.790 --> 00:48:54.200 align:middle line:90% this is our output-- 00:48:54.200 --> 00:48:55.950 align:middle line:84% STUDENT: Would give you more than desired. 00:48:55.950 --> 00:48:58.930 align:middle line:84% HAZHIR RAHMANDAD: This will lead to more than max, 00:48:58.930 --> 00:48:59.930 align:middle line:90% not more than desired. 00:48:59.930 --> 00:49:01.170 align:middle line:90% So this is the problem. 00:49:01.170 --> 00:49:05.630 align:middle line:84% So if you call this thing y, then you 00:49:05.630 --> 00:49:09.970 align:middle line:84% are going to multiply y by the desired. 00:49:09.970 --> 00:49:13.750 align:middle line:90% 00:49:13.750 --> 00:49:16.790 align:middle line:84% And if you multiply y by the desired-- 00:49:16.790 --> 00:49:22.150 align:middle line:84% let's, for a second, think of a situation where y equals x. 00:49:22.150 --> 00:49:24.270 align:middle line:90% And what is x? 00:49:24.270 --> 00:49:27.990 align:middle line:84% We see it on the right-hand side of the screen. 00:49:27.990 --> 00:49:31.030 align:middle line:90% It is max over desired. 00:49:31.030 --> 00:49:38.390 align:middle line:84% So that means these two lead to max over desired multiplied 00:49:38.390 --> 00:49:42.630 align:middle line:90% by desired is the total. 00:49:42.630 --> 00:49:44.950 align:middle line:90% Let's call this z. 00:49:44.950 --> 00:49:48.590 align:middle line:90% This is the consumption. 00:49:48.590 --> 00:49:55.390 align:middle line:84% So that means the consumption will be equal to what? 00:49:55.390 --> 00:49:55.970 align:middle line:90% Max. 00:49:55.970 --> 00:50:00.860 align:middle line:90% 00:50:00.860 --> 00:50:11.310 align:middle line:84% So this is saying that whatever is the output of this function 00:50:11.310 --> 00:50:18.190 align:middle line:84% should not go over the line y equals x. 00:50:18.190 --> 00:50:20.890 align:middle line:84% So that's another constraint that we have, 00:50:20.890 --> 00:50:25.870 align:middle line:84% which we didn't think about before this, for making 00:50:25.870 --> 00:50:27.430 align:middle line:90% a robust formulation. 00:50:27.430 --> 00:50:28.990 align:middle line:84% STUDENT: That is where the constraint 00:50:28.990 --> 00:50:31.390 align:middle line:90% comes from this [INAUDIBLE]. 00:50:31.390 --> 00:50:33.710 align:middle line:90% HAZHIR RAHMANDAD: Yes. 00:50:33.710 --> 00:50:36.110 align:middle line:90% STUDENT: [INAUDIBLE] 00:50:36.110 --> 00:50:37.270 align:middle line:90% HAZHIR RAHMANDAD: 0.5. 00:50:37.270 --> 00:50:43.450 align:middle line:84% So I said, if x is 0.5, x divided by 1 plus x-- oh sorry, 00:50:43.450 --> 00:50:44.850 align:middle line:90% if x is 1-- 00:50:44.850 --> 00:50:45.750 align:middle line:90% STUDENT: [INAUDIBLE] 00:50:45.750 --> 00:50:47.790 align:middle line:90% HAZHIR RAHMANDAD: It's 0.5, yes. 00:50:47.790 --> 00:50:50.970 align:middle line:90% So it is below. 00:50:50.970 --> 00:50:52.490 align:middle line:90% So it is fine for that one. 00:50:52.490 --> 00:50:53.765 align:middle line:90% You're right. 00:50:53.765 --> 00:50:54.390 align:middle line:90% STUDENT: Right. 00:50:54.390 --> 00:50:56.670 align:middle line:84% The problem is, though, if you want anything stronger, 00:50:56.670 --> 00:50:58.990 align:middle line:90% you wind up with stuff above it. 00:50:58.990 --> 00:51:01.310 align:middle line:84% HAZHIR RAHMANDAD: At some point, it could go above. 00:51:01.310 --> 00:51:05.270 align:middle line:84% That is, at the very beginning, at the slope 00:51:05.270 --> 00:51:10.910 align:middle line:84% of-- at the starting point, the slope 00:51:10.910 --> 00:51:15.950 align:middle line:84% is right now x at the very beginning. 00:51:15.950 --> 00:51:20.820 align:middle line:84% And so that gives a 45-degree line at the beginning. 00:51:20.820 --> 00:51:24.860 align:middle line:84% So this, without alpha, will never go above 45-degree line. 00:51:24.860 --> 00:51:30.900 align:middle line:84% But as soon as you add alphas that are below 1, for example, 00:51:30.900 --> 00:51:35.640 align:middle line:84% to go up faster, then it goes above the 45-degree line. 00:51:35.640 --> 00:51:38.260 align:middle line:90% So that's not a big problem. 00:51:38.260 --> 00:51:40.460 align:middle line:90% You can solve that. 00:51:40.460 --> 00:51:42.980 align:middle line:84% And the way to solve that-- so actually, first of all, 00:51:42.980 --> 00:51:50.800 align:middle line:84% you need alphas that are smaller than 1 for a sharper increase 00:51:50.800 --> 00:51:53.800 align:middle line:84% because that gets you closer to 1 more quickly. 00:51:53.800 --> 00:52:03.300 align:middle line:84% And then you can have, basically, the minimum of x-- 00:52:03.300 --> 00:52:05.920 align:middle line:90% 00:52:05.920 --> 00:52:11.720 align:middle line:84% this is y equals x and this function 00:52:11.720 --> 00:52:26.560 align:middle line:84% that you just came up with, so x divided by 1 plus x 00:52:26.560 --> 00:52:28.480 align:middle line:90% to the power of alpha. 00:52:28.480 --> 00:52:31.400 align:middle line:90% This does the job OK. 00:52:31.400 --> 00:52:34.260 align:middle line:84% STUDENT: But it still has some pretty sharp-- 00:52:34.260 --> 00:52:35.800 align:middle line:84% or it will have some point where-- 00:52:35.800 --> 00:52:38.380 align:middle line:84% HAZHIR RAHMANDAD: Yes, there will be a discontinuity. 00:52:38.380 --> 00:52:38.840 align:middle line:84% STUDENT: There will be a kink in there. 00:52:38.840 --> 00:52:40.673 align:middle line:84% HAZHIR RAHMANDAD: There will be a kink, yes. 00:52:40.673 --> 00:52:43.080 align:middle line:84% And actually, it is not very easy to come up 00:52:43.080 --> 00:52:45.860 align:middle line:84% with a formulation that doesn't have the kink. 00:52:45.860 --> 00:52:50.000 align:middle line:84% You can, but you basically need to build a much more 00:52:50.000 --> 00:52:51.260 align:middle line:90% complex formulation. 00:52:51.260 --> 00:52:52.920 align:middle line:90% So this is one version. 00:52:52.920 --> 00:52:54.140 align:middle line:90% You could do other ones. 00:52:54.140 --> 00:53:00.000 align:middle line:84% So you could use an exponential approach to the line rather than 00:53:00.000 --> 00:53:02.920 align:middle line:90% this polynomial approach. 00:53:02.920 --> 00:53:04.920 align:middle line:84% So if you want to do the exponential approach, 00:53:04.920 --> 00:53:08.500 align:middle line:84% you still need the mean of x and something. 00:53:08.500 --> 00:53:14.160 align:middle line:84% But that something could be 1 minus e 00:53:14.160 --> 00:53:24.170 align:middle line:90% to the power of minus alpha x. 00:53:24.170 --> 00:53:27.410 align:middle line:84% So if x is a large number, 1 minus e 00:53:27.410 --> 00:53:31.250 align:middle line:84% to the power of minus alpha x becomes-- 00:53:31.250 --> 00:53:35.060 align:middle line:90% 00:53:35.060 --> 00:53:36.505 align:middle line:90% STUDENT: [INAUDIBLE] 00:53:36.505 --> 00:53:38.630 align:middle line:84% HAZHIR RAHMANDAD: If x is a large number-- that is, 00:53:38.630 --> 00:53:41.330 align:middle line:84% we are on the right-hand side of this graph-- 00:53:41.330 --> 00:53:45.690 align:middle line:84% this e to the power of minus something becomes 0. 00:53:45.690 --> 00:53:49.310 align:middle line:84% And therefore, you get to 1 as the total. 00:53:49.310 --> 00:53:52.300 align:middle line:84% So you're exponentially getting close to that. 00:53:52.300 --> 00:53:53.550 align:middle line:90% You're approaching that curve. 00:53:53.550 --> 00:53:54.592 align:middle line:90% So that's an alternative. 00:53:54.592 --> 00:53:56.183 align:middle line:84% I'm not sure which one is better. 00:53:56.183 --> 00:53:57.850 align:middle line:84% To see which one is better, you actually 00:53:57.850 --> 00:54:00.090 align:middle line:84% need to graph them for different values of alpha 00:54:00.090 --> 00:54:02.610 align:middle line:84% and see which ones are qualitatively closer to what 00:54:02.610 --> 00:54:03.230 align:middle line:90% you want. 00:54:03.230 --> 00:54:04.170 align:middle line:90% Alexander? 00:54:04.170 --> 00:54:06.170 align:middle line:84% STUDENT: So I mean, if you were going 00:54:06.170 --> 00:54:08.110 align:middle line:84% to have a kink in this anyway no matter what, 00:54:08.110 --> 00:54:10.677 align:middle line:84% why not just do what Gerardo did? 00:54:10.677 --> 00:54:13.010 align:middle line:84% HAZHIR RAHMANDAD: Because this gives you the flexibility 00:54:13.010 --> 00:54:17.530 align:middle line:84% to capture the other factor that you talked about, 00:54:17.530 --> 00:54:20.490 align:middle line:90% which is important in this case. 00:54:20.490 --> 00:54:23.650 align:middle line:84% So that's, I would say, the primary thing. 00:54:23.650 --> 00:54:26.410 align:middle line:84% And the other thing is that it doesn't completely 00:54:26.410 --> 00:54:28.910 align:middle line:90% remove the effect. 00:54:28.910 --> 00:54:31.310 align:middle line:84% It just changes the speed of the effect, 00:54:31.310 --> 00:54:34.750 align:middle line:84% whereas the min function completely removes the effect. 00:54:34.750 --> 00:54:37.990 align:middle line:84% It has two very different regions. 00:54:37.990 --> 00:54:41.130 align:middle line:84% So it shuts off some feedback loop, 00:54:41.130 --> 00:54:45.330 align:middle line:84% whereas this version leaves the feedback loop on, but weakens it 00:54:45.330 --> 00:54:45.990 align:middle line:90% over time. 00:54:45.990 --> 00:54:48.230 align:middle line:84% So as such, the kink is not as strong. 00:54:48.230 --> 00:54:50.530 align:middle line:90% The impact is not as strong. 00:54:50.530 --> 00:54:53.810 align:middle line:84% STUDENT: More of a process question-- 00:54:53.810 --> 00:54:56.650 align:middle line:84% I can see where you would come up with this if you're starting 00:54:56.650 --> 00:55:02.230 align:middle line:84% from the point of I'm going [INAUDIBLE] 00:55:02.230 --> 00:55:06.290 align:middle line:84% But if you start from the point of hunting effectiveness 00:55:06.290 --> 00:55:11.650 align:middle line:84% [INAUDIBLE] it takes a while to land here on your own. 00:55:11.650 --> 00:55:15.610 align:middle line:84% HAZHIR RAHMANDAD: So I think the logic to get 00:55:15.610 --> 00:55:19.350 align:middle line:84% gets you here very quickly is I have predators, 00:55:19.350 --> 00:55:23.410 align:middle line:84% they want to eat prey, they have a desired value. 00:55:23.410 --> 00:55:24.910 align:middle line:90% So that is the driver. 00:55:24.910 --> 00:55:28.850 align:middle line:84% That's like the demand in an inventory model. 00:55:28.850 --> 00:55:31.750 align:middle line:84% But then if I don't have enough prey, 00:55:31.750 --> 00:55:34.810 align:middle line:84% I need to bring down their ability, their fulfillment 00:55:34.810 --> 00:55:37.910 align:middle line:84% ratio, their ability to get the prey they want. 00:55:37.910 --> 00:55:40.930 align:middle line:84% So I just need a formulation to capture that. 00:55:40.930 --> 00:55:45.810 align:middle line:84% That's the fast route to get to close to this step. 00:55:45.810 --> 00:55:49.110 align:middle line:84% Then you need to go back into your memory of, OK, 00:55:49.110 --> 00:55:53.210 align:middle line:84% what are the ways to capture these consumption of something 00:55:53.210 --> 00:55:58.490 align:middle line:84% out of a stock that is satiated on the one hand by the demand 00:55:58.490 --> 00:56:00.910 align:middle line:84% and on the other hand by the availability of the stock. 00:56:00.910 --> 00:56:04.770 align:middle line:84% And that's where, if you remember your assignment 6 00:56:04.770 --> 00:56:11.020 align:middle line:84% from 873, you would be able to do that fairly straightforward 00:56:11.020 --> 00:56:12.860 align:middle line:90% using that formulation. 00:56:12.860 --> 00:56:15.000 align:middle line:84% STUDENT: So you start from the story design. 00:56:15.000 --> 00:56:15.917 align:middle line:90% HAZHIR RAHMANDAD: Yes. 00:56:15.917 --> 00:56:18.060 align:middle line:90% Hassan? 00:56:18.060 --> 00:56:21.820 align:middle line:84% STUDENT: Can we use a second [INAUDIBLE]? 00:56:21.820 --> 00:56:24.420 align:middle line:84% Then for that formulation, we actually know 00:56:24.420 --> 00:56:27.542 align:middle line:84% where the alpha is that will create [INAUDIBLE] 00:56:27.542 --> 00:56:29.500 align:middle line:84% So we can limit the alpha so that it will never 00:56:29.500 --> 00:56:31.820 align:middle line:90% pass [INAUDIBLE] 00:56:31.820 --> 00:56:34.780 align:middle line:84% HAZHIR RAHMANDAD: So yes, but you 00:56:34.780 --> 00:56:41.460 align:middle line:84% want it to also start at the slope of 45 degrees. 00:56:41.460 --> 00:56:43.480 align:middle line:84% STUDENT: Yeah, so the second half of it. 00:56:43.480 --> 00:56:45.740 align:middle line:84% So we'll put the inflection point at 0. 00:56:45.740 --> 00:56:48.860 align:middle line:84% HAZHIR RAHMANDAD: Yes, but the sigmoid at the second half 00:56:48.860 --> 00:56:53.280 align:middle line:84% at the inflection point of the slope is flexible. 00:56:53.280 --> 00:56:56.435 align:middle line:84% STUDENT: Yeah, but we know where that limit of that slope is. 00:56:56.435 --> 00:56:57.560 align:middle line:90% HAZHIR RAHMANDAD: Yes, yes. 00:56:57.560 --> 00:56:58.840 align:middle line:90% We know limit of the slope. 00:56:58.840 --> 00:57:00.620 align:middle line:84% But I'm saying that you actually want 00:57:00.620 --> 00:57:05.720 align:middle line:84% it to be at the max consumption rate at the very beginning. 00:57:05.720 --> 00:57:11.100 align:middle line:84% So this is also an interesting but likely, reasonable approach 00:57:11.100 --> 00:57:13.900 align:middle line:84% that basically, if people are starving 00:57:13.900 --> 00:57:17.260 align:middle line:84% or the predators are starving and there is only 00:57:17.260 --> 00:57:21.200 align:middle line:84% so much prey, they will try to get as much as they can. 00:57:21.200 --> 00:57:22.478 align:middle line:90% That's the idea. 00:57:22.478 --> 00:57:24.020 align:middle line:84% STUDENT: [INAUDIBLE] there's just one 00:57:24.020 --> 00:57:25.375 align:middle line:90% value that fits these limits. 00:57:25.375 --> 00:57:27.000 align:middle line:84% Then we don't have any control over it. 00:57:27.000 --> 00:57:28.292 align:middle line:90% HAZHIR RAHMANDAD: Yes, exactly. 00:57:28.292 --> 00:57:32.620 align:middle line:84% So this min function actually brings that second constraint 00:57:32.620 --> 00:57:36.002 align:middle line:84% into your equation, but has the kink that Alexander highlighted. 00:57:36.002 --> 00:57:37.460 align:middle line:84% STUDENT: Could you explain what you 00:57:37.460 --> 00:57:41.080 align:middle line:84% were saying a minute ago about weakening the balancing feedback 00:57:41.080 --> 00:57:41.580 align:middle line:90% loop? 00:57:41.580 --> 00:57:42.497 align:middle line:90% HAZHIR RAHMANDAD: Yes. 00:57:42.497 --> 00:57:47.980 align:middle line:84% So here, we have one balancing feedback loop 00:57:47.980 --> 00:57:54.580 align:middle line:84% that is more deer, more consumption, and therefore less 00:57:54.580 --> 00:57:55.180 align:middle line:90% deer. 00:57:55.180 --> 00:57:59.020 align:middle line:84% So that's the first order control 00:57:59.020 --> 00:58:02.980 align:middle line:84% that we need to avoid the stock going to 0. 00:58:02.980 --> 00:58:10.060 align:middle line:84% So that feedback loop in the min function, in Gerardo's version, 00:58:10.060 --> 00:58:15.760 align:middle line:84% would only operate where the x is between 0 and 1, OK? 00:58:15.760 --> 00:58:19.020 align:middle line:84% And as soon as it is above 1, that feedback loop 00:58:19.020 --> 00:58:24.100 align:middle line:84% goes away because, in this formulation, 00:58:24.100 --> 00:58:28.820 align:middle line:84% the predator gets as much prey as it wants. 00:58:28.820 --> 00:58:31.980 align:middle line:84% And therefore, that feedback loop is not active anymore. 00:58:31.980 --> 00:58:34.980 align:middle line:84% Now, this is not fundamentally wrong. 00:58:34.980 --> 00:58:37.460 align:middle line:84% Those feedback loops saturate at some point 00:58:37.460 --> 00:58:39.140 align:middle line:90% and are not active anymore. 00:58:39.140 --> 00:58:45.180 align:middle line:84% But making it a very quick shift between those two domains, 00:58:45.180 --> 00:58:48.580 align:middle line:84% for all kinds of analysis, becomes 00:58:48.580 --> 00:58:53.320 align:middle line:84% a little bit less easy to trace why this is happening. 00:58:53.320 --> 00:58:55.140 align:middle line:84% Even in some optimizations, it becomes 00:58:55.140 --> 00:58:58.420 align:middle line:84% helpful to have those feedbacks active so 00:58:58.420 --> 00:59:01.460 align:middle line:84% that it pushes the optimization engine to find 00:59:01.460 --> 00:59:02.600 align:middle line:90% the right balance. 00:59:02.600 --> 00:59:06.670 align:middle line:84% Otherwise, that parameter never works. 00:59:06.670 --> 00:59:07.830 align:middle line:90% STUDENT: So OK. 00:59:07.830 --> 00:59:11.070 align:middle line:90% When we have this kink-- 00:59:11.070 --> 00:59:11.570 align:middle line:90% I see. 00:59:11.570 --> 00:59:14.390 align:middle line:84% So then when we have this kink to the left, 00:59:14.390 --> 00:59:18.470 align:middle line:84% when we do this thing, then it's always active. 00:59:18.470 --> 00:59:21.190 align:middle line:84% HAZHIR RAHMANDAD: It is always active, exactly. 00:59:21.190 --> 00:59:22.910 align:middle line:84% STUDENT: Yeah, that makes a lot of sense. 00:59:22.910 --> 00:59:25.623 align:middle line:84% HAZHIR RAHMANDAD: And in fact, in a table function, 00:59:25.623 --> 00:59:27.790 align:middle line:84% you don't have it always active because, ultimately, 00:59:27.790 --> 00:59:30.110 align:middle line:84% you put a value that says maybe at level 00:59:30.110 --> 00:59:31.730 align:middle line:90% x equals 3 it saturates. 00:59:31.730 --> 00:59:35.030 align:middle line:84% So table functions are not, in this case, necessarily 00:59:35.030 --> 00:59:36.150 align:middle line:90% the best tool. 00:59:36.150 --> 00:59:39.147 align:middle line:84% You define it and you can't control it as easily. 00:59:39.147 --> 00:59:40.730 align:middle line:84% You have to go and manually change it. 00:59:40.730 --> 00:59:42.350 align:middle line:84% So I like the analytical function 00:59:42.350 --> 00:59:45.370 align:middle line:90% for all of those reasons. 00:59:45.370 --> 00:59:45.870 align:middle line:90% Good. 00:59:45.870 --> 00:59:49.390 align:middle line:84% So if you build this part, this can 00:59:49.390 --> 00:59:52.750 align:middle line:84% go both into the consumption of prey 00:59:52.750 --> 00:59:59.270 align:middle line:84% and the consumption of the vegetation by the prey. 00:59:59.270 --> 01:00:02.870 align:middle line:84% So both of the stocks can be constrained 01:00:02.870 --> 01:00:06.720 align:middle line:84% by a mechanism like this, by a formulation like this. 01:00:06.720 --> 01:00:11.650 align:middle line:84% And that will basically do the kink thing. 01:00:11.650 --> 01:00:16.170 align:middle line:84% But Gerardo's formulation is a decent starting point. 01:00:16.170 --> 01:00:17.590 align:middle line:84% It is not fundamental-- it doesn't 01:00:17.590 --> 01:00:19.050 align:middle line:90% have any fundamental problems. 01:00:19.050 --> 01:00:27.510 align:middle line:84% It's just has this non-trivial jump. 01:00:27.510 --> 01:00:28.010 align:middle line:90% OK. 01:00:28.010 --> 01:00:31.470 align:middle line:84% So now that we saw that, let's actually look a little bit more 01:00:31.470 --> 01:00:37.150 align:middle line:84% at Gerardo's formulation and see if there is anything else 01:00:37.150 --> 01:00:39.510 align:middle line:90% we want to discuss there. 01:00:39.510 --> 01:00:41.290 align:middle line:84% And Gerardo, we haven't heard you. 01:00:41.290 --> 01:00:43.490 align:middle line:84% We just have been discussing your model. 01:00:43.490 --> 01:00:49.070 align:middle line:90% 01:00:49.070 --> 01:00:51.250 align:middle line:84% STUDENT: I was out of time this week. 01:00:51.250 --> 01:00:55.930 align:middle line:84% And I think that there are a lot of things wrong in this model. 