WEBVTT 00:00:00.060 --> 00:00:02.500 The following content is provided under a Creative 00:00:02.500 --> 00:00:04.019 Commons license. 00:00:04.019 --> 00:00:06.360 Your support will help MIT OpenCourseWare 00:00:06.360 --> 00:00:10.730 continue to offer high quality educational resources for free. 00:00:10.730 --> 00:00:13.330 To make a donation, or view additional materials 00:00:13.330 --> 00:00:17.236 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.236 --> 00:00:17.861 at ocw.mit.edu. 00:00:21.670 --> 00:00:25.290 PROFESSOR: Today, our goal is to discuss some concepts 00:00:25.290 --> 00:00:28.670 in this question of a kind of population or ecosystem level 00:00:28.670 --> 00:00:32.996 stability, resilience, and associated sudden transitions 00:00:32.996 --> 00:00:33.620 in populations. 00:00:40.070 --> 00:00:42.640 And something about diversity of populations. 00:00:42.640 --> 00:00:45.540 We will revisit this question of diversity 00:00:45.540 --> 00:00:47.861 more on the last class, where we're 00:00:47.861 --> 00:00:49.360 going to discuss some of these ideas 00:00:49.360 --> 00:00:52.090 from the neutral theory in ecology, 00:00:52.090 --> 00:00:55.350 but it'll be a bit of a preview today. 00:00:59.027 --> 00:01:00.610 So what we're going to do is start out 00:01:00.610 --> 00:01:05.030 by just considering the dynamics of just a single population, 00:01:05.030 --> 00:01:07.810 where you have maybe a one population that's 00:01:07.810 --> 00:01:11.600 growing either logistically, or via some cooperative kind 00:01:11.600 --> 00:01:12.820 of dynamic. 00:01:12.820 --> 00:01:17.380 So, single population. 00:01:17.380 --> 00:01:19.530 And in here, we're going to understand something 00:01:19.530 --> 00:01:21.550 about this idea of either logistic growth 00:01:21.550 --> 00:01:24.030 as compared to something called the Allee effect, whereby 00:01:24.030 --> 00:01:26.720 the population has some cooperative type dynamic. 00:01:26.720 --> 00:01:30.920 So this is maybe logistic versus the Allee effect. 00:01:30.920 --> 00:01:34.290 And try to understand how this Allee effect can 00:01:34.290 --> 00:01:37.010 lead to the sudden transitions that you read about 00:01:37.010 --> 00:01:39.390 in the review. 00:01:39.390 --> 00:01:42.760 And then we'll say something about the Lotka-Volterra 00:01:42.760 --> 00:01:44.899 competition model. 00:01:44.899 --> 00:01:46.440 So this is a slightly different model 00:01:46.440 --> 00:01:48.231 from the Lotka-Volterra predator-prey model 00:01:48.231 --> 00:01:51.590 that we've all been thinking about recently 00:01:51.590 --> 00:01:54.000 over the last week or so. 00:01:54.000 --> 00:01:56.390 But it's again, kind of the simplest model 00:01:56.390 --> 00:01:58.000 you might write down that captures 00:01:58.000 --> 00:02:00.550 this idea of interactions between species. 00:02:00.550 --> 00:02:02.930 In particular, competitive interactions. 00:02:06.880 --> 00:02:09.580 Then at the end, and depending on how much time we have, 00:02:09.580 --> 00:02:10.830 we'll either finish it or not. 00:02:10.830 --> 00:02:13.163 We'll talk about some these non transitive interactions. 00:02:15.220 --> 00:02:18.640 The so-called rock, paper, scissors, 00:02:18.640 --> 00:02:22.380 RPS type interactions. 00:02:22.380 --> 00:02:24.380 And some of the measurements that 00:02:24.380 --> 00:02:28.120 have been made both in male mating strategies in lizards 00:02:28.120 --> 00:02:31.900 in the California mountains, as well as kind of a rock, paper, 00:02:31.900 --> 00:02:34.670 scissors type dynamic that was explored experimentally 00:02:34.670 --> 00:02:39.050 by Benjamin Kerr in the context of bacterial toxin production, 00:02:39.050 --> 00:02:41.160 or chemical warfare bacteria. 00:02:41.160 --> 00:02:44.800 And this subject does come in a bit 00:02:44.800 --> 00:02:47.060 into this question of spatial structure, 00:02:47.060 --> 00:02:50.960 and how spatial structure may help facilitate 00:02:50.960 --> 00:02:52.455 the stability of populations. 00:02:52.455 --> 00:02:54.830 So there's this idea of that non-transitive interactions, 00:02:54.830 --> 00:02:57.430 whether you have Rock, Paper, Scissor type interactions, 00:02:57.430 --> 00:03:01.810 may facilitate coexistence of either genotypes or species. 00:03:01.810 --> 00:03:03.460 But in some of these studies, there's 00:03:03.460 --> 00:03:06.899 an argument that sometimes this non-transitive interaction 00:03:06.899 --> 00:03:08.690 may not be enough, but then in the presence 00:03:08.690 --> 00:03:11.190 of spatial structure, maybe it does help. 00:03:11.190 --> 00:03:13.640 On Thursday, the primary topic of the class 00:03:13.640