01:00:55.930 --> 01:00:58.790 align:middle line:90% 01:00:58.790 --> 01:01:00.360 align:middle line:84% HAZHIR RAHMANDAD: But actually, you 01:01:00.360 --> 01:01:02.160 align:middle line:84% had the better formulation so far 01:01:02.160 --> 01:01:04.020 align:middle line:90% for the most challenging part. 01:01:04.020 --> 01:01:04.900 align:middle line:90% So that's good. 01:01:04.900 --> 01:01:08.600 align:middle line:90% 01:01:08.600 --> 01:01:09.720 align:middle line:90% STUDENT: [INAUDIBLE] 01:01:09.720 --> 01:01:13.142 align:middle line:90% 01:01:13.142 --> 01:01:14.100 align:middle line:90% HAZHIR RAHMANDAD: Yeah. 01:01:14.100 --> 01:01:17.000 align:middle line:84% STUDENT: Jason was suggesting doesn't necessarily 01:01:17.000 --> 01:01:19.660 align:middle line:84% have a max or a minutes baked into it. 01:01:19.660 --> 01:01:23.560 align:middle line:84% HAZHIR RAHMANDAD: Yes, let's look at it. 01:01:23.560 --> 01:01:28.760 align:middle line:84% STUDENT: [INAUDIBLE] current consumption over 1 01:01:28.760 --> 01:01:32.240 align:middle line:90% plus a max consumption. 01:01:32.240 --> 01:01:34.920 align:middle line:90% HAZHIR RAHMANDAD: Where am I? 01:01:34.920 --> 01:01:37.927 align:middle line:90% So this is your-- 01:01:37.927 --> 01:01:38.760 align:middle line:90% STUDENT: [INAUDIBLE] 01:01:38.760 --> 01:01:40.880 align:middle line:84% HAZHIR RAHMANDAD: Yes, let's go for the deer. 01:01:40.880 --> 01:01:42.025 align:middle line:90% STUDENT: [INAUDIBLE] 01:01:42.025 --> 01:01:42.900 align:middle line:90% HAZHIR RAHMANDAD: OK. 01:01:42.900 --> 01:01:46.420 align:middle line:84% So for the deer, we have the effective deer lifespan. 01:01:46.420 --> 01:01:49.300 align:middle line:84% And then we have predation of mature deer. 01:01:49.300 --> 01:01:52.020 align:middle line:84% And you do have the predation of juvenile deer, which is good. 01:01:52.020 --> 01:01:54.400 align:middle line:84% If you have juvenile deer, they will be eaten. 01:01:54.400 --> 01:01:57.083 align:middle line:84% STUDENT: I think current consumption is kind of the one. 01:01:57.083 --> 01:01:58.000 align:middle line:90% HAZHIR RAHMANDAD: Yes. 01:01:58.000 --> 01:02:00.820 align:middle line:84% So let's look at the current consumption. 01:02:00.820 --> 01:02:03.040 align:middle line:84% STUDENT: So I have that idea of there's 01:02:03.040 --> 01:02:08.100 align:middle line:84% a success rate in hunting, which admittedly is just static here. 01:02:08.100 --> 01:02:13.100 align:middle line:84% But it is made into something that is like [INAUDIBLE] 01:02:13.100 --> 01:02:16.880 align:middle line:84% HAZHIR RAHMANDAD: Yeah, so let's go to the current consumption 01:02:16.880 --> 01:02:18.640 align:middle line:90% and then-- 01:02:18.640 --> 01:02:19.140 align:middle line:90% OK. 01:02:19.140 --> 01:02:25.400 align:middle line:84% So attack rate multiplied by hunt success rate-- 01:02:25.400 --> 01:02:27.460 align:middle line:84% those two together are always together. 01:02:27.460 --> 01:02:30.220 align:middle line:84% So think of that as just a single parameter. 01:02:30.220 --> 01:02:33.665 align:middle line:84% So it makes the world a little bit easier to make sense of. 01:02:33.665 --> 01:02:35.040 align:middle line:84% STUDENT: But you could change it. 01:02:35.040 --> 01:02:37.332 align:middle line:84% HAZHIR RAHMANDAD: Yes, you could change it, absolutely. 01:02:37.332 --> 01:02:39.780 align:middle line:84% And if you made one of them dynamic, 01:02:39.780 --> 01:02:43.240 align:middle line:84% you can see the value of having the parameters. 01:02:43.240 --> 01:02:47.200 align:middle line:84% So one parameter multiplied by the deer population 01:02:47.200 --> 01:02:49.240 align:middle line:90% raised to the power of 2. 01:02:49.240 --> 01:02:53.820 align:middle line:84% So the idea is that, as the deer population grows, 01:02:53.820 --> 01:02:55.880 align:middle line:84% the consumption rate can go up divided 01:02:55.880 --> 01:03:02.500 align:middle line:90% by 1 plus some time constant. 01:03:02.500 --> 01:03:04.080 align:middle line:84% STUDENT: It's not a time constant. 01:03:04.080 --> 01:03:06.520 align:middle line:84% So if you look at time to consume here, 01:03:06.520 --> 01:03:09.720 align:middle line:84% it's basically setting the maximum amount 01:03:09.720 --> 01:03:11.682 align:middle line:90% of deer consumed per day. 01:03:11.682 --> 01:03:12.640 align:middle line:90% HAZHIR RAHMANDAD: Yeah. 01:03:12.640 --> 01:03:16.360 align:middle line:84% So basically, you have the numerator 01:03:16.360 --> 01:03:21.240 align:middle line:84% divided by 1 plus some parameter multiplied by that numerator. 01:03:21.240 --> 01:03:23.780 align:middle line:84% So let's think about that formulation. 01:03:23.780 --> 01:03:26.800 align:middle line:90% So x divided by 1 plus alpha x. 01:03:26.800 --> 01:03:28.600 align:middle line:90% STUDENT: And alpha is the max. 01:03:28.600 --> 01:03:32.000 align:middle line:90% Alpha is like 1 over the max. 01:03:32.000 --> 01:03:35.000 align:middle line:90% HAZHIR RAHMANDAD: OK. 01:03:35.000 --> 01:03:39.500 align:middle line:84% So that formulation, if x is 0, it goes to 0. 01:03:39.500 --> 01:03:42.040 align:middle line:84% That is, if you don't have any deer, 01:03:42.040 --> 01:03:46.120 align:middle line:84% there is no consumption per predator. 01:03:46.120 --> 01:03:51.920 align:middle line:84% If x is a very large number, then this 01:03:51.920 --> 01:03:59.050 align:middle line:90% converges towards 1 over alpha. 01:03:59.050 --> 01:04:08.850 align:middle line:84% So if alpha is then your 1 over maximum rate that you want, 01:04:08.850 --> 01:04:14.890 align:middle line:84% you get what you want out of this formulation. 01:04:14.890 --> 01:04:17.950 align:middle line:84% So tell me about where the power 2 came from. 01:04:17.950 --> 01:04:20.250 align:middle line:84% STUDENT: The power 2-- so I originally 01:04:20.250 --> 01:04:25.730 align:middle line:84% wanted something that depended on density a lot 01:04:25.730 --> 01:04:30.650 align:middle line:84% and depended on basically interactions between wolves 01:04:30.650 --> 01:04:33.330 align:middle line:90% and deer and their success rate. 01:04:33.330 --> 01:04:37.890 align:middle line:84% And then through a lot of back and forth thinking 01:04:37.890 --> 01:04:41.450 align:middle line:84% about that specifically, I originally just had deer. 01:04:41.450 --> 01:04:46.970 align:middle line:84% And then I was thinking back through, admittedly with AI, 01:04:46.970 --> 01:04:50.173 align:middle line:84% this particular idea of I want it to depend more on [INAUDIBLE] 01:04:50.173 --> 01:04:51.090 align:middle line:90% HAZHIR RAHMANDAD: Yes. 01:04:51.090 --> 01:04:54.790 align:middle line:84% STUDENT: And so it suggested the idea [INAUDIBLE] 01:04:54.790 --> 01:04:55.772 align:middle line:90% 01:04:55.772 --> 01:04:56.730 align:middle line:90% HAZHIR RAHMANDAD: Yeah. 01:04:56.730 --> 01:05:00.010 align:middle line:84% So basically, you used a more mathematical way 01:05:00.010 --> 01:05:02.130 align:middle line:84% of trying to get to a function that 01:05:02.130 --> 01:05:03.630 align:middle line:90% does what you want it to do. 01:05:03.630 --> 01:05:05.690 align:middle line:84% That is, when the density is low, 01:05:05.690 --> 01:05:11.390 align:middle line:84% you don't get as many deer hunted per predator. 01:05:11.390 --> 01:05:14.870 align:middle line:84% And as the density goes up, you get more and more, 01:05:14.870 --> 01:05:16.050 align:middle line:90% but then it saturates. 01:05:16.050 --> 01:05:20.930 align:middle line:84% And this does all of those things. 01:05:20.930 --> 01:05:22.930 align:middle line:84% So what are the problems with this? 01:05:22.930 --> 01:05:26.930 align:middle line:84% So the first problem is that the units don't match. 01:05:26.930 --> 01:05:32.958 align:middle line:84% And the reason the units don't match is that time to consume-- 01:05:32.958 --> 01:05:34.750 align:middle line:84% STUDENT: Well, I called it time to consume. 01:05:34.750 --> 01:05:36.250 align:middle line:84% HAZHIR RAHMANDAD: It doesn't matter. 01:05:36.250 --> 01:05:39.290 align:middle line:84% Whatever is the unit, because it is 1 plus this thing, 01:05:39.290 --> 01:05:43.267 align:middle line:90% 1 is dimensionless. 01:05:43.267 --> 01:05:43.850 align:middle line:90% STUDENT: Yeah. 01:05:43.850 --> 01:05:46.370 align:middle line:90% 01:05:46.370 --> 01:05:48.255 align:middle line:84% So is that term, the second term. 01:05:48.255 --> 01:05:49.130 align:middle line:90% HAZHIR RAHMANDAD: Ah. 01:05:49.130 --> 01:05:51.490 align:middle line:84% STUDENT: So the numerator is like consumption. 01:05:51.490 --> 01:05:55.230 align:middle line:84% And then time consuming deer cancels out that consumption. 01:05:55.230 --> 01:05:57.770 align:middle line:84% HAZHIR RAHMANDAD: I see, I see, I see. 01:05:57.770 --> 01:05:59.790 align:middle line:90% Then the units can work. 01:05:59.790 --> 01:06:02.570 align:middle line:90% You're right, you're right. 01:06:02.570 --> 01:06:04.330 align:middle line:90% OK. 01:06:04.330 --> 01:06:07.370 align:middle line:84% So then the main problem that remains 01:06:07.370 --> 01:06:11.090 align:middle line:84% is just it is hard to consume this equation in terms 01:06:11.090 --> 01:06:14.810 align:middle line:90% of the micro-mechanisms. 01:06:14.810 --> 01:06:21.370 align:middle line:84% And if we go through the actual exact equation and we compare it 01:06:21.370 --> 01:06:24.730 align:middle line:84% with the one that we designed, the question of whether 01:06:24.730 --> 01:06:29.150 align:middle line:84% the 45-degree line constraint gets violated here or not, 01:06:29.150 --> 01:06:31.770 align:middle line:84% with an alternative conceptualization that thinks 01:06:31.770 --> 01:06:35.090 align:middle line:90% of deer availability as-- 01:06:35.090 --> 01:06:38.970 align:middle line:84% or max consumption rate of deer as another constraint-- 01:06:38.970 --> 01:06:42.250 align:middle line:84% that I don't know how it will pan out here. 01:06:42.250 --> 01:06:46.810 align:middle line:84% That is, in this function of 1 over 1 plus alpha x-- 01:06:46.810 --> 01:06:50.740 align:middle line:84% or x2 divided by 1 plus alpha x2, 01:06:50.740 --> 01:06:56.460 align:middle line:84% whether you may go over the 45-degree line in the range that 01:06:56.460 --> 01:06:57.560 align:middle line:90% we are considering. 01:06:57.560 --> 01:06:59.940 align:middle line:84% It's not something I can mentally do. 01:06:59.940 --> 01:07:01.320 align:middle line:90% We would have to write it down. 01:07:01.320 --> 01:07:07.340 align:middle line:84% And that's why having a story that breaks down the logic 01:07:07.340 --> 01:07:11.620 align:middle line:84% will help a little bit more with getting good formulations. 01:07:11.620 --> 01:07:12.540 align:middle line:90% STUDENT: I mean-- 01:07:12.540 --> 01:07:14.207 align:middle line:84% HAZHIR RAHMANDAD: But this does the job. 01:07:14.207 --> 01:07:17.420 align:middle line:84% So this is good because it does the things that we want. 01:07:17.420 --> 01:07:21.860 align:middle line:84% It shuts off consumption when there is no deer. 01:07:21.860 --> 01:07:25.620 align:middle line:84% It allows for max consumption per predator 01:07:25.620 --> 01:07:28.260 align:middle line:84% or desired consumption per predator 01:07:28.260 --> 01:07:30.780 align:middle line:90% when we have plenty of deer. 01:07:30.780 --> 01:07:32.920 align:middle line:90% It has the softmax story. 01:07:32.920 --> 01:07:35.480 align:middle line:84% It has a free parameter, that alpha 1. 01:07:35.480 --> 01:07:38.900 align:middle line:84% So it has all of those characteristics. 01:07:38.900 --> 01:07:41.820 align:middle line:84% So it is good on all of those fronts. 01:07:41.820 --> 01:07:46.820 align:middle line:84% STUDENT: But I guess my thing is, isn't there 01:07:46.820 --> 01:07:49.580 align:middle line:90% a more reasonable story to-- 01:07:49.580 --> 01:07:51.700 align:middle line:90% HAZHIR RAHMANDAD: To match this? 01:07:51.700 --> 01:07:56.860 align:middle line:84% STUDENT: To this than to something that has a kink in it? 01:07:56.860 --> 01:08:02.100 align:middle line:84% HAZHIR RAHMANDAD: So this one, again because we haven't done 01:08:02.100 --> 01:08:05.680 align:middle line:84% the analysis with the 45-degree line, may violate that. 01:08:05.680 --> 01:08:08.860 align:middle line:84% I don't have the intuition right now. 01:08:08.860 --> 01:08:11.420 align:middle line:84% STUDENT: What would it mean practically [INAUDIBLE] 01:08:11.420 --> 01:08:13.120 align:middle line:84% HAZHIR RAHMANDAD: So if it violates it, 01:08:13.120 --> 01:08:18.620 align:middle line:84% it means that at some levels of prey-- 01:08:18.620 --> 01:08:24.939 align:middle line:84% the rate of consumption of prey exceeds the maximum rate 01:08:24.939 --> 01:08:28.040 align:middle line:84% that, based on random interactions, 01:08:28.040 --> 01:08:31.939 align:middle line:84% is even feasible to happen in that-- 01:08:31.939 --> 01:08:34.160 align:middle line:84% with the density of deer that is available. 01:08:34.160 --> 01:08:38.819 align:middle line:84% So that becomes-- so let's say it takes some time 01:08:38.819 --> 01:08:40.720 align:middle line:90% to prey any deer. 01:08:40.720 --> 01:08:44.359 align:middle line:84% There is a literally physical time that you need. 01:08:44.359 --> 01:08:49.370 align:middle line:84% And it takes some time to find the deer. 01:08:49.370 --> 01:08:51.130 align:middle line:90% You add those two together. 01:08:51.130 --> 01:08:57.910 align:middle line:84% There is a minimum time to catch a deer, so to speak. 01:08:57.910 --> 01:09:01.830 align:middle line:84% So if you have very few deer, the total predation 01:09:01.830 --> 01:09:03.590 align:middle line:90% should not exceed that. 01:09:03.590 --> 01:09:06.470 align:middle line:84% Whether that will happen here or not we cannot tell from 01:09:06.470 --> 01:09:07.189 align:middle line:90% the equation. 01:09:07.189 --> 01:09:09.064 align:middle line:84% STUDENT: I think that might be what my attack 01:09:09.064 --> 01:09:14.330 align:middle line:90% rate is because it's like-- 01:09:14.330 --> 01:09:15.830 align:middle line:84% HAZHIR RAHMANDAD: It is a parameter. 01:09:15.830 --> 01:09:20.790 align:middle line:84% I'm just saying-- so the attack rate right now 01:09:20.790 --> 01:09:23.090 align:middle line:90% is trying to capture that idea. 01:09:23.090 --> 01:09:26.790 align:middle line:84% But because you have just deer to the power of 2 playing 01:09:26.790 --> 01:09:31.950 align:middle line:84% something, it is not clear how it actually ends up downstream 01:09:31.950 --> 01:09:34.430 align:middle line:84% when you get the total consumption. 01:09:34.430 --> 01:09:36.290 align:middle line:90% But it does the job. 01:09:36.290 --> 01:09:39.470 align:middle line:84% In fact, so this one actually doesn't 01:09:39.470 --> 01:09:44.790 align:middle line:84% have the flexibility to change the slope of how fast you 01:09:44.790 --> 01:09:46.149 align:middle line:90% converge. 01:09:46.149 --> 01:09:52.630 align:middle line:84% You just change the height of where the total consumption 01:09:52.630 --> 01:09:56.910 align:middle line:90% per predator converges to. 01:09:56.910 --> 01:09:59.210 align:middle line:84% If you want to make that flexible, 01:09:59.210 --> 01:10:01.210 align:middle line:84% you would play with the power here. 01:10:01.210 --> 01:10:03.850 align:middle line:84% So you can go with power of 2 or 3 or whatever. 01:10:03.850 --> 01:10:06.070 align:middle line:84% So that becomes the alpha equivalent 01:10:06.070 --> 01:10:09.430 align:middle line:84% for the other formulation we saw. 01:10:09.430 --> 01:10:13.050 align:middle line:84% OK, this is a nice alternative to think through. 01:10:13.050 --> 01:10:15.910 align:middle line:84% And thanks for bringing that up [? and ?] never. 01:10:15.910 --> 01:10:18.430 align:middle line:90% 01:10:18.430 --> 01:10:21.190 align:middle line:90% OK. 01:10:21.190 --> 01:10:25.090 align:middle line:84% Anybody else wants to look at their formulation for this part? 01:10:25.090 --> 01:10:28.110 align:middle line:90% 01:10:28.110 --> 01:10:30.630 align:middle line:84% So Kathy also wants-- let's go to Kathy's. 01:10:30.630 --> 01:10:32.750 align:middle line:90% Let me see. 01:10:32.750 --> 01:10:33.690 align:middle line:90% OK, good. 01:10:33.690 --> 01:10:36.650 align:middle line:84% STUDENT: So it's a different analytical formulation. 01:10:36.650 --> 01:10:39.750 align:middle line:84% It has the same problem, where it needs to be-- 01:10:39.750 --> 01:10:41.730 align:middle line:84% it goes above the 45-degree line. 01:10:41.730 --> 01:10:42.870 align:middle line:90% So that-- 01:10:42.870 --> 01:10:46.550 align:middle line:84% HAZHIR RAHMANDAD: Yeah, so let's start 01:10:46.550 --> 01:10:48.130 align:middle line:90% looking at it step by step. 01:10:48.130 --> 01:10:51.050 align:middle line:84% So you have also separated the starvation from natural deaths. 01:10:51.050 --> 01:10:57.470 align:middle line:84% And as I said, it's a little bit-- 01:10:57.470 --> 01:10:59.290 align:middle line:84% we can get to that equation as well. 01:10:59.290 --> 01:11:01.270 align:middle line:84% So I think that's a separate equation 01:11:01.270 --> 01:11:03.050 align:middle line:90% than eating of the deer. 01:11:03.050 --> 01:11:04.450 align:middle line:90% So let's focus on this one. 01:11:04.450 --> 01:11:06.450 align:middle line:84% And this has four things going into it. 01:11:06.450 --> 01:11:10.270 align:middle line:84% So I have to open it and see what it's doing. 01:11:10.270 --> 01:11:15.650 align:middle line:84% So minimum effective RDD and deer. 01:11:15.650 --> 01:11:16.670 align:middle line:90% What is RDD? 01:11:16.670 --> 01:11:20.390 align:middle line:84% STUDENT: Relative deer density, so how many deer 01:11:20.390 --> 01:11:23.190 align:middle line:84% over some reference, like full density amount of deer. 01:11:23.190 --> 01:11:27.750 align:middle line:84% HAZHIR RAHMANDAD: OK, multiplied by indicated deer 01:11:27.750 --> 01:11:32.130 align:middle line:84% eaten versus deer divided by minimum time to eat deer. 01:11:32.130 --> 01:11:37.870 align:middle line:84% So this is similar to what we had with Gerardo's formulation. 01:11:37.870 --> 01:11:41.860 align:middle line:84% And this first part is a little bit more involved. 01:11:41.860 --> 01:11:44.480 align:middle line:84% So let's look at what that is doing. 01:11:44.480 --> 01:11:49.360 align:middle line:84% So indicated deer eaten comes from wolf 01:11:49.360 --> 01:11:51.060 align:middle line:90% multiplied by wolf appetite. 01:11:51.060 --> 01:11:54.320 align:middle line:90% So this is the desired, OK. 01:11:54.320 --> 01:12:02.680 align:middle line:84% And then this relative RDD is adding another constraint 01:12:02.680 --> 01:12:07.560 align:middle line:90% on consumption that is-- 01:12:07.560 --> 01:12:08.060 align:middle line:90% OK. 01:12:08.060 --> 01:12:10.460 align:middle line:84% So let's look at this equation and make sense of it. 01:12:10.460 --> 01:12:11.800 align:middle line:90% So it is max of 0. 01:12:11.800 --> 01:12:12.300 align:middle line:90% OK. 01:12:12.300 --> 01:12:14.280 align:middle line:90% It shouldn't go negative. 01:12:14.280 --> 01:12:19.920 align:middle line:84% Some parameter A multiplied by 1 minus e 01:12:19.920 --> 01:12:25.020 align:middle line:84% to the power of minus B relative deer density. 01:12:25.020 --> 01:12:32.300 align:middle line:84% So if deer density is very high, this thing turns into-- 01:12:32.300 --> 01:12:36.440 align:middle line:84% the exponent goes to a large negative number. 01:12:36.440 --> 01:12:38.520 align:middle line:84% e to the power of that goes to 0. 01:12:38.520 --> 01:12:40.380 align:middle line:90% 1 minus that becomes 1. 01:12:40.380 --> 01:12:42.520 align:middle line:90% It goes to A. 01:12:42.520 --> 01:12:44.360 align:middle line:90% But in this case-- 01:12:44.360 --> 01:12:44.860 align:middle line:90% OK. 01:12:44.860 --> 01:12:47.880 align:middle line:84% So that's for high deer densities. 01:12:47.880 --> 01:12:53.800 align:middle line:84% This goes to a number A. And for lower deer densities, 01:12:53.800 --> 01:12:57.640 align:middle line:90% it can go down to 0, basically. 01:12:57.640 --> 01:13:04.100 align:middle line:84% So if it is 0 deer density, that e to the power of blah, 01:13:04.100 --> 01:13:05.760 align:middle line:90% blah becomes 1. 01:13:05.760 --> 01:13:08.200 align:middle line:90% So this thing is-- 01:13:08.200 --> 01:13:10.360 align:middle line:84% so it's very similar to the equation 01:13:10.360 --> 01:13:13.920 align:middle line:90% that I wrote for the other part. 01:13:13.920 --> 01:13:21.880 align:middle line:90% And this is saying per predator. 01:13:21.880 --> 01:13:23.880 align:middle line:90% So what's the unit for A? 01:13:23.880 --> 01:13:26.400 align:middle line:90% All of these are dimensionless. 01:13:26.400 --> 01:13:29.720 align:middle line:90% So why do you need A here then? 01:13:29.720 --> 01:13:31.500 align:middle line:90% STUDENT: So A is 1 currently. 01:13:31.500 --> 01:13:34.660 align:middle line:84% I was thinking though that there is kind of a gluttony effect. 01:13:34.660 --> 01:13:38.420 align:middle line:84% And maybe I want want it to go to 1.2 or something like that. 01:13:38.420 --> 01:13:40.820 align:middle line:84% HAZHIR RAHMANDAD: But why not capture that already? 01:13:40.820 --> 01:13:44.720 align:middle line:90% 01:13:44.720 --> 01:13:47.000 align:middle line:84% You see, human learning is very slow. 01:13:47.000 --> 01:13:51.545 align:middle line:84% OK, so why don't you capture that involved appetite? 01:13:51.545 --> 01:13:52.920 align:middle line:84% Because ultimately, this is going 01:13:52.920 --> 01:13:55.200 align:middle line:90% to be multiplied by appetite. 01:13:55.200 --> 01:13:57.080 align:middle line:84% So it is just one degree of freedom 01:13:57.080 --> 01:13:59.400 align:middle line:84% that is not well-defined and doesn't 01:13:59.400 --> 01:14:01.140 align:middle line:90% do anything extra for you. 01:14:01.140 --> 01:14:04.483 align:middle line:90% So don't add that type of a-- 01:14:04.483 --> 01:14:06.400 align:middle line:84% STUDENT: Yeah, I was thinking of wolf appetite 01:14:06.400 --> 01:14:09.120 align:middle line:90% as the equilibrium. 01:14:09.120 --> 01:14:12.460 align:middle line:84% But if they have 100 deer in front of them, 01:14:12.460 --> 01:14:16.380 align:middle line:84% maybe they'll eat extra, I guess, over and above that. 01:14:16.380 --> 01:14:18.658 align:middle line:84% But right, it doesn't need to be there. 01:14:18.658 --> 01:14:20.700 align:middle line:84% HAZHIR RAHMANDAD: Yeah, because whatever it does, 01:14:20.700 --> 01:14:22.820 align:middle line:84% it will ultimately be multiplied by this. 01:14:22.820 --> 01:14:24.960 align:middle line:90% A. It will never-- 01:14:24.960 --> 01:14:30.465 align:middle line:84% and then you don't need the max with 0 thing as well, I think. 01:14:30.465 --> 01:14:31.340 align:middle line:90% STUDENT: That's true. 01:14:31.340 --> 01:14:32.620 align:middle line:84% I removed it when we were talking. 01:14:32.620 --> 01:14:34.453 align:middle line:84% HAZHIR RAHMANDAD: Yeah, you don't need that. 01:14:34.453 --> 01:14:40.030 align:middle line:84% And then yeah, beta does the thing for slowing it down. 01:14:40.030 --> 01:14:43.930 align:middle line:84% So yeah, this should work as well. 01:14:43.930 --> 01:14:49.170 align:middle line:84% So this way, you have the 45-degree constraint going 01:14:49.170 --> 01:14:51.570 align:middle line:90% through the min function. 01:14:51.570 --> 01:14:53.450 align:middle line:84% And you have the other constraints 01:14:53.450 --> 01:14:58.070 align:middle line:84% kicking in through limiting how many deer the wolf can eat. 01:14:58.070 --> 01:14:59.290 align:middle line:90% So this works fine. 01:14:59.290 --> 01:15:02.230 align:middle line:90% I think that's good. 01:15:02.230 --> 01:15:06.670 align:middle line:90% 01:15:06.670 --> 01:15:07.170 align:middle line:90% OK. 01:15:07.170 --> 01:15:09.690 align:middle line:84% Let's start looking at the starvation 01:15:09.690 --> 01:15:12.810 align:middle line:90% equations or let's take a break. 01:15:12.810 --> 01:15:13.770 align:middle line:90% STUDENT: Break. 01:15:13.770 --> 01:15:15.770 align:middle line:84% HAZHIR RAHMANDAD: Yeah, I think it's a good time 01:15:15.770 --> 01:15:19.370 align:middle line:90% to take a break, 10 minutes. 01:15:19.370 --> 01:15:21.810 align:middle line:90% So this is Jason's model. 01:15:21.810 --> 01:15:25.270 align:middle line:84% Let's take a look at a couple of items. 01:15:25.270 --> 01:15:33.250 align:middle line:84% So we had the predation of deer from desired consumption. 01:15:33.250 --> 01:15:38.850 align:middle line:84% And then you have the desired fulfillment ratio. 01:15:38.850 --> 01:15:43.210 align:middle line:84% So this is the formulation we talked about 01:15:43.210 --> 01:15:49.030 align:middle line:84% in following the same idea as the shipment from inventory. 01:15:49.030 --> 01:15:51.930 align:middle line:84% So there is a max feasible consumption, 01:15:51.930 --> 01:15:54.730 align:middle line:84% deer divided by minimum time to capture, 01:15:54.730 --> 01:15:59.130 align:middle line:84% versus the desired consumption that gives a prey sufficiency. 01:15:59.130 --> 01:16:02.150 align:middle line:84% Let's just make sure we know the formulation. 01:16:02.150 --> 01:16:04.290 align:middle line:84% This is the max divided by desired. 01:16:04.290 --> 01:16:04.790 align:middle line:90% OK. 01:16:04.790 --> 01:16:07.410 align:middle line:84% It's very similar to what we just discussed. 01:16:07.410 --> 01:16:10.530 align:middle line:84% And then this goes into a fulfillment ratio 01:16:10.530 --> 01:16:12.830 align:middle line:84% that, right now, we talked about it. 01:16:12.830 --> 01:16:15.610 align:middle line:90% You highlighted that. 01:16:15.610 --> 01:16:20.250 align:middle line:84% And we said how you can actually overcome some of the challenges 01:16:20.250 --> 01:16:25.970 align:middle line:84% with this formulation to get it to do what you want to do. 01:16:25.970 --> 01:16:30.600 align:middle line:84% And then you get the deaths just from predation. 01:16:30.600 --> 01:16:32.360 align:middle line:90% Let's go to starvation part. 01:16:32.360 --> 01:16:39.175 align:middle line:90% 01:16:39.175 --> 01:16:43.680 align:middle line:84% So you have the general death from old age. 01:16:43.680 --> 01:16:45.840 align:middle line:84% So deer, after a while, they die. 01:16:45.840 --> 01:16:47.700 align:middle line:90% That's very standard, simple. 01:16:47.700 --> 01:16:49.900 align:middle line:90% You take care of it that way. 01:16:49.900 --> 01:16:54.660 align:middle line:84% Then you're saying a fraction of the deer 01:16:54.660 --> 01:17:02.760 align:middle line:84% is at risk of dying from starvation. 01:17:02.760 --> 01:17:05.580 align:middle line:90% 01:17:05.580 --> 01:17:08.740 align:middle line:90% So to calculate that-- 01:17:08.740 --> 01:17:11.780 align:middle line:84% I'm not following the units here. 01:17:11.780 --> 01:17:13.240 align:middle line:90% Oh, you have deer also here. 01:17:13.240 --> 01:17:14.980 align:middle line:90% OK. 01:17:14.980 --> 01:17:17.350 align:middle line:90% Where did the deer come from? 01:17:17.350 --> 01:17:18.600 align:middle line:90% STUDENT: It's the first order. 01:17:18.600 --> 01:17:19.900 align:middle line:90% So there's some minimum-- 01:17:19.900 --> 01:17:20.320 align:middle line:90% HAZHIR RAHMANDAD: Oh, yeah. 01:17:20.320 --> 01:17:21.560 align:middle line:90% There is deer inside here. 01:17:21.560 --> 01:17:24.340 align:middle line:84% STUDENT: So if they don't eat for two weeks, they'll die. 01:17:24.340 --> 01:17:29.260 align:middle line:84% And then that flow gets closer and closer to 0, 01:17:29.260 --> 01:17:30.920 align:middle line:84% depending on how satiated they are. 01:17:30.920 --> 01:17:34.300 align:middle line:84% HAZHIR RAHMANDAD: OK, so you're saying deer divided by deer time 01:17:34.300 --> 01:17:41.820 align:middle line:84% to starve is the death rate from starvation. 01:17:41.820 --> 01:17:49.035 align:middle line:84% If satiation is 1, then nothing happens. 01:17:49.035 --> 01:17:49.660 align:middle line:90% STUDENT: Right. 01:17:49.660 --> 01:17:53.680 align:middle line:84% HAZHIR RAHMANDAD: So that's how you close this valve off, 01:17:53.680 --> 01:17:54.440 align:middle line:90% basically. 01:17:54.440 --> 01:17:55.540 align:middle line:90% STUDENT: Yeah. 01:17:55.540 --> 01:17:58.440 align:middle line:84% HAZHIR RAHMANDAD: OK, that's fine. 01:17:58.440 --> 01:18:03.240 align:middle line:84% I think the risk I see with that is the deer satiation equation. 01:18:03.240 --> 01:18:08.700 align:middle line:84% It needs to be turned on only in relatively extreme conditions 01:18:08.700 --> 01:18:10.620 align:middle line:90% or cases. 01:18:10.620 --> 01:18:14.500 align:middle line:84% That is, you have significant lack of malnutrition. 01:18:14.500 --> 01:18:18.260 align:middle line:84% Otherwise, it starts killing deer quite rapidly. 01:18:18.260 --> 01:18:21.420 align:middle line:90% STUDENT: [INAUDIBLE] 01:18:21.420 --> 01:18:23.182 align:middle line:84% HAZHIR RAHMANDAD: But in this formulation, 01:18:23.182 --> 01:18:25.140 align:middle line:84% we can't see that, because it's the calculation 01:18:25.140 --> 01:18:28.160 align:middle line:84% of deer satiation, that determines it. 01:18:28.160 --> 01:18:31.240 align:middle line:84% And in this case, you have just a grass fulfillment ratio, 01:18:31.240 --> 01:18:35.620 align:middle line:84% which is the ratio of what the grass is 01:18:35.620 --> 01:18:39.057 align:middle line:84% available for the desired consumption for the deer. 01:18:39.057 --> 01:18:40.140 align:middle line:90% STUDENT: Yeah, [INAUDIBLE] 01:18:40.140 --> 01:18:40.920 align:middle line:90% HAZHIR RAHMANDAD: That's good. 01:18:40.920 --> 01:18:43.120 align:middle line:84% So this is actually a very important piece. 01:18:43.120 --> 01:18:45.440 align:middle line:84% And I was looking at some of the other models. 01:18:45.440 --> 01:18:47.120 align:middle line:84% Some of them didn't follow this idea. 01:18:47.120 --> 01:18:52.300 align:middle line:84% So note here, the grass fulfillment ratio-- 01:18:52.300 --> 01:18:56.460 align:middle line:84% that is, what fraction of the desired consumption by deer 01:18:56.460 --> 01:19:00.580 align:middle line:84% is actually happening because of the availability of grass-- 01:19:00.580 --> 01:19:03.820 align:middle line:84% is having an impact on the grass consumption. 01:19:03.820 --> 01:19:08.500 align:middle line:84% But it also feeds back into deer starvation. 01:19:08.500 --> 01:19:11.580 align:middle line:84% It is very important to use the same concept, 01:19:11.580 --> 01:19:15.040 align:middle line:84% rather than defining this concept in another way, 01:19:15.040 --> 01:19:18.300 align:middle line:84% in another part of the model, because the underlying 01:19:18.300 --> 01:19:22.260 align:middle line:84% concept is deer does not get to eat grass 01:19:22.260 --> 01:19:24.050 align:middle line:90% or does get to eat grass. 01:19:24.050 --> 01:19:25.210 align:middle line:90% It's the same thing. 01:19:25.210 --> 01:19:26.930 align:middle line:84% It's the two sides of the same coin. 01:19:26.930 --> 01:19:31.270 align:middle line:84% STUDENT: It's easy to see it, if you scroll down to the wolves. 01:19:31.270 --> 01:19:34.030 align:middle line:84% It's the same thing, but I just broke it down 01:19:34.030 --> 01:19:36.750 align:middle line:84% into more steps-- actual deer consumption, 01:19:36.750 --> 01:19:39.950 align:middle line:90% and then wolf satiation, i.e. 01:19:39.950 --> 01:19:43.450 align:middle line:84% it's the fraction that they ate divided by what they desired, 01:19:43.450 --> 01:19:47.350 align:middle line:84% which is equivalent to the desired ratio. 01:19:47.350 --> 01:19:49.110 align:middle line:84% But that story, that is [INAUDIBLE] 01:19:49.110 --> 01:19:50.027 align:middle line:90% HAZHIR RAHMANDAD: Yes. 01:19:50.027 --> 01:19:53.050 align:middle line:84% So both of those give the same answer mathematically. 01:19:53.050 --> 01:19:55.810 align:middle line:84% But the version down here is saying, OK, 01:19:55.810 --> 01:20:00.570 align:middle line:84% this is the desired consumption versus actual consumption. 01:20:00.570 --> 01:20:02.670 align:middle line:90% That ratio gives satiation. 01:20:02.670 --> 01:20:05.490 align:middle line:84% And it is a good practice to spell out those steps 01:20:05.490 --> 01:20:08.510 align:middle line:84% because if, for whatever reason, you add another mechanism that 01:20:08.510 --> 01:20:12.550 align:middle line:84% changes the actual consumption, for example, 01:20:12.550 --> 01:20:16.250 align:middle line:84% it would not be captured in that fulfillment ratio alone. 01:20:16.250 --> 01:20:18.550 align:middle line:84% It will be captured in something else. 01:20:18.550 --> 01:20:21.150 align:middle line:84% So it is fine to add this extra variable, 01:20:21.150 --> 01:20:25.110 align:middle line:84% even though it is ultimately the same as desired fulfillment 01:20:25.110 --> 01:20:25.770 align:middle line:90% ratio. 01:20:25.770 --> 01:20:29.470 align:middle line:84% Is the logic here clear what we are talking about? 01:20:29.470 --> 01:20:30.180 align:middle line:90% STUDENT: Yeah. 01:20:30.180 --> 01:20:32.430 align:middle line:84% HAZHIR RAHMANDAD: If you feel anything is, OK, I'm not 01:20:32.430 --> 01:20:34.030 align:middle line:90% following, raise your hands. 01:20:34.030 --> 01:20:35.690 align:middle line:90% I wouldn't know. 01:20:35.690 --> 01:20:38.170 align:middle line:84% But we would pass it too quickly, then. 01:20:38.170 --> 01:20:43.910 align:middle line:84% OK, so the only place that I'm a little bit unsure 01:20:43.910 --> 01:20:47.070 align:middle line:84% is that this wolf satiation should 01:20:47.070 --> 01:20:51.110 align:middle line:84% be the same thing that goes into here 01:20:51.110 --> 01:20:53.437 align:middle line:90% and get 1 minus that satiation. 01:20:53.437 --> 01:20:54.270 align:middle line:90% STUDENT: [INAUDIBLE] 01:20:54.270 --> 01:20:55.478 align:middle line:90% HAZHIR RAHMANDAD: No, I know. 01:20:55.478 --> 01:20:58.870 align:middle line:84% But I'm saying, it may be too strong 01:20:58.870 --> 01:21:03.350 align:middle line:84% a feedback of starvation because time to starve 01:21:03.350 --> 01:21:05.190 align:middle line:90% is relatively short. 01:21:05.190 --> 01:21:09.310 align:middle line:84% And then basically, OK, biologically, what happens? 01:21:09.310 --> 01:21:14.830 align:middle line:84% If you don't get to eat what you desire to eat, you get thinner. 01:21:14.830 --> 01:21:17.110 align:middle line:84% And your desire for eating actually 01:21:17.110 --> 01:21:21.110 align:middle line:84% goes down because your metabolic rate goes down. 01:21:21.110 --> 01:21:24.510 align:middle line:84% And then you reduce your activity level. 01:21:24.510 --> 01:21:25.870 align:middle line:90% And you do all these things. 01:21:25.870 --> 01:21:27.387 align:middle line:90% STUDENT: [INAUDIBLE] 01:21:27.387 --> 01:21:29.470 align:middle line:84% HAZHIR RAHMANDAD: So there are all these balancing 01:21:29.470 --> 01:21:31.970 align:middle line:84% loops that you wouldn't start dying right away. 01:21:31.970 --> 01:21:39.070 align:middle line:84% In fact, for anybody like us, we may actually live for a while 01:21:39.070 --> 01:21:43.410 align:middle line:84% without eating at 100% of our desire-- 01:21:43.410 --> 01:21:46.010 align:middle line:84% 2,200 calories or whatever it is. 01:21:46.010 --> 01:21:48.890 align:middle line:84% If I bring that down to 2,000 calories, 01:21:48.890 --> 01:21:51.750 align:middle line:84% it doesn't mean that I'll start to die. 01:21:51.750 --> 01:21:56.410 align:middle line:84% It just means I'll probably equilibriate at 2,000 calories. 01:21:56.410 --> 01:21:59.270 align:middle line:84% Even if you bring it down to 1,500, 01:21:59.270 --> 01:22:01.990 align:middle line:84% you may equilibrate at that and not die. 01:22:01.990 --> 01:22:05.350 align:middle line:84% Now, if you bring it to 1,000, then you will probably die after 01:22:05.350 --> 01:22:07.230 align:middle line:90% maybe a few months or something. 01:22:07.230 --> 01:22:10.450 align:middle line:84% So that's the thinking that gives me the feeling, 01:22:10.450 --> 01:22:13.270 align:middle line:84% OK, this wolf satiation shouldn't directly 01:22:13.270 --> 01:22:15.270 align:middle line:90% go into the death. 01:22:15.270 --> 01:22:17.790 align:middle line:84% It needs to be transformed somehow. 01:22:17.790 --> 01:22:23.600 align:middle line:84% And the transformation should make it keep it closer to 1. 01:22:23.600 --> 01:22:32.600 align:middle line:84% That is, if the satiation is 0.7, I should push it up to 0.95 01:22:32.600 --> 01:22:35.040 align:middle line:90% or something like that. 01:22:35.040 --> 01:22:38.320 align:middle line:90% You see the logic why-- 01:22:38.320 --> 01:22:39.980 align:middle line:90% the way to think about this? 01:22:39.980 --> 01:22:41.700 align:middle line:90% So then how would you do that? 01:22:41.700 --> 01:22:45.560 align:middle line:84% So what is the right or easy way to do this 01:22:45.560 --> 01:22:47.860 align:middle line:84% with the current formulation that Jason has? 01:22:47.860 --> 01:22:56.440 align:middle line:90% 01:22:56.440 --> 01:22:57.780 align:middle line:90% Any suggestions? 01:22:57.780 --> 01:22:58.400 align:middle line:90% Anybody? 01:22:58.400 --> 01:23:03.200 align:middle line:90% 01:23:03.200 --> 01:23:11.352 align:middle line:84% STUDENT: I mean, when you have wolf satiation [INAUDIBLE] 01:23:11.352 --> 01:23:12.560 align:middle line:90% HAZHIR RAHMANDAD: Yeah, yeah. 01:23:12.560 --> 01:23:15.480 align:middle line:90% Well, not desired consumption. 01:23:15.480 --> 01:23:20.447 align:middle line:84% Wolf satiation-- OK, so the story I was telling 01:23:20.447 --> 01:23:21.280 align:middle line:90% STUDENT: [INAUDIBLE] 01:23:21.280 --> 01:23:23.657 align:middle line:84% HAZHIR RAHMANDAD: Included a feedback into desired 01:23:23.657 --> 01:23:24.740 align:middle line:90% consumption, you're right. 01:23:24.740 --> 01:23:26.320 align:middle line:90% But that's too much. 01:23:26.320 --> 01:23:28.120 align:middle line:90% That's messy. 01:23:28.120 --> 01:23:30.180 align:middle line:84% So that was not what I was asking. 01:23:30.180 --> 01:23:33.400 align:middle line:84% I was asking for a simple formulation that 01:23:33.400 --> 01:23:36.800 align:middle line:84% gets wolf satiation and gives you effective wolf 01:23:36.800 --> 01:23:39.160 align:middle line:90% satiation on death rate. 01:23:39.160 --> 01:23:43.600 align:middle line:84% Basically, right now, Jason is equating wolf satiation-- 01:23:43.600 --> 01:23:47.200 align:middle line:84% or 1 minus wolf satiation with death effect. 01:23:47.200 --> 01:23:56.440 align:middle line:84% STUDENT: [INAUDIBLE] [? x ?] the wolf satiation to some exponent. 01:23:56.440 --> 01:23:59.460 align:middle line:84% HAZHIR RAHMANDAD: OK, so what should the exponent range be? 01:23:59.460 --> 01:24:03.060 align:middle line:84% So let's say we have wolf satiation to the power of alpha. 01:24:03.060 --> 01:24:05.647 align:middle line:84% Should the alpha be small or big? 01:24:05.647 --> 01:24:06.480 align:middle line:90% STUDENT: [INAUDIBLE] 01:24:06.480 --> 01:24:08.460 align:middle line:84% HAZHIR RAHMANDAD: It should be smaller than 1. 01:24:08.460 --> 01:24:13.520 align:middle line:84% So if alpha is 0, you shot off this feedback loop. 01:24:13.520 --> 01:24:16.290 align:middle line:84% That is, basically, there is no starvation 01:24:16.290 --> 01:24:20.170 align:middle line:84% because anything to the power of 0 is still 1. 01:24:20.170 --> 01:24:22.610 align:middle line:84% And therefore, this thing goes away. 01:24:22.610 --> 01:24:26.570 align:middle line:84% If alpha is 1, it is the current formulation. 01:24:26.570 --> 01:24:30.870 align:middle line:84% So somewhere between that range is where you want alpha to be. 01:24:30.870 --> 01:24:32.850 align:middle line:84% And that's a good way of formulating this 01:24:32.850 --> 01:24:35.010 align:middle line:84% because wolf satiation will never 01:24:35.010 --> 01:24:37.490 align:middle line:84% go above 1 the way it is formulated here 01:24:37.490 --> 01:24:41.110 align:middle line:84% because it is always actual divided by desired. 01:24:41.110 --> 01:24:42.590 align:middle line:90% So at max, it will be 1. 01:24:42.590 --> 01:24:44.870 align:middle line:90% So I'm fine with that range. 01:24:44.870 --> 01:24:46.650 align:middle line:90% It will all work. 01:24:46.650 --> 01:24:48.930 align:middle line:90% Is the idea clear? 01:24:48.930 --> 01:24:50.770 align:middle line:90% So that's a good formulation. 01:24:50.770 --> 01:24:55.090 align:middle line:84% I think, with that correction, this does a decent job. 01:24:55.090 --> 01:24:57.410 align:middle line:84% We talked about the issue of whether we 01:24:57.410 --> 01:24:59.270 align:middle line:84% want to combine them or separate them. 01:24:59.270 --> 01:25:01.470 align:middle line:84% Now that I'm seeing how you have separated them-- 01:25:01.470 --> 01:25:04.170 align:middle line:84% and I think some of the others have done a similar thing-- 01:25:04.170 --> 01:25:04.910 align:middle line:90% I like it. 01:25:04.910 --> 01:25:05.590 align:middle line:90% It is fine. 01:25:05.590 --> 01:25:09.290 align:middle line:84% I don't have anything against it because the logic 01:25:09.290 --> 01:25:12.810 align:middle line:84% of natural death from old age is easy to capture 01:25:12.810 --> 01:25:14.370 align:middle line:90% using this lifespan. 01:25:14.370 --> 01:25:17.270 align:middle line:84% And then you can separately capture it here. 01:25:17.270 --> 01:25:19.170 align:middle line:90% So this is fine. 01:25:19.170 --> 01:25:20.610 align:middle line:90% Ali? 01:25:20.610 --> 01:25:23.330 align:middle line:84% STUDENT: [INAUDIBLE] I had this issue also. 01:25:23.330 --> 01:25:25.810 align:middle line:90% And I found it very last minute. 01:25:25.810 --> 01:25:29.490 align:middle line:84% So I think, if you don't have any vegetation, 01:25:29.490 --> 01:25:33.010 align:middle line:84% deer would grow because there is no control 01:25:33.010 --> 01:25:37.210 align:middle line:84% on the [? earthworms. ?] So what I'm trying to say 01:25:37.210 --> 01:25:40.210 align:middle line:84% is that the outflow from vegetation 01:25:40.210 --> 01:25:43.570 align:middle line:84% should be the inflow to the deer. 01:25:43.570 --> 01:25:51.970 align:middle line:84% [INAUDIBLE] of the vegetation should be the inflow 01:25:51.970 --> 01:25:53.370 align:middle line:90% to the birth rate of the deer. 01:25:53.370 --> 01:25:54.953 align:middle line:84% HAZHIR RAHMANDAD: OK, very good point. 01:25:54.953 --> 01:25:58.890 align:middle line:84% So now, this is an important biological question for us 01:25:58.890 --> 01:26:03.850 align:middle line:84% who are not biologists and have no clue about how deer life 01:26:03.850 --> 01:26:04.810 align:middle line:90% spans are. 01:26:04.810 --> 01:26:08.350 align:middle line:84% So the question is, if people are-- or deer are starving, 01:26:08.350 --> 01:26:10.290 align:middle line:90% would they have babies? 01:26:10.290 --> 01:26:13.170 align:middle line:84% And the short answer is not as many. 01:26:13.170 --> 01:26:16.010 align:middle line:84% Basically, the stress of starvation 01:26:16.010 --> 01:26:18.730 align:middle line:90% brings down their birth rate. 01:26:18.730 --> 01:26:24.370 align:middle line:84% So that is another way to potentially have a feedback 01:26:24.370 --> 01:26:25.730 align:middle line:90% into the system. 01:26:25.730 --> 01:26:32.290 align:middle line:84% So that definitely is a logical, biologically sound idea. 01:26:32.290 --> 01:26:36.770 align:middle line:84% Now, there is a question, do I even want to do that? 01:26:36.770 --> 01:26:39.530 align:middle line:84% And the reason for asking that question 01:26:39.530 --> 01:26:42.750 align:middle line:84% is, this outflow is pretty strong. 01:26:42.750 --> 01:26:44.690 align:middle line:90% And you have control over it. 01:26:44.690 --> 01:26:47.470 align:middle line:84% And it is proportional to the size of the deer. 01:26:47.470 --> 01:26:51.050 align:middle line:84% Basically, you have deer divided by some time constant and so on. 01:26:51.050 --> 01:26:58.250 align:middle line:84% So you can say I'm capturing those unborn babies as death. 01:26:58.250 --> 01:27:04.130 align:middle line:84% So you are a pro-life person or something. 01:27:04.130 --> 01:27:10.340 align:middle line:84% So that's one way of making the argument 01:27:10.340 --> 01:27:13.600 align:middle line:84% that I don't need to separately capture it 01:27:13.600 --> 01:27:15.900 align:middle line:84% because then I would add some other parameter 01:27:15.900 --> 01:27:17.160 align:middle line:90% and some other structure. 01:27:17.160 --> 01:27:21.360 align:middle line:84% And I'm perfectly fine with that recognition, 01:27:21.360 --> 01:27:23.820 align:middle line:84% and then saying I'm not doing it. 01:27:23.820 --> 01:27:27.580 align:middle line:84% But it's good to have that recognition. 01:27:27.580 --> 01:27:30.120 align:middle line:90% OK, great. 01:27:30.120 --> 01:27:33.980 align:middle line:84% So I think this is working fine with the couple of ideas 01:27:33.980 --> 01:27:36.620 align:middle line:90% that we mentioned. 01:27:36.620 --> 01:27:40.700 align:middle line:90% We didn't have the-- 01:27:40.700 --> 01:27:43.400 align:middle line:84% yeah, so the grass is growing exponentially here. 01:27:43.400 --> 01:27:47.340 align:middle line:84% It doesn't have any balancing loop. 01:27:47.340 --> 01:27:57.300 align:middle line:84% So if you leave the grass unchecked with the deer, 01:27:57.300 --> 01:28:00.360 align:middle line:90% then it grows exponentially. 01:28:00.360 --> 01:28:05.220 align:middle line:84% And that is a problem that this formulation doesn't capture. 01:28:05.220 --> 01:28:10.580 align:middle line:84% Now, again, for that, you could have a grass death separately. 01:28:10.580 --> 01:28:14.740 align:middle line:84% Or you could use Alexander's or some alternative version 01:28:14.740 --> 01:28:21.340 align:middle line:84% that uses the single flow for grass and saturates that. 01:28:21.340 --> 01:28:22.900 align:middle line:90% Great. 01:28:22.900 --> 01:28:23.860 align:middle line:90% Let's see. 01:28:23.860 --> 01:28:28.240 align:middle line:84% Anybody else want to bring a question from their models? 01:28:28.240 --> 01:28:31.000 align:middle line:84% I think we have really discussed the key issues. 01:28:31.000 --> 01:28:34.297 align:middle line:90% So we can open it to any models. 01:28:34.297 --> 01:28:35.880 align:middle line:84% Alexander, do you want to take a look? 01:28:35.880 --> 01:28:38.220 align:middle line:90% Sure. 01:28:38.220 --> 01:28:43.120 align:middle line:84% STUDENT: About the predation rate and the death rate. 01:28:43.120 --> 01:28:45.820 align:middle line:90% So I completely-- 01:28:45.820 --> 01:28:48.060 align:middle line:84% I guess I implicitly assumed that animals 01:28:48.060 --> 01:28:49.612 align:middle line:90% have insatiable appetites. 01:28:49.612 --> 01:28:51.820 align:middle line:84% So I forgot about the whole capacity thing and things 01:28:51.820 --> 01:28:52.320 align:middle line:90% like that. 01:28:52.320 --> 01:28:56.298 align:middle line:84% And I focused a lot on the hunting efficiency. 01:28:56.298 --> 01:28:57.340 align:middle line:90% HAZHIR RAHMANDAD: Uh-huh. 01:28:57.340 --> 01:29:00.460 align:middle line:84% STUDENT: And so basically, what I wanted to capture 01:29:00.460 --> 01:29:08.420 align:middle line:84% was that, as there's fewer deer, wolves can't hunt as well, 01:29:08.420 --> 01:29:09.400 align:middle line:90% can't hunt as well. 01:29:09.400 --> 01:29:11.920 align:middle line:84% And then their efficiency maxes out at some point. 01:29:11.920 --> 01:29:14.580 align:middle line:84% So that's the top half of the model. 01:29:14.580 --> 01:29:20.300 align:middle line:84% HAZHIR RAHMANDAD: OK, so let's look at it. 01:29:20.300 --> 01:29:23.380 align:middle line:84% So you have the hunting efficiency multiplied 01:29:23.380 --> 01:29:25.880 align:middle line:90% by the number of the predators. 01:29:25.880 --> 01:29:28.340 align:middle line:84% So this hunting efficiency is what we already 01:29:28.340 --> 01:29:31.100 align:middle line:90% have seen in various formats. 01:29:31.100 --> 01:29:34.420 align:middle line:84% And that comes from some reference 01:29:34.420 --> 01:29:38.420 align:middle line:84% multiplied by some effective availability of prey. 01:29:38.420 --> 01:29:45.880 align:middle line:84% And that effect is a fraction of carrying capacity, 01:29:45.880 --> 01:29:50.340 align:middle line:84% so prey divided by carrying capacity of prey. 01:29:50.340 --> 01:29:51.880 align:middle line:90% It should be fine. 01:29:51.880 --> 01:29:56.500 align:middle line:84% So that number, if the model works well on other fronts, 01:29:56.500 --> 01:29:59.000 align:middle line:90% would always be between 0 and 1. 01:29:59.000 --> 01:30:01.740 align:middle line:84% But you can never be sure because you can actually start 01:30:01.740 --> 01:30:04.170 align:middle line:90% from beyond carrying capacity. 01:30:04.170 --> 01:30:10.790 align:middle line:84% You can throw in too many prey in a natural environment. 01:30:10.790 --> 01:30:15.790 align:middle line:84% And then effective prey population and efficiency is-- 01:30:15.790 --> 01:30:19.830 align:middle line:84% STUDENT: So it's similar to what Jason was suggesting earlier. 01:30:19.830 --> 01:30:24.070 align:middle line:84% It's x over 1 plus x to some alpha. 01:30:24.070 --> 01:30:26.950 align:middle line:84% And then I put that 2 in the beginning so it's such that, 01:30:26.950 --> 01:30:29.670 align:middle line:84% when that fraction is equal to 1, 01:30:29.670 --> 01:30:34.410 align:middle line:84% then the efficiency is [INAUDIBLE] So without the 2, 01:30:34.410 --> 01:30:38.230 align:middle line:84% and let's say the sensitivity is 1, then if the fraction is 1, 01:30:38.230 --> 01:30:40.350 align:middle line:84% then this whole effect would be 1/2. 01:30:40.350 --> 01:30:41.830 align:middle line:84% And so I put in the 2 so that way, 01:30:41.830 --> 01:30:43.590 align:middle line:84% when the fraction is 1 [INAUDIBLE] 01:30:43.590 --> 01:30:45.470 align:middle line:90% HAZHIR RAHMANDAD: Yeah. 01:30:45.470 --> 01:30:47.790 align:middle line:84% STUDENT: So I just wanted to bring this up 01:30:47.790 --> 01:30:50.190 align:middle line:84% because I approached this differently 01:30:50.190 --> 01:30:51.490 align:middle line:90% than a lot of other people. 01:30:51.490 --> 01:30:53.610 align:middle line:84% And I just wanted to know what the issue is. 01:30:53.610 --> 01:30:55.610 align:middle line:84% HAZHIR RAHMANDAD: Yeah, so let's think about it. 01:30:55.610 --> 01:30:59.170 align:middle line:84% So we already have talked a little bit about some of these. 01:30:59.170 --> 01:31:06.350 align:middle line:84% So basically, this one is saying, 01:31:06.350 --> 01:31:11.710 align:middle line:84% at carrying capacity levels of prey population, 01:31:11.710 --> 01:31:17.950 align:middle line:84% we give predators what they want. 01:31:17.950 --> 01:31:28.550 align:middle line:84% And higher levels wouldn't happen 01:31:28.550 --> 01:31:30.950 align:middle line:84% because this fraction of prey carrying capacity 01:31:30.950 --> 01:31:34.190 align:middle line:90% should not go above 1. 01:31:34.190 --> 01:31:40.430 align:middle line:84% So anything below that will lead to less consumption 01:31:40.430 --> 01:31:44.150 align:middle line:90% per predator. 01:31:44.150 --> 01:31:48.370 align:middle line:84% And how much less depends on this sensitivity parameter. 01:31:48.370 --> 01:31:50.910 align:middle line:84% You're currently putting it at 2. 01:31:50.910 --> 01:31:56.230 align:middle line:84% So this leads to a relatively sharp reduction. 01:31:56.230 --> 01:31:59.400 align:middle line:84% That is, so the table function-ish view 01:31:59.400 --> 01:32:00.540 align:middle line:90% is more like this. 01:32:00.540 --> 01:32:01.660 align:middle line:90% This is 0 to 1. 01:32:01.660 --> 01:32:08.060 align:middle line:84% It goes quickly up, which means it is very sensitive. 01:32:08.060 --> 01:32:11.960 align:middle line:84% If you don't have enough deer, the amount of consumption 01:32:11.960 --> 01:32:14.720 align:middle line:90% goes down non-linearly. 01:32:14.720 --> 01:32:19.600 align:middle line:84% So you wouldn't have a problem with hitting the 45-degree line 01:32:19.600 --> 01:32:23.360 align:middle line:84% in our other version of the formulation, 01:32:23.360 --> 01:32:27.540 align:middle line:84% as long as your sensitivity parameter is above 1. 01:32:27.540 --> 01:32:34.360 align:middle line:90% 01:32:34.360 --> 01:32:41.520 align:middle line:84% We do have the issue that maybe that is too harsh, maybe-- 01:32:41.520 --> 01:32:44.140 align:middle line:90% that is, to predators. 01:32:44.140 --> 01:32:46.780 align:middle line:84% That is, they could consume more. 01:32:46.780 --> 01:32:52.600 align:middle line:90% 01:32:52.600 --> 01:32:57.040 align:middle line:84% So we said that, at the 0 level of prey, 01:32:57.040 --> 01:33:03.240 align:middle line:84% the 45-degree line in the other conception is a potentially good 01:33:03.240 --> 01:33:04.960 align:middle line:90% reference curve. 01:33:04.960 --> 01:33:11.540 align:middle line:84% And the argument there was that, if there is prey to be eaten 01:33:11.540 --> 01:33:13.820 align:middle line:84% and there is plenty of predators, 01:33:13.820 --> 01:33:16.800 align:middle line:84% they'll be eating the prey with the maximum rate 01:33:16.800 --> 01:33:22.560 align:middle line:84% feasible that comes from prey divided by number of-- 01:33:22.560 --> 01:33:25.000 align:middle line:90% minimum time to consume prey. 01:33:25.000 --> 01:33:29.640 align:middle line:84% So that was the argument for targeting the 45-degree line 01:33:29.640 --> 01:33:31.720 align:middle line:90% as a reference line. 01:33:31.720 --> 01:33:34.660 align:middle line:84% Now, you may argue that is not the case. 01:33:34.660 --> 01:33:46.320 align:middle line:84% That is, the search process for the prey scales 01:33:46.320 --> 01:33:48.220 align:middle line:90% non-linearly with its density. 01:33:48.220 --> 01:33:52.880 align:middle line:84% That is, if the density is very small, 01:33:52.880 --> 01:33:57.120 align:middle line:84% it doesn't go down the rate of consumption-- 01:33:57.120 --> 01:33:58.980 align:middle line:90% linearly, it goes non-linearly. 01:33:58.980 --> 01:34:02.400 align:middle line:84% And I think your equation, Jake, also 01:34:02.400 --> 01:34:05.480 align:middle line:84% had that alternative way of justifying it. 01:34:05.480 --> 01:34:09.000 align:middle line:84% So I want us to be clear about what the formulations are 01:34:09.000 --> 01:34:11.120 align:middle line:90% doing in both of these cases. 01:34:11.120 --> 01:34:14.640 align:middle line:84% In your case, I'm not sure because we haven't connected 01:34:14.640 --> 01:34:16.380 align:middle line:90% it fully to that formulation. 01:34:16.380 --> 01:34:18.760 align:middle line:84% In this case, it is making the assumption 01:34:18.760 --> 01:34:24.100 align:middle line:84% that the non-linearity in preyability is significant. 01:34:24.100 --> 01:34:26.620 align:middle line:84% And therefore, even if there is some prey, 01:34:26.620 --> 01:34:30.420 align:middle line:84% they would not be hunted because they can't be found. 01:34:30.420 --> 01:34:32.240 align:middle line:84% They go into some holes and something. 01:34:32.240 --> 01:34:32.980 align:middle line:90% I don't know. 01:34:32.980 --> 01:34:36.840 align:middle line:84% And that's OK, as long as you know that's the assumption 01:34:36.840 --> 01:34:39.583 align:middle line:90% you're making. 01:34:39.583 --> 01:34:41.000 align:middle line:84% STUDENT: So yeah, I was thinking-- 01:34:41.000 --> 01:34:42.600 align:middle line:84% I just want to bring this up because [INAUDIBLE] I 01:34:42.600 --> 01:34:44.142 align:middle line:84% was thinking about the search process 01:34:44.142 --> 01:34:45.420 align:middle line:90% as opposed to the [INAUDIBLE] 01:34:45.420 --> 01:34:50.680 align:middle line:84% HAZHIR RAHMANDAD: Yes, but you also have basically the desire. 01:34:50.680 --> 01:34:54.370 align:middle line:84% It is built into your parameter reference hunting efficiency. 01:34:54.370 --> 01:34:58.970 align:middle line:84% That's basically the desired parameter in your setup. 01:34:58.970 --> 01:35:02.450 align:middle line:84% And its units should be in deer per wolf per month. 01:35:02.450 --> 01:35:04.510 align:middle line:90% So it is really just the desire. 01:35:04.510 --> 01:35:08.210 align:middle line:90% 01:35:08.210 --> 01:35:08.710 align:middle line:90% STUDENT: OK. 01:35:08.710 --> 01:35:12.990 align:middle line:84% So I think, in that case, my mistake was is that I-- 01:35:12.990 --> 01:35:13.490 align:middle line:90% OK. 01:35:13.490 --> 01:35:14.865 align:middle line:84% I think my mistake, in that case, 01:35:14.865 --> 01:35:16.570 align:middle line:90% was that I didn't realize that. 01:35:16.570 --> 01:35:18.070 align:middle line:84% I wasn't thinking about [INAUDIBLE]. 01:35:18.070 --> 01:35:20.028 align:middle line:84% So then when I talk about the effect of hunting 01:35:20.028 --> 01:35:23.183 align:middle line:84% on the wolf death rate in the lower half, 01:35:23.183 --> 01:35:24.850 align:middle line:84% I'll have a separate parameter, which is 01:35:24.850 --> 01:35:26.430 align:middle line:90% the required prey per predator. 01:35:26.430 --> 01:35:28.090 align:middle line:84% And now, I see that they're [INAUDIBLE] 01:35:28.090 --> 01:35:29.007 align:middle line:90% HAZHIR RAHMANDAD: Yes. 01:35:29.007 --> 01:35:32.070 align:middle line:84% Conceptually, these two are very similar to each other. 01:35:32.070 --> 01:35:37.070 align:middle line:84% And basically, you shouldn't be creating yet another process. 01:35:37.070 --> 01:35:40.810 align:middle line:84% Now, this yet another process will give you 01:35:40.810 --> 01:35:45.050 align:middle line:84% a number that is very comparable probably 01:35:45.050 --> 01:35:46.670 align:middle line:84% with the parameters you have chosen. 01:35:46.670 --> 01:35:50.090 align:middle line:84% And then you have another sensitivity thingy here. 01:35:50.090 --> 01:35:53.810 align:middle line:84% So maybe, ultimately, it doesn't matter, 01:35:53.810 --> 01:35:55.468 align:middle line:90% in terms of degrees of freedom. 01:35:55.468 --> 01:35:57.010 align:middle line:84% You have plenty of degrees of freedom 01:35:57.010 --> 01:35:58.790 align:middle line:90% to get the behavior you want. 01:35:58.790 --> 01:36:02.975 align:middle line:84% But it reduces the clarity of the formulation. 01:36:02.975 --> 01:36:04.850 align:middle line:84% STUDENT: I guess I was thinking maybe there's 01:36:04.850 --> 01:36:07.430 align:middle line:84% a predator who really wants to eat a lot of prey, 01:36:07.430 --> 01:36:08.590 align:middle line:90% but is really bad at it. 01:36:08.590 --> 01:36:10.370 align:middle line:90% And that's why I had-- 01:36:10.370 --> 01:36:11.930 align:middle line:90% I mean, unironically. 01:36:11.930 --> 01:36:15.318 align:middle line:90% And that's why I had separate-- 01:36:15.318 --> 01:36:16.610 align:middle line:90% it kind of ran through my mind. 01:36:16.610 --> 01:36:18.870 align:middle line:84% And that was my haphazard justification. 01:36:18.870 --> 01:36:21.290 align:middle line:84% But yeah, so I see the-- and then 01:36:21.290 --> 01:36:23.810 align:middle line:84% I also see how the hunting adequacy, which 01:36:23.810 --> 01:36:26.610 align:middle line:84% is just prey hunted divided by required, 01:36:26.610 --> 01:36:29.772 align:middle line:90% is similar to that effect. 01:36:29.772 --> 01:36:30.730 align:middle line:90% HAZHIR RAHMANDAD: Yeah. 01:36:30.730 --> 01:36:34.090 align:middle line:84% So basically, this sensitivity parameter 01:36:34.090 --> 01:36:39.770 align:middle line:84% can take care of the efficiency consideration. 01:36:39.770 --> 01:36:44.410 align:middle line:84% And then the desired parameter should mean the desired. 01:36:44.410 --> 01:36:46.330 align:middle line:84% STUDENT: Yeah, so I see how it's very similar. 01:36:46.330 --> 01:36:47.050 align:middle line:90% HAZHIR RAHMANDAD: Yeah. 01:36:47.050 --> 01:36:49.175 align:middle line:84% STUDENT: And then I guess I just wanted to bring up 01:36:49.175 --> 01:36:51.190 align:middle line:90% as well how it affects it. 01:36:51.190 --> 01:36:55.250 align:middle line:84% So I had predator deaths be the same. 01:36:55.250 --> 01:36:58.490 align:middle line:84% And then what I had is that this effect, which we discussed 01:36:58.490 --> 01:36:59.930 align:middle line:84% the issues with, basically changes 01:36:59.930 --> 01:37:03.810 align:middle line:84% how long hunters live with logistics so that hunter 01:37:03.810 --> 01:37:04.852 align:middle line:90% lifespan is weighted. 01:37:04.852 --> 01:37:05.810 align:middle line:90% HAZHIR RAHMANDAD: Yeah. 01:37:05.810 --> 01:37:13.010 align:middle line:84% So now, basically, you have the version with a single outflow 01:37:13.010 --> 01:37:14.670 align:middle line:84% rather than two separate outflows. 01:37:14.670 --> 01:37:16.470 align:middle line:90% So we saw Jason's two outflow. 01:37:16.470 --> 01:37:18.970 align:middle line:84% Let's look at this single outflow version 01:37:18.970 --> 01:37:20.230 align:middle line:90% and see how it works. 01:37:20.230 --> 01:37:22.050 align:middle line:90% So you have the hunter lifespan. 01:37:22.050 --> 01:37:24.430 align:middle line:84% And you are just reducing the lifespan. 01:37:24.430 --> 01:37:27.370 align:middle line:84% So that's a very natural, good alternative 01:37:27.370 --> 01:37:30.050 align:middle line:84% because, even if you're dying from starvation, 01:37:30.050 --> 01:37:31.630 align:middle line:90% you're dying after a while. 01:37:31.630 --> 01:37:33.150 align:middle line:90% So your lifespan has ended. 01:37:33.150 --> 01:37:36.130 align:middle line:84% So as such, this is also a good formulation, 01:37:36.130 --> 01:37:37.670 align:middle line:90% the basic starting point. 01:37:37.670 --> 01:37:40.830 align:middle line:84% Now, we have to get what is the hunter lifespan. 01:37:40.830 --> 01:37:43.330 align:middle line:84% You have the minimum hunter lifespan and the maximum hunter 01:37:43.330 --> 01:37:43.950 align:middle line:90% lifespan. 01:37:43.950 --> 01:37:45.470 align:middle line:90% This is the natural. 01:37:45.470 --> 01:37:46.850 align:middle line:84% The maximum is the natural if you 01:37:46.850 --> 01:37:49.260 align:middle line:90% have all of the food you need. 01:37:49.260 --> 01:37:52.940 align:middle line:84% And then the minimum is the starvation time horizon 01:37:52.940 --> 01:37:55.280 align:middle line:90% that Jason was talking about. 01:37:55.280 --> 01:37:57.820 align:middle line:84% And then this is some kind of a weighted average. 01:37:57.820 --> 01:38:00.140 align:middle line:84% So this effect multiplied by maximum, 01:38:00.140 --> 01:38:03.440 align:middle line:84% 1 minus effect multiplied by minimum, this is fine. 01:38:03.440 --> 01:38:09.140 align:middle line:84% I like that way of shifting between those two life spans. 01:38:09.140 --> 01:38:14.400 align:middle line:84% Then the key question is, how does this effect get formulated? 01:38:14.400 --> 01:38:18.980 align:middle line:84% It needs to be between 0 and 1 because that switch requires 01:38:18.980 --> 01:38:20.440 align:middle line:90% you to be between 0 and 1. 01:38:20.440 --> 01:38:21.860 align:middle line:90% Otherwise, it doesn't work. 01:38:21.860 --> 01:38:24.700 align:middle line:84% So you have hunting adequacy, which 01:38:24.700 --> 01:38:30.060 align:middle line:84% is how much you eat versus how much you are requiring. 01:38:30.060 --> 01:38:35.860 align:middle line:84% Again, keep in mind that this required value should probably 01:38:35.860 --> 01:38:38.740 align:middle line:84% be just hunting efficiency multiplied 01:38:38.740 --> 01:38:40.000 align:middle line:90% by predator or something. 01:38:40.000 --> 01:38:43.140 align:middle line:84% But that put aside, this is also a decent input 01:38:43.140 --> 01:38:45.860 align:middle line:90% into the formulation. 01:38:45.860 --> 01:38:49.660 align:middle line:84% And this haunting adequacy should not go above 1, 01:38:49.660 --> 01:38:51.700 align:middle line:90% if it is correctly formulated. 01:38:51.700 --> 01:38:56.060 align:middle line:84% Now, in this version, it may go above 1 depending 01:38:56.060 --> 01:38:57.600 align:middle line:90% on the parameters chosen. 01:38:57.600 --> 01:39:00.620 align:middle line:84% That is, if the required prey per predators 01:39:00.620 --> 01:39:04.100 align:middle line:84% is smaller than this hunting efficiency parameter-- 01:39:04.100 --> 01:39:07.960 align:middle line:84% and it's probably not the case in Alexander's So this is 2. 01:39:07.960 --> 01:39:10.040 align:middle line:90% That one was 1.5. 01:39:10.040 --> 01:39:11.480 align:middle line:90% So it wouldn't go above 1. 01:39:11.480 --> 01:39:14.300 align:middle line:84% But technically, it could if somebody comes and plays 01:39:14.300 --> 01:39:16.140 align:middle line:90% with the parameters. 01:39:16.140 --> 01:39:19.880 align:middle line:84% Everybody is following what we are saying? 01:39:19.880 --> 01:39:24.340 align:middle line:84% So you see, on the top, hunting efficiency 01:39:24.340 --> 01:39:28.220 align:middle line:84% is multiplied, after some adjustments, 01:39:28.220 --> 01:39:29.680 align:middle line:90% by the number of predators. 01:39:29.680 --> 01:39:31.780 align:middle line:90% That's the actual prey hunted. 01:39:31.780 --> 01:39:36.140 align:middle line:84% In the bottom, hunting required as the predators multiplied 01:39:36.140 --> 01:39:38.780 align:middle line:90% by this required prey. 01:39:38.780 --> 01:39:42.940 align:middle line:84% So this gives a different number than the maximum up there. 01:39:42.940 --> 01:39:49.670 align:middle line:84% And if the maximum up there is bigger than this number, 01:39:49.670 --> 01:39:53.630 align:middle line:84% the ratio of hunting adequacy exceeds 1. 01:39:53.630 --> 01:39:58.150 align:middle line:84% And if that exceeds 1, things get messed up 01:39:58.150 --> 01:40:01.910 align:middle line:90% because now this has-- 01:40:01.910 --> 01:40:06.070 align:middle line:90% this ratio should not exceed 1. 01:40:06.070 --> 01:40:10.550 align:middle line:84% Now, in this formulation, maybe it is kept between 0 and 1 01:40:10.550 --> 01:40:11.690 align:middle line:90% using the exponential. 01:40:11.690 --> 01:40:13.310 align:middle line:90% So let's look at this equation. 01:40:13.310 --> 01:40:14.380 align:middle line:90% STUDENT: [INAUDIBLE] 01:40:14.380 --> 01:40:16.630 align:middle line:84% HAZHIR RAHMANDAD: Yeah, so this is a logistic one that 01:40:16.630 --> 01:40:23.070 align:middle line:84% says e to the power of some alpha hunting adequacy minus 1 01:40:23.070 --> 01:40:26.768 align:middle line:90% divided by same thing plus 1. 01:40:26.768 --> 01:40:28.810 align:middle line:84% STUDENT: Yeah, I also screwed up the parentheses, 01:40:28.810 --> 01:40:29.477 align:middle line:90% I realized here. 01:40:29.477 --> 01:40:30.352 align:middle line:90% HAZHIR RAHMANDAD: OK. 01:40:30.352 --> 01:40:32.870 align:middle line:84% STUDENT: The entire denominator is taken to the exponent. 01:40:32.870 --> 01:40:34.690 align:middle line:84% But my intent was to do logistic. 01:40:34.690 --> 01:40:38.550 align:middle line:84% HAZHIR RAHMANDAD: Yes, you need the parameter-- 01:40:38.550 --> 01:40:41.570 align:middle line:90% sorry you need another-- 01:40:41.570 --> 01:40:48.430 align:middle line:84% STUDENT: [INAUDIBLE] I think the second-- 01:40:48.430 --> 01:40:50.910 align:middle line:90% HAZHIR RAHMANDAD: Is extra? 01:40:50.910 --> 01:40:55.430 align:middle line:84% No, this is fine, e to the power of whatever 01:40:55.430 --> 01:41:02.390 align:middle line:84% you have up there divided by e to the power of sensitivity. 01:41:02.390 --> 01:41:05.750 align:middle line:90% I think it is fine. 01:41:05.750 --> 01:41:07.150 align:middle line:90% Maybe this one is-- 01:41:07.150 --> 01:41:08.150 align:middle line:90% STUDENT: Yeah, that one. 01:41:08.150 --> 01:41:09.470 align:middle line:84% HAZHIR RAHMANDAD: This one is misplaced, yes. 01:41:09.470 --> 01:41:10.570 align:middle line:90% So this should be here. 01:41:10.570 --> 01:41:11.070 align:middle line:90% OK. 01:41:11.070 --> 01:41:15.330 align:middle line:84% But the basic idea works well here. 01:41:15.330 --> 01:41:19.110 align:middle line:84% So it is saying, if hunting adequacy is 1-- that is, 01:41:19.110 --> 01:41:22.070 align:middle line:84% you have plenty, that you have what you wanted-- 01:41:22.070 --> 01:41:30.430 align:middle line:84% then e to the power of 0 divided by-- so you get a value of 0.5 01:41:30.430 --> 01:41:32.690 align:middle line:90% out of this expression. 01:41:32.690 --> 01:41:34.210 align:middle line:90% Is that what you wanted to get? 01:41:34.210 --> 01:41:36.230 align:middle line:84% STUDENT: I was thinking of hunting adequacy 01:41:36.230 --> 01:41:40.230 align:middle line:90% as basically this is like-- 01:41:40.230 --> 01:41:45.110 align:middle line:84% I was thinking of it more of like this is some minimal level. 01:41:45.110 --> 01:41:47.590 align:middle line:84% And that if you go below this level, 01:41:47.590 --> 01:41:49.395 align:middle line:84% your lifespan is going to go really down. 01:41:49.395 --> 01:41:50.270 align:middle line:90% HAZHIR RAHMANDAD: OK. 01:41:50.270 --> 01:41:53.170 align:middle line:84% So this is an important consideration. 01:41:53.170 --> 01:41:55.590 align:middle line:90% So Alexander talked about it. 01:41:55.590 --> 01:41:58.310 align:middle line:84% And he tried to anchor the operating 01:41:58.310 --> 01:42:05.150 align:middle line:84% point of this relationship to the value of hunting adequacy 01:42:05.150 --> 01:42:09.430 align:middle line:84% equals to 1 means some lifespan that 01:42:09.430 --> 01:42:15.710 align:middle line:84% is the middle of the starving time and the full lifetime. 01:42:15.710 --> 01:42:19.550 align:middle line:90% That is not intuitive for me. 01:42:19.550 --> 01:42:22.890 align:middle line:84% Hearing that, I would say, OK, I can follow your logic. 01:42:22.890 --> 01:42:26.990 align:middle line:84% But if hunting adequacy-- the way I think of the concept 01:42:26.990 --> 01:42:30.210 align:middle line:84% is, if it is 1, then I should get my full lifespan. 01:42:30.210 --> 01:42:32.870 align:middle line:90% Why do you kill me halfway? 01:42:32.870 --> 01:42:35.570 align:middle line:90% So it is not wrong. 01:42:35.570 --> 01:42:38.080 align:middle line:84% Mathematically, it probably works OK. 01:42:38.080 --> 01:42:41.160 align:middle line:84% But conceptually, it is harder to justify it. 01:42:41.160 --> 01:42:45.600 align:middle line:84% There is also, technically, you want your operating point 01:42:45.600 --> 01:42:48.640 align:middle line:84% for your model, when you are normalizing your model 01:42:48.640 --> 01:42:53.160 align:middle line:84% and trying to set your baseline equilibrium level, not 01:42:53.160 --> 01:42:57.720 align:middle line:84% to be somewhere in the middle of these time constants or things 01:42:57.720 --> 01:42:58.260 align:middle line:90% like that. 01:42:58.260 --> 01:43:02.120 align:middle line:84% It just becomes much harder to put the model in equilibrium 01:43:02.120 --> 01:43:04.140 align:middle line:90% and get it to behave well. 01:43:04.140 --> 01:43:08.640 align:middle line:84% You want it to be somewhere in a more normal range. 01:43:08.640 --> 01:43:13.000 align:middle line:84% Here, more normal means you have your adequate life. 01:43:13.000 --> 01:43:15.520 align:middle line:90% The deer is plentiful, probably. 01:43:15.520 --> 01:43:18.660 align:middle line:84% And grass is plentiful or something like that. 01:43:18.660 --> 01:43:20.200 align:middle line:90% Now, this is a complex model. 01:43:20.200 --> 01:43:22.580 align:middle line:84% Putting it into equilibrium is not trivial. 01:43:22.580 --> 01:43:25.700 align:middle line:84% So you may need to think more about it. 01:43:25.700 --> 01:43:27.800 align:middle line:90% And I haven't looked at it. 01:43:27.800 --> 01:43:31.500 align:middle line:84% So this works, but has those characteristics. 01:43:31.500 --> 01:43:32.000 align:middle line:90% Great. 01:43:32.000 --> 01:43:36.040 align:middle line:90% It was good to look at this. 01:43:36.040 --> 01:43:41.000 align:middle line:84% But the idea of a single flow with this type of feedback 01:43:41.000 --> 01:43:43.340 align:middle line:90% all works, all meaningful. 01:43:43.340 --> 01:43:45.960 align:middle line:90% 01:43:45.960 --> 01:43:52.400 align:middle line:84% Also, the formulation here is a kind of S-shape 01:43:52.400 --> 01:43:56.850 align:middle line:90% when you have the logistic. 01:43:56.850 --> 01:43:58.600 align:middle line:84% STUDENT: And that was kind of intentional. 01:43:58.600 --> 01:44:03.360 align:middle line:84% My point was saying that, when you're crossing over 01:44:03.360 --> 01:44:05.760 align:middle line:84% that threshold of what you need, your lifespan 01:44:05.760 --> 01:44:10.760 align:middle line:84% improves quite a lot in that short little range. 01:44:10.760 --> 01:44:12.260 align:middle line:90% HAZHIR RAHMANDAD: Yes. 01:44:12.260 --> 01:44:17.580 align:middle line:84% So You were making that very responsive. 01:44:17.580 --> 01:44:23.640 align:middle line:90% 01:44:23.640 --> 01:44:28.280 align:middle line:84% So I would have probably thought of hunting adequacy 01:44:28.280 --> 01:44:32.820 align:middle line:84% to keep it between 0 and 1, and then normalize around 01:44:32.820 --> 01:44:37.360 align:middle line:84% the value of 1-- that is, make that the equilibrium 01:44:37.360 --> 01:44:40.240 align:middle line:90% region or the baseline region. 01:44:40.240 --> 01:44:47.020 align:middle line:84% And then you can have a sigmoid function 01:44:47.020 --> 01:44:49.380 align:middle line:84% if you think there is non-linearity, 01:44:49.380 --> 01:44:51.600 align:middle line:90% which there might be. 01:44:51.600 --> 01:44:54.740 align:middle line:84% Yeah, if I have 90% of what I want to eat, 01:44:54.740 --> 01:44:57.500 align:middle line:84% my life span wouldn't come down as fast. 01:44:57.500 --> 01:45:00.280 align:middle line:84% And then there is a point that it does come down very fast. 01:45:00.280 --> 01:45:04.340 align:middle line:84% And then it doesn't matter if I'm getting 10% or 20%. 01:45:04.340 --> 01:45:05.720 align:middle line:90% I'm going to die. 01:45:05.720 --> 01:45:07.500 align:middle line:90% So that is fine. 01:45:07.500 --> 01:45:10.120 align:middle line:84% I think the S-shaped logic here does 01:45:10.120 --> 01:45:15.355 align:middle line:84% have a good biological support for it. 01:45:15.355 --> 01:45:17.480 align:middle line:84% STUDENT: But you were saying that the anchor should 01:45:17.480 --> 01:45:21.092 align:middle line:84% be on the ends of these distributions? 01:45:21.092 --> 01:45:22.300 align:middle line:90% HAZHIR RAHMANDAD: Not always. 01:45:22.300 --> 01:45:24.280 align:middle line:84% So in this case, it is hard because, when 01:45:24.280 --> 01:45:26.860 align:middle line:84% you put it to the end of the distribution, 01:45:26.860 --> 01:45:28.720 align:middle line:90% it wouldn't saturate. 01:45:28.720 --> 01:45:32.730 align:middle line:84% So you have to go a lot, to a lot of deer. 01:45:32.730 --> 01:45:34.810 align:middle line:84% And a lot of deer may not give you 01:45:34.810 --> 01:45:36.790 align:middle line:84% equilibrium on the other parts of the model. 01:45:36.790 --> 01:45:39.450 align:middle line:84% So that's why I'm saying putting this model into equilibrium 01:45:39.450 --> 01:45:41.630 align:middle line:90% is not actually always easy. 01:45:41.630 --> 01:45:47.550 align:middle line:84% You need to-- so in terms of the explaining and the logic, 01:45:47.550 --> 01:45:51.810 align:middle line:84% it is easier to put your anchor at the point 01:45:51.810 --> 01:45:54.810 align:middle line:84% where you're getting what you want. 01:45:54.810 --> 01:45:57.590 align:middle line:84% In terms of getting the analytical equilibrium, 01:45:57.590 --> 01:45:59.250 align:middle line:90% it may or may not be. 01:45:59.250 --> 01:46:05.890 align:middle line:84% You have to write it and see if you can get [INAUDIBLE] Good. 01:46:05.890 --> 01:46:09.410 align:middle line:84% Anybody else wants to look at their model? 01:46:09.410 --> 01:46:12.290 align:middle line:84% Anything else that is here that we want to look at? 01:46:12.290 --> 01:46:15.133 align:middle line:90% STUDENT: [INAUDIBLE] 01:46:15.133 --> 01:46:16.050 align:middle line:90% HAZHIR RAHMANDAD: Yes. 01:46:16.050 --> 01:46:18.770 align:middle line:84% STUDENT: In my mind there's a requirement [INAUDIBLE]. 01:46:18.770 --> 01:46:22.010 align:middle line:90% 01:46:22.010 --> 01:46:25.690 align:middle line:84% What was the bit about hunting efficiency should actually be-- 01:46:25.690 --> 01:46:27.970 align:middle line:90% HAZHIR RAHMANDAD: Oh, yes. 01:46:27.970 --> 01:46:32.570 align:middle line:84% So here, the hunting efficiency is a parameter 01:46:32.570 --> 01:46:36.530 align:middle line:84% that is basically saying how many deer per wolf per month 01:46:36.530 --> 01:46:39.370 align:middle line:90% are being hunted. 01:46:39.370 --> 01:46:40.490 align:middle line:90% OK. 01:46:40.490 --> 01:46:44.370 align:middle line:84% And then there is the required prey per predator. 01:46:44.370 --> 01:46:49.890 align:middle line:84% That is too how many deer per wolf per month 01:46:49.890 --> 01:46:51.690 align:middle line:90% is desired to be hunted. 01:46:51.690 --> 01:46:55.150 align:middle line:84% And the two are conceptually very close to each other. 01:46:55.150 --> 01:46:57.690 align:middle line:84% Now, Alexander had this idea in mind 01:46:57.690 --> 01:47:01.610 align:middle line:84% that I want-- maybe my hunters are not good hunters. 01:47:01.610 --> 01:47:04.890 align:middle line:84% And in fact, in this version, then they 01:47:04.890 --> 01:47:07.650 align:middle line:90% should die in some way. 01:47:07.650 --> 01:47:09.110 align:middle line:90% I'm not simulating it. 01:47:09.110 --> 01:47:10.950 align:middle line:90% There are other reasons for it. 01:47:10.950 --> 01:47:12.110 align:middle line:90% They're not going extinct. 01:47:12.110 --> 01:47:18.170 align:middle line:84% But if they can't even hunt what they desire in their best case, 01:47:18.170 --> 01:47:19.430 align:middle line:90% then they are in big trouble. 01:47:19.430 --> 01:47:24.290 align:middle line:90% That's challenging for them. 01:47:24.290 --> 01:47:27.250 align:middle line:84% But I was saying, it is better not to separate these two 01:47:27.250 --> 01:47:29.420 align:middle line:84% parameters, to have a single one that 01:47:29.420 --> 01:47:33.580 align:middle line:90% is desired prey per predator. 01:47:33.580 --> 01:47:37.780 align:middle line:84% That desired will determine both the required hunting 01:47:37.780 --> 01:47:39.400 align:middle line:90% and hunting efficiency. 01:47:39.400 --> 01:47:41.120 align:middle line:84% That's the same thing, basically, 01:47:41.120 --> 01:47:42.740 align:middle line:84% except that hunting efficiency also 01:47:42.740 --> 01:47:46.900 align:middle line:84% has the effect of prey population and efficiency. 01:47:46.900 --> 01:47:54.560 align:middle line:84% STUDENT: [INAUDIBLE] I need 15 deer a year to survive. 01:47:54.560 --> 01:47:58.540 align:middle line:84% But at this base rate, I'm going to eat 25. 01:47:58.540 --> 01:48:00.340 align:middle line:90% HAZHIR RAHMANDAD: Yes. 01:48:00.340 --> 01:48:03.980 align:middle line:84% So that difference comes from effective prey population. 01:48:03.980 --> 01:48:07.380 align:middle line:90% So that is captured here. 01:48:07.380 --> 01:48:10.580 align:middle line:84% But otherwise, it should be the same. 01:48:10.580 --> 01:48:11.700 align:middle line:90% OK. 01:48:11.700 --> 01:48:13.060 align:middle line:90% [? Albert? ?] 01:48:13.060 --> 01:48:15.020 align:middle line:90% STUDENT: [INAUDIBLE] 01:48:15.020 --> 01:48:33.300 align:middle line:90% 01:48:33.300 --> 01:48:35.520 align:middle line:84% HAZHIR RAHMANDAD: So we have the predator, prey. 01:48:35.520 --> 01:48:39.940 align:middle line:90% And let's look at a few things. 01:48:39.940 --> 01:48:44.980 align:middle line:84% So consumption rate of prey comes 01:48:44.980 --> 01:48:50.220 align:middle line:90% from consumption per predator. 01:48:50.220 --> 01:48:52.460 align:middle line:84% Sorry, consumption is per predator. 01:48:52.460 --> 01:48:54.660 align:middle line:84% And that comes from normal consumption 01:48:54.660 --> 01:48:59.180 align:middle line:84% per predator and effect of density on consumption. 01:48:59.180 --> 01:49:02.140 align:middle line:84% So effect of density on consumption 01:49:02.140 --> 01:49:08.620 align:middle line:84% comes from density of prey versus a reference density, 01:49:08.620 --> 01:49:14.580 align:middle line:84% and then normalized by sensitivity of consumption rate 01:49:14.580 --> 01:49:16.280 align:middle line:90% to density. 01:49:16.280 --> 01:49:17.800 align:middle line:90% So we can see this. 01:49:17.800 --> 01:49:21.340 align:middle line:90% 01:49:21.340 --> 01:49:26.880 align:middle line:84% So this is if y density equals reference-- 01:49:26.880 --> 01:49:29.660 align:middle line:84% so let's think of the reference density of prey 01:49:29.660 --> 01:49:33.780 align:middle line:90% to be maximum carrying capacity. 01:49:33.780 --> 01:49:37.020 align:middle line:84% Is that what you had in mind, [? Albert? ?] 01:49:37.020 --> 01:49:37.860 align:middle line:90% STUDENT: No. 01:49:37.860 --> 01:49:38.735 align:middle line:90% HAZHIR RAHMANDAD: No. 01:49:38.735 --> 01:49:41.580 align:middle line:90% So what is that? 01:49:41.580 --> 01:49:49.380 align:middle line:84% STUDENT: I was anchoring from the prey death rate. 01:49:49.380 --> 01:49:53.540 align:middle line:84% So there's a death rate that is if you have ample food, that's 01:49:53.540 --> 01:49:55.460 align:middle line:90% a fraction of death rate. 01:49:55.460 --> 01:50:02.180 align:middle line:84% And that implies the normal consumption rate, so 01:50:02.180 --> 01:50:04.340 align:middle line:90% the desired consumption rate. 01:50:04.340 --> 01:50:08.180 align:middle line:84% And that anchored the reference density. 01:50:08.180 --> 01:50:16.900 align:middle line:84% That is, in that density, you will have that consumption rate. 01:50:16.900 --> 01:50:19.560 align:middle line:84% HAZHIR RAHMANDAD: But this is not anchored to anything else. 01:50:19.560 --> 01:50:24.510 align:middle line:84% This is just a number reference by density. 01:50:24.510 --> 01:50:32.190 align:middle line:84% So it's just how many prey per unit of land you can have. 01:50:32.190 --> 01:50:34.230 align:middle line:84% STUDENT: So in that kind of density, 01:50:34.230 --> 01:50:41.130 align:middle line:84% then the predator can have catch how many-- 01:50:41.130 --> 01:50:42.890 align:middle line:84% HAZHIR RAHMANDAD: Yes, they can get, 01:50:42.890 --> 01:50:45.270 align:middle line:90% how many deer they want to get. 01:50:45.270 --> 01:50:47.910 align:middle line:90% So that's fine. 01:50:47.910 --> 01:50:53.770 align:middle line:84% But if this is not the maximum for predator population-- 01:50:53.770 --> 01:50:55.230 align:middle line:90% for prey population, sorry. 01:50:55.230 --> 01:50:58.270 align:middle line:84% It's not the maximum density for prey population, 01:50:58.270 --> 01:51:03.030 align:middle line:84% then this ratio of y divided by reference y-- 01:51:03.030 --> 01:51:04.030 align:middle line:90% STUDENT: Can go beyond-- 01:51:04.030 --> 01:51:05.447 align:middle line:84% HAZHIR RAHMANDAD: Can go beyond 1. 01:51:05.447 --> 01:51:09.430 align:middle line:84% And then that ratio raised to this sensitivity power 01:51:09.430 --> 01:51:15.710 align:middle line:84% can go beyond 1, which means that consumption per predator 01:51:15.710 --> 01:51:19.650 align:middle line:84% can go above the normal consumption per predator. 01:51:19.650 --> 01:51:22.590 align:middle line:84% And that you don't want to happen. 01:51:22.590 --> 01:51:25.190 align:middle line:84% STUDENT: I was in the similar realm with Kathy 01:51:25.190 --> 01:51:28.110 align:middle line:84% that, if you have more density, then 01:51:28.110 --> 01:51:31.040 align:middle line:84% you might go above your normal consumption. 01:51:31.040 --> 01:51:32.290 align:middle line:90% HAZHIR RAHMANDAD: Yeah, maybe. 01:51:32.290 --> 01:51:37.490 align:middle line:84% But even then it should saturate. 01:51:37.490 --> 01:51:41.290 align:middle line:84% That is, yeah maybe somehow you could argue for 20% more, 01:51:41.290 --> 01:51:43.130 align:middle line:90% but it shouldn't go to infinity. 01:51:43.130 --> 01:51:46.430 align:middle line:84% And this allows it to go to large numbers. 01:51:46.430 --> 01:51:49.110 align:middle line:90% And I'm not seeing yet-- 01:51:49.110 --> 01:51:49.890 align:middle line:90% we can look at it. 01:51:49.890 --> 01:51:53.150 align:middle line:84% But I'm not yet seeing how the prey population would remain 01:51:53.150 --> 01:51:54.650 align:middle line:90% below the reference density. 01:51:54.650 --> 01:51:58.970 align:middle line:84% And therefore, I suspect this ratio-- 01:51:58.970 --> 01:52:01.430 align:middle line:84% and therefore, this effect-- could exceed 1 very easily 01:52:01.430 --> 01:52:04.030 align:middle line:90% in this model. 01:52:04.030 --> 01:52:08.110 align:middle line:84% So that would be perhaps the primary issue. 01:52:08.110 --> 01:52:10.190 align:middle line:84% Now, there are different solutions. 01:52:10.190 --> 01:52:13.010 align:middle line:84% And we discussed different ways of formulating it. 01:52:13.010 --> 01:52:17.150 align:middle line:84% But ultimately, this effect should saturate at 1. 01:52:17.150 --> 01:52:20.450 align:middle line:84% And then you multiply it by normal consumption. 01:52:20.450 --> 01:52:22.690 align:middle line:84% And you can call that maximum consumption 01:52:22.690 --> 01:52:27.010 align:middle line:84% so that when you multiply it by maximum, you get-- 01:52:27.010 --> 01:52:29.130 align:middle line:84% by this effect, you get maximum at the end. 01:52:29.130 --> 01:52:32.150 align:middle line:90% You can't go above that. 01:52:32.150 --> 01:52:34.830 align:middle line:90% But the rest makes sense. 01:52:34.830 --> 01:52:39.710 align:middle line:84% And then you have consumption per predator. 01:52:39.710 --> 01:52:43.310 align:middle line:84% That creates a food scarcity effect 01:52:43.310 --> 01:52:46.150 align:middle line:84% compared to the normal consumption per predator. 01:52:46.150 --> 01:52:46.650 align:middle line:90% That's good. 01:52:46.650 --> 01:52:49.190 align:middle line:84% That's the same idea we discussed. 01:52:49.190 --> 01:52:52.730 align:middle line:84% And it is good to get it from this part of the formulation, 01:52:52.730 --> 01:52:57.450 align:middle line:84% rather than another parameter being defined for that. 01:52:57.450 --> 01:53:01.430 align:middle line:84% Then you have the effect of food on death rate of predator. 01:53:01.430 --> 01:53:06.750 align:middle line:90% So that effect is-- 01:53:06.750 --> 01:53:07.490 align:middle line:90% let's see. 01:53:07.490 --> 01:53:17.920 align:middle line:84% This is also-- so this is e to the power 01:53:17.920 --> 01:53:22.680 align:middle line:84% of some parameter multiplied by this food scarcity. 01:53:22.680 --> 01:53:25.160 align:middle line:84% And for now, let's think about the active range 01:53:25.160 --> 01:53:28.860 align:middle line:84% of food scarcity to be mostly between 0 and 1, 01:53:28.860 --> 01:53:34.360 align:middle line:84% even though it could go above 1 in our formulation. 01:53:34.360 --> 01:53:36.940 align:middle line:84% We can look at the equation for food scarcity. 01:53:36.940 --> 01:53:40.760 align:middle line:90% 01:53:40.760 --> 01:53:45.880 align:middle line:84% OK, so maximum of 0 and normal consumption 01:53:45.880 --> 01:53:51.920 align:middle line:84% minus actual consumption divided by normal consumption. 01:53:51.920 --> 01:53:58.600 align:middle line:90% So if normal minus-- 01:53:58.600 --> 01:53:59.500 align:middle line:90% OK, I see. 01:53:59.500 --> 01:54:03.760 align:middle line:84% So 1 here means you don't get anything to eat. 01:54:03.760 --> 01:54:06.960 align:middle line:84% You don't get any of what you want. 01:54:06.960 --> 01:54:10.320 align:middle line:84% STUDENT: Food scarcity of 1 means-- 01:54:10.320 --> 01:54:12.860 align:middle line:84% HAZHIR RAHMANDAD: Yes, food scarcity of 1 means famine. 01:54:12.860 --> 01:54:15.920 align:middle line:84% And because of the max 0, even if y consumption 01:54:15.920 --> 01:54:22.980 align:middle line:84% goes above desired or normal, it doesn't go below 0. 01:54:22.980 --> 01:54:26.520 align:middle line:90% So it is robust on that part. 01:54:26.520 --> 01:54:27.920 align:middle line:90% So this works fine. 01:54:27.920 --> 01:54:32.200 align:middle line:84% So now, with that understanding, let's go back to the equation 01:54:32.200 --> 01:54:35.420 align:middle line:84% for effective-- so this number is between 0 and 1. 01:54:35.420 --> 01:54:39.320 align:middle line:90% 1 means significant scarcity. 01:54:39.320 --> 01:54:41.920 align:middle line:84% So then that significant scarcity 01:54:41.920 --> 01:54:52.200 align:middle line:84% is multiplied by some multiplier, 01:54:52.200 --> 01:54:53.940 align:middle line:90% e to the power of that. 01:54:53.940 --> 01:54:57.680 align:middle line:90% 01:54:57.680 --> 01:55:00.580 align:middle line:84% That can lead to large death rates. 01:55:00.580 --> 01:55:04.040 align:middle line:90% 01:55:04.040 --> 01:55:06.560 align:middle line:84% And then 1 minus fractional death 01:55:06.560 --> 01:55:13.510 align:middle line:84% rate of r with ample food, it should not go-- 01:55:13.510 --> 01:55:19.410 align:middle line:90% 01:55:19.410 --> 01:55:23.370 align:middle line:84% So I have a unit problem here that I'm not sure-- 01:55:23.370 --> 01:55:27.153 align:middle line:84% STUDENT: The 1 should be the max fractional death rate. 01:55:27.153 --> 01:55:28.570 align:middle line:84% HAZHIR RAHMANDAD: Ah, it should be 01:55:28.570 --> 01:55:30.410 align:middle line:90% max fractional death rate, yes. 01:55:30.410 --> 01:55:32.690 align:middle line:84% STUDENT: Because we don't want it to go beyond 1. 01:55:32.690 --> 01:55:33.607 align:middle line:90% HAZHIR RAHMANDAD: Yes. 01:55:33.607 --> 01:55:36.410 align:middle line:84% So this is both a unit problem and a little bit 01:55:36.410 --> 01:55:39.050 align:middle line:90% of a conceptual issue because-- 01:55:39.050 --> 01:55:43.730 align:middle line:84% so fractional death rate can go above 1. 01:55:43.730 --> 01:55:48.490 align:middle line:84% That is, per unit of time, more than 100% can die. 01:55:48.490 --> 01:55:50.890 align:middle line:84% That is, if the unit of time is one year 01:55:50.890 --> 01:55:53.650 align:middle line:84% and you're going to die in two weeks, 01:55:53.650 --> 01:55:59.230 align:middle line:84% the fractional death rate could be 50 rather than 1. 01:55:59.230 --> 01:56:02.850 align:middle line:84% So you don't need to worry about 1 being a-- 01:56:02.850 --> 01:56:05.410 align:middle line:84% and therefore, you need to explicitly put 01:56:05.410 --> 01:56:09.410 align:middle line:84% the max fractional death rate because its unit is 1 01:56:09.410 --> 01:56:12.490 align:middle line:84% over year in this population because your, I think, 01:56:12.490 --> 01:56:14.610 align:middle line:90% time is year rather than month. 01:56:14.610 --> 01:56:16.870 align:middle line:84% So that's the source of confusion. 01:56:16.870 --> 01:56:19.970 align:middle line:84% You may come into this and you put 1 because you say, yes, 01:56:19.970 --> 01:56:22.130 align:middle line:90% death rate cannot go above 1. 01:56:22.130 --> 01:56:23.490 align:middle line:90% But it can, actually. 01:56:23.490 --> 01:56:26.390 align:middle line:84% If the unit is 1 over year, it is not a probability. 01:56:26.390 --> 01:56:28.570 align:middle line:90% It is per unit of time. 01:56:28.570 --> 01:56:32.210 align:middle line:84% But if we had max fractional death rate divided 01:56:32.210 --> 01:56:35.370 align:middle line:84% by fractional death rate here, that 01:56:35.370 --> 01:56:44.170 align:middle line:84% becomes OK because now you're saying we switch to the regime 01:56:44.170 --> 01:56:49.270 align:middle line:84% where you are killing the deer with the maximum rate possible. 01:56:49.270 --> 01:56:51.930 align:middle line:90% This is 1 over-- 01:56:51.930 --> 01:56:55.990 align:middle line:84% it's related to their starvation period. 01:56:55.990 --> 01:56:59.050 align:middle line:90% 01:56:59.050 --> 01:56:59.830 align:middle line:90% This works. 01:56:59.830 --> 01:57:02.850 align:middle line:84% But it is a very clunky formulation 01:57:02.850 --> 01:57:06.050 align:middle line:84% because this exponential leads to things 01:57:06.050 --> 01:57:09.890 align:middle line:84% going up, and then getting hit by this other ceiling 01:57:09.890 --> 01:57:10.990 align:middle line:90% that you are imposing. 01:57:10.990 --> 01:57:16.970 align:middle line:84% So it creates this weird non-linearity that I don't like. 01:57:16.970 --> 01:57:23.270 align:middle line:84% It's not doing what you really want to see in the real world. 01:57:23.270 --> 01:57:25.730 align:middle line:90% It doesn't happen that way. 01:57:25.730 --> 01:57:34.050 align:middle line:84% But it puts meaningful feedback from consumption of deer 01:57:34.050 --> 01:57:37.530 align:middle line:90% to the death of predators. 01:57:37.530 --> 01:57:39.970 align:middle line:90% OK, that was good to look at. 01:57:39.970 --> 01:57:44.672 align:middle line:84% And then the fractional death rate is straightforward. 01:57:44.672 --> 01:57:46.630 align:middle line:84% And then you have [? exogenous ?] [? deaths. ?] 01:57:46.630 --> 01:57:48.130 align:middle line:84% You have [? assumptions ?] for that. 01:57:48.130 --> 01:57:49.970 align:middle line:90% That's good. 01:57:49.970 --> 01:57:55.050 align:middle line:84% You have the natural death rate, I assume, fairly similar story 01:57:55.050 --> 01:57:57.550 align:middle line:90% down here for this. 01:57:57.550 --> 01:58:00.890 align:middle line:84% And then regeneration right now is exponential. 01:58:00.890 --> 01:58:05.930 align:middle line:90% OK, good. 01:58:05.930 --> 01:58:10.640 align:middle line:84% So maybe we have time to look, rather than at the model, 01:58:10.640 --> 01:58:12.600 align:middle line:90% look at some model analysis. 01:58:12.600 --> 01:58:16.460 align:middle line:84% Anybody wants to look at their model 01:58:16.460 --> 01:58:19.160 align:middle line:84% and how its behavior is explained? 01:58:19.160 --> 01:58:21.740 align:middle line:84% Anybody who has given it some thought? 01:58:21.740 --> 01:58:25.540 align:middle line:84% I saw a couple of you had submitted some simulation 01:58:25.540 --> 01:58:26.560 align:middle line:90% analysis as well. 01:58:26.560 --> 01:58:30.660 align:middle line:90% So Ross, you had some? 01:58:30.660 --> 01:58:33.060 align:middle line:84% STUDENT: I did, but my model was basically 01:58:33.060 --> 01:58:36.580 align:middle line:90% the mathematical model, so yeah. 01:58:36.580 --> 01:58:38.960 align:middle line:84% HAZHIR RAHMANDAD: OK, let's quickly look at your model. 01:58:38.960 --> 01:58:43.860 align:middle line:84% If it is too mathematical, I may not be able to-- 01:58:43.860 --> 01:58:47.560 align:middle line:84% too mathematical, in the sense that it is hiding the stories. 01:58:47.560 --> 01:58:49.080 align:middle line:90% No, this is Jake's. 01:58:49.080 --> 01:58:53.420 align:middle line:90% 01:58:53.420 --> 01:58:57.420 align:middle line:90% OK, this one is Ross. 01:58:57.420 --> 01:58:59.000 align:middle line:90% No, it is not too mathematical. 01:58:59.000 --> 01:59:02.140 align:middle line:84% It has prey and vegetation and so on. 01:59:02.140 --> 01:59:03.800 align:middle line:90% OK, let's run this. 01:59:03.800 --> 01:59:09.580 align:middle line:90% 01:59:09.580 --> 01:59:10.700 align:middle line:90% STUDENT: [INAUDIBLE] 01:59:10.700 --> 01:59:12.020 align:middle line:90% HAZHIR RAHMANDAD: OK, good. 01:59:12.020 --> 01:59:13.000 align:middle line:90% Good practice. 01:59:13.000 --> 01:59:14.840 align:middle line:84% Everything starts from equilibrium. 01:59:14.840 --> 01:59:16.100 align:middle line:90% Great job. 01:59:16.100 --> 01:59:18.297 align:middle line:84% Is it analytical equilibrium or is it-- 01:59:18.297 --> 01:59:19.380 align:middle line:90% STUDENT: It is analytical. 01:59:19.380 --> 01:59:21.260 align:middle line:90% HAZHIR RAHMANDAD: OK, perfect. 01:59:21.260 --> 01:59:24.393 align:middle line:90% STUDENT: [INAUDIBLE] 01:59:24.393 --> 01:59:26.060 align:middle line:84% HAZHIR RAHMANDAD: And what's the switch? 01:59:26.060 --> 01:59:29.260 align:middle line:84% STUDENT: Oh, it's for deforestation shock. 01:59:29.260 --> 01:59:33.540 align:middle line:84% I specified for four years that's going to happen. 01:59:33.540 --> 01:59:36.260 align:middle line:84% HAZHIR RAHMANDAD: Oh OK, so you have a shock for deforestation, 01:59:36.260 --> 01:59:37.120 align:middle line:90% for example? 01:59:37.120 --> 01:59:37.740 align:middle line:90% STUDENT: Yeah. 01:59:37.740 --> 01:59:38.740 align:middle line:90% HAZHIR RAHMANDAD: I see. 01:59:38.740 --> 01:59:40.260 align:middle line:84% So you can turn it on to see what 01:59:40.260 --> 01:59:45.860 align:middle line:90% happens if you take out some-- 01:59:45.860 --> 01:59:46.480 align:middle line:90% oh, OK. 01:59:46.480 --> 01:59:49.220 align:middle line:90% 01:59:49.220 --> 01:59:50.580 align:middle line:90% Huh. 01:59:50.580 --> 01:59:52.040 align:middle line:90% STUDENT: [INAUDIBLE] 01:59:52.040 --> 01:59:55.340 align:middle line:90% 01:59:55.340 --> 01:59:57.660 align:middle line:84% HAZHIR RAHMANDAD: So you are turning 01:59:57.660 --> 02:00:01.700 align:middle line:90% on this shock for how long? 02:00:01.700 --> 02:00:04.660 align:middle line:90% For 48 months. 02:00:04.660 --> 02:00:05.997 align:middle line:90% Is that correct? 02:00:05.997 --> 02:00:06.580 align:middle line:90% STUDENT: Yeah. 02:00:06.580 --> 02:00:08.740 align:middle line:84% HAZHIR RAHMANDAD: So let's bring it down. 02:00:08.740 --> 02:00:13.860 align:middle line:90% 02:00:13.860 --> 02:00:15.060 align:middle line:90% STUDENT: It's a step. 02:00:15.060 --> 02:00:17.720 align:middle line:84% HAZHIR RAHMANDAD: So there is a pulse. 02:00:17.720 --> 02:00:20.300 align:middle line:84% So you can see the pulse in the fractional death 02:00:20.300 --> 02:00:21.800 align:middle line:90% rate of the vegetation. 02:00:21.800 --> 02:00:24.860 align:middle line:90% Basically, more vegetation is-- 02:00:24.860 --> 02:00:27.300 align:middle line:84% and perhaps, this model-- yeah, this model 02:00:27.300 --> 02:00:31.920 align:middle line:90% has a built-in oscillatory mode. 02:00:31.920 --> 02:00:36.100 align:middle line:84% So even if you bring it to a very small pulse, 02:00:36.100 --> 02:00:38.200 align:middle line:84% it is still would create something. 02:00:38.200 --> 02:00:40.040 align:middle line:90% But it is interesting. 02:00:40.040 --> 02:00:44.020 align:middle line:84% So with the smaller magnitudes, it 02:00:44.020 --> 02:00:48.620 align:middle line:84% takes more time for the pulse to become stronger, yeah. 02:00:48.620 --> 02:00:53.480 align:middle line:84% But ultimately, I think it will get to a limit cycle. 02:00:53.480 --> 02:00:55.320 align:middle line:84% I mean, I don't know the formulation. 02:00:55.320 --> 02:00:57.540 align:middle line:84% We didn't go through your model's formulations. 02:00:57.540 --> 02:00:58.180 align:middle line:90% OK. 02:00:58.180 --> 02:01:00.540 align:middle line:90% But this is good. 02:01:00.540 --> 02:01:02.590 align:middle line:84% Now, we need to actually understand 02:01:02.590 --> 02:01:05.550 align:middle line:84% what is happening in a model like this. 02:01:05.550 --> 02:01:09.910 align:middle line:84% And you don't have the start time for this. 02:01:09.910 --> 02:01:15.603 align:middle line:84% So one challenge is, with an oscillatory mode of behavior, 02:01:15.603 --> 02:01:16.770 align:middle line:90% where do you get it started? 02:01:16.770 --> 02:01:20.370 align:middle line:84% Because things are moving too rapidly. 02:01:20.370 --> 02:01:24.710 align:middle line:84% Have you gone through trying to understand it? 02:01:24.710 --> 02:01:28.630 align:middle line:84% STUDENT: I guess I have rates underneath the graph. 02:01:28.630 --> 02:01:33.790 align:middle line:84% And I tried to look at more initial when the shock starts 02:01:33.790 --> 02:01:36.030 align:middle line:90% in the beginning. 02:01:36.030 --> 02:01:38.730 align:middle line:84% HAZHIR RAHMANDAD: When the shock starts, where do you have it? 02:01:38.730 --> 02:01:39.230 align:middle line:90% Oops. 02:01:39.230 --> 02:01:45.910 align:middle line:90% 02:01:45.910 --> 02:01:49.233 align:middle line:84% STUDENT: Looking at the first 0 to 20 maybe time horizon. 02:01:49.233 --> 02:01:50.150 align:middle line:90% HAZHIR RAHMANDAD: Yes. 02:01:50.150 --> 02:01:52.830 align:middle line:84% So one of the good things to do is 02:01:52.830 --> 02:01:54.710 align:middle line:84% to reduce the time horizon because you 02:01:54.710 --> 02:01:57.150 align:middle line:90% don't want to see everything. 02:01:57.150 --> 02:02:02.710 align:middle line:84% And you can do that in the time axis part in Vensim. 02:02:02.710 --> 02:02:06.270 align:middle line:84% In fact, you get the shock around time 10. 02:02:06.270 --> 02:02:09.310 align:middle line:84% So we don't really care about before that. 02:02:09.310 --> 02:02:15.870 align:middle line:84% And by time 20, 30, 30-ish, we have the core dynamics 02:02:15.870 --> 02:02:16.530 align:middle line:90% unfolding. 02:02:16.530 --> 02:02:18.790 align:middle line:84% So we don't need to go above that. 02:02:18.790 --> 02:02:22.850 align:middle line:90% So let's make those as our-- 02:02:22.850 --> 02:02:27.670 align:middle line:90% 02:02:27.670 --> 02:02:28.270 align:middle line:90% what happened? 02:02:28.270 --> 02:02:29.270 align:middle line:90% End time is 30? 02:02:29.270 --> 02:02:30.190 align:middle line:90% STUDENT: Yeah. 02:02:30.190 --> 02:02:31.990 align:middle line:90% HAZHIR RAHMANDAD: OK. 02:02:31.990 --> 02:02:41.230 align:middle line:84% Now, typically, Vensim would not update the graphs 02:02:41.230 --> 02:02:42.550 align:middle line:90% when we change this. 02:02:42.550 --> 02:02:45.470 align:middle line:84% But if we now look at the vegetation, 02:02:45.470 --> 02:02:49.470 align:middle line:84% it hopefully shows us, yes, between these two times. 02:02:49.470 --> 02:02:50.210 align:middle line:90% OK. 02:02:50.210 --> 02:02:52.790 align:middle line:84% So let's go through the causal tracing 02:02:52.790 --> 02:02:56.350 align:middle line:84% and see what is happening that creates these oscillations 02:02:56.350 --> 02:02:59.640 align:middle line:84% or at least come up with a hypothesis for it. 02:02:59.640 --> 02:03:03.020 align:middle line:90% This is an important process. 02:03:03.020 --> 02:03:06.160 align:middle line:84% So we don't know what is generating the oscillations. 02:03:06.160 --> 02:03:07.500 align:middle line:90% How do we figure that out? 02:03:07.500 --> 02:03:10.000 align:middle line:84% So start with whichever you want. 02:03:10.000 --> 02:03:15.760 align:middle line:84% In this case, vegetation would be a good starting place. 02:03:15.760 --> 02:03:20.120 align:middle line:90% The vegetation comes from what? 02:03:20.120 --> 02:03:24.720 align:middle line:84% You can look at the causal trace graph. 02:03:24.720 --> 02:03:27.480 align:middle line:84% It comes from restoration and destruction. 02:03:27.480 --> 02:03:30.140 align:middle line:84% Restoration depends on vegetation alone. 02:03:30.140 --> 02:03:32.900 align:middle line:84% And you know it is a reinforcing loop. 02:03:32.900 --> 02:03:35.380 align:middle line:84% It's not going to create oscillations for you. 02:03:35.380 --> 02:03:39.160 align:middle line:84% So you don't even need to go down the path of figuring out 02:03:39.160 --> 02:03:41.040 align:middle line:90% restoration. 02:03:41.040 --> 02:03:42.940 align:middle line:84% There is another signal to it as well. 02:03:42.940 --> 02:03:46.500 align:middle line:84% The destruction rate initially goes up. 02:03:46.500 --> 02:03:50.160 align:middle line:84% You see, this small jump up is what 02:03:50.160 --> 02:03:52.980 align:middle line:84% is triggering the initial decline in vegetation. 02:03:52.980 --> 02:03:56.535 align:middle line:84% So that is the driver of the behavior that gets 02:03:56.535 --> 02:03:57.660 align:middle line:90% us started with everything. 02:03:57.660 --> 02:04:01.600 align:middle line:84% So you would know I need to actually trace destruction rate 02:04:01.600 --> 02:04:04.160 align:middle line:90% and see where that comes from. 02:04:04.160 --> 02:04:06.800 align:middle line:84% Now, as I'm thinking about that, I'm 02:04:06.800 --> 02:04:10.220 align:middle line:84% worried about this time horizon because this is too messy. 02:04:10.220 --> 02:04:11.540 align:middle line:90% I can't easily see. 02:04:11.540 --> 02:04:15.720 align:middle line:84% So let me limit the time even further. 02:04:15.720 --> 02:04:22.720 align:middle line:84% So rather than 30, let's go to 15. 02:04:22.720 --> 02:04:29.320 align:middle line:84% And maybe-- yeah, let's keep 9 and see how things go. 02:04:29.320 --> 02:04:35.840 align:middle line:84% So now, we are going to look at the drivers of destruction. 02:04:35.840 --> 02:04:39.700 align:middle line:84% And now, you can see a clear oscillatory mode. 02:04:39.700 --> 02:04:43.280 align:middle line:90% 02:04:43.280 --> 02:04:45.300 align:middle line:90% So what is driving it? 02:04:45.300 --> 02:04:48.320 align:middle line:90% It is partly the vegetation. 02:04:48.320 --> 02:04:51.680 align:middle line:84% But that cannot be the main driver. 02:04:51.680 --> 02:04:55.680 align:middle line:84% It has to be prey population that 02:04:55.680 --> 02:05:00.720 align:middle line:84% is driving the oscillation because vegetation was coming 02:05:00.720 --> 02:05:04.000 align:middle line:90% from the destruction rate. 02:05:04.000 --> 02:05:09.560 align:middle line:84% So then we need to figure out why the prey population is 02:05:09.560 --> 02:05:10.660 align:middle line:90% going down. 02:05:10.660 --> 02:05:17.360 align:middle line:90% 02:05:17.360 --> 02:05:24.880 align:middle line:84% Then prey population is based on birth and death. 02:05:24.880 --> 02:05:28.720 align:middle line:90% So it's interesting. 02:05:28.720 --> 02:05:34.760 align:middle line:90% 02:05:34.760 --> 02:05:39.280 align:middle line:84% STUDENT: In this formulation, the birth rate [INAUDIBLE] 02:05:39.280 --> 02:05:41.580 align:middle line:84% HAZHIR RAHMANDAD: Yeah, so I haven't looked at that. 02:05:41.580 --> 02:05:44.960 align:middle line:84% But my first reaction was that the shock here-- 02:05:44.960 --> 02:05:47.760 align:middle line:84% the discontinuity here is very sharp. 02:05:47.760 --> 02:05:49.400 align:middle line:90% STUDENT: If the birth rate is-- 02:05:49.400 --> 02:05:52.730 align:middle line:90% 02:05:52.730 --> 02:05:54.970 align:middle line:84% HAZHIR RAHMANDAD: But before we look at it-- 02:05:54.970 --> 02:05:58.410 align:middle line:84% because normally, you wouldn't-- you shouldn't be needing 02:05:58.410 --> 02:06:01.090 align:middle line:84% to think about what the formulations were as much 02:06:01.090 --> 02:06:04.710 align:middle line:84% as looking at these and deciding which of these two drivers, 02:06:04.710 --> 02:06:08.890 align:middle line:84% birth or death, is more likely to explain the prey population 02:06:08.890 --> 02:06:10.130 align:middle line:90% dynamics. 02:06:10.130 --> 02:06:11.930 align:middle line:90% That's the key question. 02:06:11.930 --> 02:06:15.490 align:middle line:84% And the prey population is going down. 02:06:15.490 --> 02:06:21.050 align:middle line:84% But the question is, which of these two is driving it down? 02:06:21.050 --> 02:06:25.090 align:middle line:84% Is it because birth is going down or death is going down? 02:06:25.090 --> 02:06:25.985 align:middle line:90% What's the answer? 02:06:25.985 --> 02:06:26.610 align:middle line:90% STUDENT: Birth. 02:06:26.610 --> 02:06:29.490 align:middle line:84% HAZHIR RAHMANDAD: It should be birth because death going down 02:06:29.490 --> 02:06:34.450 align:middle line:84% doesn't lead to prey population going down. 02:06:34.450 --> 02:06:37.550 align:middle line:90% So death cannot be the driver. 02:06:37.550 --> 02:06:40.370 align:middle line:84% It should be the birth rate that is driving it. 02:06:40.370 --> 02:06:44.450 align:middle line:84% So now, we can click on that and see 02:06:44.450 --> 02:06:47.650 align:middle line:90% what are the drivers of that. 02:06:47.650 --> 02:06:48.850 align:middle line:90% OK. 02:06:48.850 --> 02:06:51.430 align:middle line:90% Prey birth rate is going down. 02:06:51.430 --> 02:06:54.130 align:middle line:84% It could be because of vegetation going down 02:06:54.130 --> 02:06:58.450 align:middle line:84% or could be because of the prey population going down. 02:06:58.450 --> 02:07:01.970 align:middle line:84% This is a little bit more tricky at this point 02:07:01.970 --> 02:07:05.910 align:middle line:84% because, yes, if the prey population goes down, 02:07:05.910 --> 02:07:07.770 align:middle line:90% birth rate will also go down. 02:07:07.770 --> 02:07:10.830 align:middle line:84% So how do I know which one is driving it? 02:07:10.830 --> 02:07:17.870 align:middle line:84% Let me look at the valley, The, minimum in the prey birth rate. 02:07:17.870 --> 02:07:21.970 align:middle line:84% That is where my cursor is now on all of the graphs. 02:07:21.970 --> 02:07:27.850 align:middle line:84% That valley, where we have the minimum birth rate, 02:07:27.850 --> 02:07:32.450 align:middle line:84% is happening after the minimization of vegetation, 02:07:32.450 --> 02:07:36.370 align:middle line:84% but happening before the minimization of the prey 02:07:36.370 --> 02:07:38.530 align:middle line:90% population. 02:07:38.530 --> 02:07:39.370 align:middle line:90% You see that? 02:07:39.370 --> 02:07:42.890 align:middle line:84% That's a very important signal for which of these two 02:07:42.890 --> 02:07:46.490 align:middle line:84% is more likely to be driving the behavior. 02:07:46.490 --> 02:07:47.610 align:middle line:90% Which of the two? 02:07:47.610 --> 02:07:49.010 align:middle line:90% STUDENT: Vegetation. 02:07:49.010 --> 02:07:50.530 align:middle line:90% HAZHIR RAHMANDAD: Vegetation. 02:07:50.530 --> 02:07:54.210 align:middle line:90% So that takes us where? 02:07:54.210 --> 02:07:56.850 align:middle line:90% Back to where we started. 02:07:56.850 --> 02:08:01.450 align:middle line:90% So what does that mean? 02:08:01.450 --> 02:08:04.450 align:middle line:90% What does it mean? 02:08:04.450 --> 02:08:05.390 align:middle line:90% What does it mean? 02:08:05.390 --> 02:08:09.530 align:middle line:90% 02:08:09.530 --> 02:08:14.070 align:middle line:84% It means we have a candidate hypothesis 02:08:14.070 --> 02:08:17.530 align:middle line:84% for what feedback loop is generating 02:08:17.530 --> 02:08:19.450 align:middle line:90% the oscillations here. 02:08:19.450 --> 02:08:24.570 align:middle line:84% The candidate hypothesis is that more vegetation-- 02:08:24.570 --> 02:08:29.010 align:middle line:84% or less vegetation leads to less prey population, 02:08:29.010 --> 02:08:35.150 align:middle line:84% which leads to more vegetation-- less vegetation consumption, 02:08:35.150 --> 02:08:38.150 align:middle line:84% which leads to more vegetation, which reduces this. 02:08:38.150 --> 02:08:41.650 align:middle line:84% So this is a balancing feedback loop. 02:08:41.650 --> 02:08:45.850 align:middle line:84% It has two stocks on its way-- vegetation and prey population-- 02:08:45.850 --> 02:08:49.520 align:middle line:84% which means it is a higher order balancing feedback loop. 02:08:49.520 --> 02:08:51.720 align:middle line:90% So it can generate oscillations. 02:08:51.720 --> 02:08:54.380 align:middle line:84% The first order balancing feedbacks cannot generate 02:08:54.380 --> 02:08:56.940 align:middle line:90% oscillations, but this one can. 02:08:56.940 --> 02:08:57.880 align:middle line:90% So this is good. 02:08:57.880 --> 02:09:00.120 align:middle line:84% You already have a good hypothesis. 02:09:00.120 --> 02:09:02.980 align:middle line:84% But remember, it is still a hypothesis. 02:09:02.980 --> 02:09:08.820 align:middle line:84% So then the question is, how can we disrupt this? 02:09:08.820 --> 02:09:14.500 align:middle line:84% Or how can we change this hypothesis's strength 02:09:14.500 --> 02:09:17.540 align:middle line:84% and see whether it changes the behavior? 02:09:17.540 --> 02:09:21.180 align:middle line:84% So any suggestions for how would you 02:09:21.180 --> 02:09:25.660 align:middle line:84% change the strength of this hypothesis? 02:09:25.660 --> 02:09:26.745 align:middle line:90% STUDENT: [INAUDIBLE] 02:09:26.745 --> 02:09:28.120 align:middle line:84% HAZHIR RAHMANDAD: Anne, go ahead. 02:09:28.120 --> 02:09:34.340 align:middle line:90% 02:09:34.340 --> 02:09:36.220 align:middle line:90% STUDENT: OK, cool. 02:09:36.220 --> 02:09:41.460 align:middle line:84% I think I missed the explanation for why death rate-- 02:09:41.460 --> 02:09:44.780 align:middle line:84% or rather, why birth rate and not death rate rates cannot 02:09:44.780 --> 02:09:46.125 align:middle line:90% affect prey population. 02:09:46.125 --> 02:09:48.500 align:middle line:84% HAZHIR RAHMANDAD: Sure, let's look at the prey population 02:09:48.500 --> 02:09:50.100 align:middle line:90% again. 02:09:50.100 --> 02:09:53.280 align:middle line:84% So the prey population, we looked at the causes. 02:09:53.280 --> 02:09:55.320 align:middle line:90% It can be either birth or death. 02:09:55.320 --> 02:09:56.820 align:middle line:84% What is it we are trying to explain? 02:09:56.820 --> 02:09:59.980 align:middle line:84% The fact that prey population starts to go down. 02:09:59.980 --> 02:10:03.220 align:middle line:84% So the fact that it starts to go down 02:10:03.220 --> 02:10:07.320 align:middle line:84% means that its birth should have been going down 02:10:07.320 --> 02:10:09.660 align:middle line:84% or its death should have been going up. 02:10:09.660 --> 02:10:11.500 align:middle line:90% The death is not going up. 02:10:11.500 --> 02:10:14.180 align:middle line:84% So that means the only plausible explanation 02:10:14.180 --> 02:10:17.100 align:middle line:84% is that the birth was going down, which is the case. 02:10:17.100 --> 02:10:21.320 align:middle line:84% You could actually use the valley thing as well here. 02:10:21.320 --> 02:10:25.200 align:middle line:84% That is, if you go to the valley of the prey population, 02:10:25.200 --> 02:10:28.420 align:middle line:84% you see it happens after the prey birth rate, 02:10:28.420 --> 02:10:30.560 align:middle line:84% the valley value of the prey birth rate. 02:10:30.560 --> 02:10:35.080 align:middle line:84% So that is the driver of what is happening to the population. 02:10:35.080 --> 02:10:38.560 align:middle line:84% Whereas, for the death, It is happening at the same time, 02:10:38.560 --> 02:10:40.640 align:middle line:84% which is coming from the stock directly. 02:10:40.640 --> 02:10:41.500 align:middle line:90% OK. 02:10:41.500 --> 02:10:43.510 align:middle line:90% Is that clear, Anne? 02:10:43.510 --> 02:10:44.690 align:middle line:90% STUDENT: Yes, thank you. 02:10:44.690 --> 02:10:45.648 align:middle line:90% HAZHIR RAHMANDAD: Sure. 02:10:45.648 --> 02:10:49.090 align:middle line:84% OK, so now, we need a test for our hypothesis. 02:10:49.090 --> 02:10:58.310 align:middle line:84% Our hypothesis was that the prey birthrate, prey population, 02:10:58.310 --> 02:11:03.070 align:middle line:84% degeneration, vegetation is the cause of the oscillations. 02:11:03.070 --> 02:11:06.630 align:middle line:84% So if that is the case, if I disrupt this feedback 02:11:06.630 --> 02:11:11.390 align:middle line:84% loop at some point in that period we are looking at, 02:11:11.390 --> 02:11:14.310 align:middle line:84% it should disrupt the oscillations. 02:11:14.310 --> 02:11:17.190 align:middle line:84% Now, I need to disrupt that feedback loop without disrupting 02:11:17.190 --> 02:11:18.430 align:middle line:90% anything else. 02:11:18.430 --> 02:11:21.330 align:middle line:90% So that part is not trivial. 02:11:21.330 --> 02:11:24.230 align:middle line:90% 02:11:24.230 --> 02:11:27.750 align:middle line:84% One way might be to stop the prey birth 02:11:27.750 --> 02:11:29.830 align:middle line:90% rate from oscillating. 02:11:29.830 --> 02:11:38.110 align:middle line:84% If I can rewrite the prey birth rate to not oscillate, 02:11:38.110 --> 02:11:41.230 align:middle line:84% to switch to a fixed number, then I 02:11:41.230 --> 02:11:44.430 align:middle line:84% have disrupted the oscillatory mode. 02:11:44.430 --> 02:11:45.730 align:middle line:90% That's one way to do it. 02:11:45.730 --> 02:11:48.830 align:middle line:84% Now, we could write equations to get that going. 02:11:48.830 --> 02:11:53.770 align:middle line:84% Or we could see what was the initial value for this variable. 02:11:53.770 --> 02:11:56.190 align:middle line:90% So it was around 15. 02:11:56.190 --> 02:11:59.250 align:middle line:90% So I'm going to say to do this. 02:11:59.250 --> 02:12:00.530 align:middle line:90% So I right click on it. 02:12:00.530 --> 02:12:03.470 align:middle line:90% I override the default behavior. 02:12:03.470 --> 02:12:10.630 align:middle line:90% And I would say at time-- 02:12:10.630 --> 02:12:20.530 align:middle line:84% well, if I make it a constant, and the value should be 15-- 02:12:20.530 --> 02:12:23.430 align:middle line:90% 02:12:23.430 --> 02:12:27.630 align:middle line:84% let's see if it is starting at 15. 02:12:27.630 --> 02:12:29.590 align:middle line:90% Yeah, m is at 15. 02:12:29.590 --> 02:12:32.190 align:middle line:90% So what are you seeing? 02:12:32.190 --> 02:12:34.630 align:middle line:84% All of the oscillations went away. 02:12:34.630 --> 02:12:38.110 align:middle line:84% And the vegetation is now going all the way to 0 02:12:38.110 --> 02:12:43.870 align:middle line:84% because we brought it below its equilibrium level. 02:12:43.870 --> 02:12:47.750 align:middle line:84% And then it cannot restore itself. 02:12:47.750 --> 02:12:53.070 align:middle line:84% That's partly a function of how these are formulated here. 02:12:53.070 --> 02:12:56.310 align:middle line:84% I haven't even looked at the equations. 02:12:56.310 --> 02:12:59.070 align:middle line:84% But now, we have a good hypothesis 02:12:59.070 --> 02:13:02.390 align:middle line:84% for what feedback loop is driving that oscillatory mode. 02:13:02.390 --> 02:13:05.530 align:middle line:84% This was the practice I wanted you to go through 02:13:05.530 --> 02:13:09.710 align:middle line:84% because this is how you identify potentially 02:13:09.710 --> 02:13:14.590 align:middle line:84% important structures that are driving 02:13:14.590 --> 02:13:16.830 align:middle line:90% the behavior you are seeing. 02:13:16.830 --> 02:13:17.750 align:middle line:90% Alexander? 02:13:17.750 --> 02:13:20.930 align:middle line:84% STUDENT: Instead of changing the value of the birth rate itself, 02:13:20.930 --> 02:13:24.230 align:middle line:84% it seems like there's that parameter that's 02:13:24.230 --> 02:13:26.750 align:middle line:84% next to it that controls the relationship between vegetation 02:13:26.750 --> 02:13:27.750 align:middle line:90% and birth rate. 02:13:27.750 --> 02:13:29.830 align:middle line:84% Could you have set that parameter to 0 02:13:29.830 --> 02:13:30.730 align:middle line:90% or whatever it is? 02:13:30.730 --> 02:13:32.772 align:middle line:84% HAZHIR RAHMANDAD: The problem with that is-- yes. 02:13:32.772 --> 02:13:34.870 align:middle line:84% But that parameter perhaps is not 02:13:34.870 --> 02:13:37.860 align:middle line:84% doing what we wanted it to do because that parameter-- 02:13:37.860 --> 02:13:41.460 align:middle line:84% my most likely guess is that it is multiplied by vegetation. 02:13:41.460 --> 02:13:47.880 align:middle line:84% So if I bring this one to 0, then my prey birth rate 02:13:47.880 --> 02:13:50.240 align:middle line:90% is going to vanish. 02:13:50.240 --> 02:13:55.000 align:middle line:84% And then I'm going to change the prey beyond its equilibrium 02:13:55.000 --> 02:13:55.800 align:middle line:90% change. 02:13:55.800 --> 02:13:58.640 align:middle line:84% So this way, at least, I'm looking at its value 02:13:58.640 --> 02:14:01.440 align:middle line:84% being at its equilibrium value, which was 15, 02:14:01.440 --> 02:14:04.480 align:middle line:84% and seeing how that intervention impacts it. 02:14:04.480 --> 02:14:10.480 align:middle line:84% If I basically kill all the prey or don't let them be born, 02:14:10.480 --> 02:14:13.840 align:middle line:84% there are more-- it is a more disruptive intervention 02:14:13.840 --> 02:14:14.820 align:middle line:90% into the dynamics. 02:14:14.820 --> 02:14:18.600 align:middle line:84% It will achieve the same thing, in the sense of disrupting 02:14:18.600 --> 02:14:21.140 align:middle line:84% that feedback loop, but it disrupts a lot of other things. 02:14:21.140 --> 02:14:23.760 align:middle line:84% Even the disruption I introduced is too much. 02:14:23.760 --> 02:14:30.080 align:middle line:84% So the best way to intervene is to keep this as it was. 02:14:30.080 --> 02:14:31.100 align:middle line:90% We could even do that. 02:14:31.100 --> 02:14:35.000 align:middle line:84% But basically, put an if/then/else equation here 02:14:35.000 --> 02:14:41.440 align:middle line:84% that before time, let's say, 12 it is whatever is the equation. 02:14:41.440 --> 02:14:44.860 align:middle line:84% At time 12, it switches to 15, to the value 02:14:44.860 --> 02:14:46.720 align:middle line:90% that was the equilibrium. 02:14:46.720 --> 02:14:49.960 align:middle line:84% And then you see whether the oscillations stop 02:14:49.960 --> 02:14:52.860 align:middle line:90% at that time 15-- at time 12. 02:14:52.860 --> 02:14:55.793 align:middle line:84% We could do that just so that we-- 02:14:55.793 --> 02:14:57.460 align:middle line:84% I mean, now a lot of things will change. 02:14:57.460 --> 02:14:59.580 align:middle line:84% So I'm not sure if we'll get what we need. 02:14:59.580 --> 02:15:10.680 align:middle line:84% But this would be if/then/else time is smaller than 12, 02:15:10.680 --> 02:15:13.240 align:middle line:90% then use this formulation. 02:15:13.240 --> 02:15:15.560 align:middle line:90% Otherwise, be 15. 02:15:15.560 --> 02:15:18.440 align:middle line:90% Now, this is all bad. 02:15:18.440 --> 02:15:20.740 align:middle line:84% I'm putting parameters into equations and so on. 02:15:20.740 --> 02:15:21.980 align:middle line:90% Don't follow this practice. 02:15:21.980 --> 02:15:27.400 align:middle line:84% I'm just showing a simple piece here. 02:15:27.400 --> 02:15:30.960 align:middle line:90% And then we come here. 02:15:30.960 --> 02:15:33.160 align:middle line:90% We go there. 02:15:33.160 --> 02:15:38.660 align:middle line:90% We just give a shock. 02:15:38.660 --> 02:15:43.320 align:middle line:90% 02:15:43.320 --> 02:15:49.080 align:middle line:90% And we make the-- 02:15:49.080 --> 02:15:54.980 align:middle line:84% yeah, we don't need all of this long and laborious shock. 02:15:54.980 --> 02:15:58.880 align:middle line:90% 02:15:58.880 --> 02:16:02.500 align:middle line:84% OK, so everything is happening as we discussed, 02:16:02.500 --> 02:16:03.980 align:middle line:90% except that this might be-- 02:16:03.980 --> 02:16:13.760 align:middle line:90% 02:16:13.760 --> 02:16:15.660 align:middle line:90% prey birth rate never goes down. 02:16:15.660 --> 02:16:17.320 align:middle line:90% So why is that happening? 02:16:17.320 --> 02:16:18.260 align:middle line:90% Anybody knows? 02:16:18.260 --> 02:16:20.120 align:middle line:90% STUDENT: When is your-- 02:16:20.120 --> 02:16:23.100 align:middle line:84% HAZHIR RAHMANDAD: Oh, because at time 12, we have the shock. 02:16:23.100 --> 02:16:24.940 align:middle line:90% And I switched this at time 12. 02:16:24.940 --> 02:16:28.540 align:middle line:90% So exactly, this was not-- 02:16:28.540 --> 02:16:33.450 align:middle line:90% 02:16:33.450 --> 02:16:37.330 align:middle line:84% if I do it as 14, then we start to see the oscillation 02:16:37.330 --> 02:16:40.450 align:middle line:90% before this goes. 02:16:40.450 --> 02:16:41.990 align:middle line:90% So that's my guess. 02:16:41.990 --> 02:16:46.490 align:middle line:90% 02:16:46.490 --> 02:16:48.010 align:middle line:90% OK. 02:16:48.010 --> 02:16:49.170 align:middle line:90% Yes. 02:16:49.170 --> 02:16:52.650 align:middle line:84% Now, you can start to see the oscillations start to happen. 02:16:52.650 --> 02:16:55.010 align:middle line:84% But then as soon as we turn them off-- 02:16:55.010 --> 02:16:58.230 align:middle line:84% turn the switch off, at this point in time, 02:16:58.230 --> 02:16:59.969 align:middle line:84% there is no more oscillation going. 02:16:59.969 --> 02:17:03.930 align:middle line:84% So this tells you that this was a good hypothesis for explaining 02:17:03.930 --> 02:17:05.795 align:middle line:90% the oscillations. 02:17:05.795 --> 02:17:07.170 align:middle line:84% STUDENT: Why do you need the step 02:17:07.170 --> 02:17:10.450 align:middle line:84% as opposed to just changing the parameter value 02:17:10.450 --> 02:17:12.090 align:middle line:90% as you're saying? 02:17:12.090 --> 02:17:14.375 align:middle line:84% Because I was thinking that you could just-- 02:17:14.375 --> 02:17:16.250 align:middle line:84% when you were doing it earlier, when you just 02:17:16.250 --> 02:17:19.090 align:middle line:84% changed the parameter value for birth rate to always be 15, 02:17:19.090 --> 02:17:21.513 align:middle line:84% you can just compare the two runs 02:17:21.513 --> 02:17:23.430 align:middle line:84% because that's the only thing you're changing. 02:17:23.430 --> 02:17:27.330 align:middle line:84% Why was that too much of a disruption? 02:17:27.330 --> 02:17:32.010 align:middle line:84% HAZHIR RAHMANDAD: So I like to see that the oscillations are 02:17:32.010 --> 02:17:37.570 align:middle line:84% starting, and then disrupt it and see that the disruption 02:17:37.570 --> 02:17:38.950 align:middle line:90% happens instantaneously. 02:17:38.950 --> 02:17:41.870 align:middle line:84% As soon as I introduce this, oscillations go away. 02:17:41.870 --> 02:17:44.490 align:middle line:84% That's a very strong signal for me. 02:17:44.490 --> 02:17:47.290 align:middle line:84% The other way, I would never see this start. 02:17:47.290 --> 02:17:49.450 align:middle line:84% So there could be other things that are happening. 02:17:49.450 --> 02:17:51.049 align:middle line:84% Now, in this case, it's easy enough 02:17:51.049 --> 02:17:53.030 align:middle line:84% that I'm comfortable going with that, 02:17:53.030 --> 02:17:58.570 align:middle line:84% but it may not always work as well. 02:17:58.570 --> 02:17:59.290 align:middle line:90% OK. 02:17:59.290 --> 02:18:01.650 align:middle line:90% I think this is good. 02:18:01.650 --> 02:18:07.950 align:middle line:84% And basically, you folks had the core structures in place. 02:18:07.950 --> 02:18:10.870 align:middle line:84% The most common problems-- so the two most common problems-- 02:18:10.870 --> 02:18:16.010 align:middle line:84% first was these outflows from the prey or from the vegetation 02:18:16.010 --> 02:18:19.110 align:middle line:90% not being constrained correctly. 02:18:19.110 --> 02:18:23.690 align:middle line:84% That is, you let things go out too much, even if you don't have 02:18:23.690 --> 02:18:27.020 align:middle line:84% predators, for example, with the multiplicative formulations that 02:18:27.020 --> 02:18:28.500 align:middle line:90% were very common. 02:18:28.500 --> 02:18:34.020 align:middle line:84% The second issue was the death rates 02:18:34.020 --> 02:18:36.120 align:middle line:90% that are due to starvation. 02:18:36.120 --> 02:18:38.200 align:middle line:84% We wanted robust formulations for those. 02:18:38.200 --> 02:18:42.620 align:middle line:84% And some of them were not very well formulated. 02:18:42.620 --> 02:18:45.500 align:middle line:84% And the third thing is to make sure they are not 02:18:45.500 --> 02:18:48.440 align:middle line:84% getting their signals from different parts of the model. 02:18:48.440 --> 02:18:53.139 align:middle line:84% That is, you don't create new parameters that are overlapping. 02:18:53.139 --> 02:18:57.100 align:middle line:84% We also learned these constrained formulations 02:18:57.100 --> 02:18:59.420 align:middle line:84% for removing things out of stocks that 02:18:59.420 --> 02:19:01.340 align:middle line:90% does not overwhelm the stock. 02:19:01.340 --> 02:19:03.320 align:middle line:84% This is something you have seen before. 02:19:03.320 --> 02:19:04.840 align:middle line:90% It is in the molecules as well. 02:19:04.840 --> 02:19:08.000 align:middle line:84% If you look at the molecules, you will see that as well. 02:19:08.000 --> 02:19:12.299 align:middle line:90% 02:19:12.299 --> 02:19:12.840 align:middle line:90% OK. 02:19:12.840 --> 02:19:15.379 align:middle line:90% 02:19:15.379 --> 02:19:18.559 align:middle line:90% Yeah, so we talked about these. 02:19:18.559 --> 02:19:22.340 align:middle line:90% 02:19:22.340 --> 02:19:27.559 align:middle line:84% Yeah, I think that's good for the predator prey. 02:19:27.559 --> 02:19:32.260 align:middle line:84% Let's spend the last few minutes on the next week's assignment. 02:19:32.260 --> 02:19:36.180 align:middle line:84% So next week, you are working on a model of drug development 02:19:36.180 --> 02:19:37.740 align:middle line:90% pipeline. 02:19:37.740 --> 02:19:42.540 align:middle line:84% So drug development has multiple stages. 02:19:42.540 --> 02:19:45.680 align:middle line:84% You have coming up with compounds. 02:19:45.680 --> 02:19:49.840 align:middle line:84% Then you have animal trials and laboratory tests. 02:19:49.840 --> 02:19:51.660 align:middle line:84% Then you get into phase I trials. 02:19:51.660 --> 02:19:55.580 align:middle line:84% And then you get to phase II trials and phase III trials. 02:19:55.580 --> 02:19:58.660 align:middle line:84% And then you launch it into the marketplace. 02:19:58.660 --> 02:20:01.540 align:middle line:84% So if you want to make some amount of money, 02:20:01.540 --> 02:20:04.740 align:middle line:84% you have to manage this pipeline from a decade 02:20:04.740 --> 02:20:08.520 align:middle line:84% or more before to have the revenue today. 02:20:08.520 --> 02:20:10.500 align:middle line:90% This is also very stochastic. 02:20:10.500 --> 02:20:14.040 align:middle line:84% For our purposes, don't worry about the stochasticity. 02:20:14.040 --> 02:20:17.840 align:middle line:84% That's too much to capture in your model. 02:20:17.840 --> 02:20:22.060 align:middle line:84% Focus on a deterministic version that basically says each stage 02:20:22.060 --> 02:20:23.620 align:middle line:90% has a failure rate. 02:20:23.620 --> 02:20:27.220 align:middle line:84% These failure rates are actually pretty high, typically. 02:20:27.220 --> 02:20:31.260 align:middle line:84% So across the three phases of trials, 02:20:31.260 --> 02:20:35.940 align:middle line:84% total failure rate is in the order of 90% or more. 02:20:35.940 --> 02:20:39.580 align:middle line:90% But it is what it is. 02:20:39.580 --> 02:20:44.540 align:middle line:84% So with that, your goal is to formulate the start 02:20:44.540 --> 02:20:50.660 align:middle line:84% rate for phase I for entering into phase I. So think 02:20:50.660 --> 02:20:53.740 align:middle line:84% of those previous steps as easy to manage. 02:20:53.740 --> 02:20:58.520 align:middle line:84% So the company can decide to have 100 new drugs per year 02:20:58.520 --> 02:21:01.860 align:middle line:84% this year started into phase I or five 02:21:01.860 --> 02:21:04.220 align:middle line:84% or whatever is the number, so that they 02:21:04.220 --> 02:21:09.740 align:middle line:84% can get the revenue that they want to have a few years down 02:21:09.740 --> 02:21:10.700 align:middle line:90% the line. 02:21:10.700 --> 02:21:15.300 align:middle line:84% So basically, they need to have some revenue projection. 02:21:15.300 --> 02:21:18.660 align:middle line:84% And that revenue projection, think of it 02:21:18.660 --> 02:21:20.960 align:middle line:90% as a exponential curve. 02:21:20.960 --> 02:21:23.750 align:middle line:84% That is, we want our current revenue, 02:21:23.750 --> 02:21:28.430 align:middle line:84% but growing at a 5% rate per year. 02:21:28.430 --> 02:21:31.130 align:middle line:84% We want our revenue to be growing 5% per year. 02:21:31.130 --> 02:21:42.910 align:middle line:84% How do we start the initial entry initial start of the phase 02:21:42.910 --> 02:21:46.190 align:middle line:84% I so that we get the desired revenue 02:21:46.190 --> 02:21:48.470 align:middle line:90% trajectory in the long term? 02:21:48.470 --> 02:21:50.530 align:middle line:84% And this is a stock management problem. 02:21:50.530 --> 02:21:53.950 align:middle line:84% That is, if you have too many items also in the pipeline, 02:21:53.950 --> 02:21:57.550 align:middle line:84% you don't want to get started too much. 02:21:57.550 --> 02:21:59.930 align:middle line:84% And it should be robust to any percentage. 02:21:59.930 --> 02:22:05.130 align:middle line:84% So if rather than 5% you go to 0%, it's a steady state. 02:22:05.130 --> 02:22:06.070 align:middle line:90% That should work fine. 02:22:06.070 --> 02:22:10.110 align:middle line:84% Even if you have a declining rate-- that is, 02:22:10.110 --> 02:22:12.670 align:middle line:84% I want to actually shrink my operation and close 02:22:12.670 --> 02:22:15.450 align:middle line:84% shop in 10 years or 20 years or whatever-- 02:22:15.450 --> 02:22:17.030 align:middle line:90% that should also work. 02:22:17.030 --> 02:22:22.932 align:middle line:84% So that's your marching order for this week. 02:22:22.932 --> 02:22:25.390 align:middle line:84% STUDENT: Resource constraints, should we think about those? 02:22:25.390 --> 02:22:29.450 align:middle line:84% HAZHIR RAHMANDAD: So no, unless you're really pushing yourself. 02:22:29.450 --> 02:22:31.950 align:middle line:84% So right now, no resource constraints 02:22:31.950 --> 02:22:35.470 align:middle line:90% on the start of the projects. 02:22:35.470 --> 02:22:39.550 align:middle line:84% In reality, there is actually a very serious optimization 02:22:39.550 --> 02:22:42.030 align:middle line:84% problem because, as you increase the number of drugs 02:22:42.030 --> 02:22:44.390 align:middle line:84% that you start, the likelihood of success 02:22:44.390 --> 02:22:49.190 align:middle line:84% actually comes down because you are pushing in lower probability 02:22:49.190 --> 02:22:50.110 align:middle line:90% items. 02:22:50.110 --> 02:22:54.550 align:middle line:84% So a good model would actually capture that. 02:22:54.550 --> 02:22:56.390 align:middle line:84% And then you have to decide how much 02:22:56.390 --> 02:22:59.750 align:middle line:84% cost there is versus how much expected 02:22:59.750 --> 02:23:04.910 align:middle line:84% benefit given this reduction in the likelihood of success. 02:23:04.910 --> 02:23:07.310 align:middle line:90% Hassan? 02:23:07.310 --> 02:23:09.110 align:middle line:90% Go ahead, Anne. 02:23:09.110 --> 02:23:11.950 align:middle line:84% STUDENT: So how does the company get the revenue 02:23:11.950 --> 02:23:14.470 align:middle line:84% at the end of the number of drugs 02:23:14.470 --> 02:23:16.533 align:middle line:90% that make it to the last stage? 02:23:16.533 --> 02:23:17.450 align:middle line:90% HAZHIR RAHMANDAD: Yes. 02:23:17.450 --> 02:23:19.290 align:middle line:90% Then they go into the market. 02:23:19.290 --> 02:23:22.610 align:middle line:84% They stay on the market for a while. 02:23:22.610 --> 02:23:24.110 align:middle line:84% And then they come out of the market 02:23:24.110 --> 02:23:27.470 align:middle line:84% because, basically-- they don't come out of the market. 02:23:27.470 --> 02:23:29.210 align:middle line:90% Their patents expire. 02:23:29.210 --> 02:23:30.470 align:middle line:90% Generics come in. 02:23:30.470 --> 02:23:32.690 align:middle line:90% You stop making money from them. 02:23:32.690 --> 02:23:35.410 align:middle line:84% Now, you don't fully stop making money out of them. 02:23:35.410 --> 02:23:37.530 align:middle line:84% But for first order approximation, 02:23:37.530 --> 02:23:40.750 align:middle line:84% let's assume, once the patents expire, 02:23:40.750 --> 02:23:43.450 align:middle line:84% you are not going to make any more money on that drug. 02:23:43.450 --> 02:23:47.296 align:middle line:84% Now, reality is a little bit more complex. 02:23:47.296 --> 02:23:49.390 align:middle line:84% STUDENT: [INAUDIBLE] the clinical trials 02:23:49.390 --> 02:23:55.350 align:middle line:84% system has threshold dynamics embedded to the last stage of 02:23:55.350 --> 02:23:57.335 align:middle line:90% [INAUDIBLE] 02:23:57.335 --> 02:23:58.210 align:middle line:90% HAZHIR RAHMANDAD: No. 02:23:58.210 --> 02:24:00.150 align:middle line:84% So the threshold dynamics in the sense 02:24:00.150 --> 02:24:02.790 align:middle line:84% that the availability of other drugs on the market 02:24:02.790 --> 02:24:05.610 align:middle line:84% impacts the acceptance rate or what? 02:24:05.610 --> 02:24:08.210 align:middle line:84% STUDENT: Yeah, so for it to get FDA approved, 02:24:08.210 --> 02:24:10.710 align:middle line:84% it has to outperform whatever's on the market in some sense. 02:24:10.710 --> 02:24:12.877 align:middle line:84% HAZHIR RAHMANDAD: Yeah, exactly, so the availability 02:24:12.877 --> 02:24:13.510 align:middle line:90% of other drugs. 02:24:13.510 --> 02:24:14.810 align:middle line:90% No, don't worry about that. 02:24:14.810 --> 02:24:17.880 align:middle line:84% That's too much to do in this model. 02:24:17.880 --> 02:24:22.060 align:middle line:84% So this is really the stock management, 02:24:22.060 --> 02:24:27.160 align:middle line:84% but with failure rate and with growing demand 02:24:27.160 --> 02:24:32.760 align:middle line:84% and how you incorporate those into your modeling. 02:24:32.760 --> 02:24:35.720 align:middle line:84% So as such, it is not very different from your stock 02:24:35.720 --> 02:24:38.580 align:middle line:90% management in assignment 6. 02:24:38.580 --> 02:24:42.680 align:middle line:84% But it has these additional pieces. 02:24:42.680 --> 02:24:48.760 align:middle line:84% So in light of the time, we have, let's just think about-- 02:24:48.760 --> 02:24:52.120 align:middle line:84% basically, you should start from what you already 02:24:52.120 --> 02:24:57.760 align:middle line:84% know for stock management, but then incorporate failure rates. 02:24:57.760 --> 02:25:00.040 align:middle line:84% Make sure you are incorporating adjustments 02:25:00.040 --> 02:25:01.340 align:middle line:90% for different stocks. 02:25:01.340 --> 02:25:03.880 align:middle line:84% So there are desired levels for each of these stocks 02:25:03.880 --> 02:25:05.740 align:middle line:90% in the work in progress. 02:25:05.740 --> 02:25:08.020 align:middle line:84% That is, you have phase I, phase II, phase III. 02:25:08.020 --> 02:25:12.280 align:middle line:84% These are your work in progress equivalents 02:25:12.280 --> 02:25:18.960 align:middle line:84% for the inventory management model that you need to adjust. 02:25:18.960 --> 02:25:23.880 align:middle line:84% And then when you get the full model in place, 02:25:23.880 --> 02:25:26.200 align:middle line:84% you want it to be able to match your desired 02:25:26.200 --> 02:25:29.480 align:middle line:90% revenue over the longer term. 02:25:29.480 --> 02:25:34.960 align:middle line:84% You want it to also get there relatively quickly. 02:25:34.960 --> 02:25:37.320 align:middle line:84% So if it is starting from a point 02:25:37.320 --> 02:25:41.980 align:middle line:84% that it is not matching the desired trend, 02:25:41.980 --> 02:25:46.400 align:middle line:84% it can quickly move towards that desired trend 02:25:46.400 --> 02:25:53.480 align:middle line:84% and not overshoot or oscillate in a weird way 02:25:53.480 --> 02:25:55.820 align:middle line:84% when you're exposing it to a step function, 02:25:55.820 --> 02:26:00.640 align:middle line:84% a step increase in desired revenue, for example. 02:26:00.640 --> 02:26:07.000 align:middle line:84% So those are the key consideration sets 02:26:07.000 --> 02:26:09.640 align:middle line:90% for a good model. 02:26:09.640 --> 02:26:10.360 align:middle line:90% Questions? 02:26:10.360 --> 02:26:13.370 align:middle line:90% 02:26:13.370 --> 02:26:15.567 align:middle line:84% STUDENT: Failure rates can happen at any stage? 02:26:15.567 --> 02:26:17.650 align:middle line:84% HAZHIR RAHMANDAD: Yes, failure rates at any stage. 02:26:17.650 --> 02:26:20.930 align:middle line:84% Put a parameter for the fraction that get failed. 02:26:20.930 --> 02:26:22.350 align:middle line:90% Use first order delays. 02:26:22.350 --> 02:26:26.410 align:middle line:84% Don't worry about the fact that these clinical trials take time 02:26:26.410 --> 02:26:27.870 align:middle line:90% and are not really first order. 02:26:27.870 --> 02:26:30.830 align:middle line:84% It is OK for our purposes to go with first order delays. 02:26:30.830 --> 02:26:32.010 align:middle line:90% STUDENT: [INAUDIBLE] 02:26:32.010 --> 02:26:35.910 align:middle line:90% 02:26:35.910 --> 02:26:37.790 align:middle line:84% HAZHIR RAHMANDAD: No, they also expire. 02:26:37.790 --> 02:26:39.478 align:middle line:90% So there is a life cycle, yeah. 02:26:39.478 --> 02:26:41.770 align:middle line:84% STUDENT: But they don't have a failure rate [INAUDIBLE] 02:26:41.770 --> 02:26:43.190 align:middle line:84% HAZHIR RAHMANDAD: No, they don't have a separate failure. 02:26:43.190 --> 02:26:45.690 align:middle line:84% So for now, assume each drug on the market 02:26:45.690 --> 02:26:50.010 align:middle line:84% creates some amount of revenue per year. 02:26:50.010 --> 02:26:53.850 align:middle line:90% And that's all you care about. 02:26:53.850 --> 02:26:55.570 align:middle line:90% Jason? 02:26:55.570 --> 02:26:57.570 align:middle line:84% STUDENT: Should we assume constant failure rates 02:26:57.570 --> 02:26:59.332 align:middle line:90% [INAUDIBLE] 02:26:59.332 --> 02:27:00.790 align:middle line:84% HAZHIR RAHMANDAD: Constant for now. 02:27:00.790 --> 02:27:04.810 align:middle line:90% So don't worry about that. 02:27:04.810 --> 02:27:06.690 align:middle line:90% Good? 02:27:06.690 --> 02:27:08.240 align:middle line:90% OK. 02:27:08.240 --> 02:27:11.000 align:middle line:90%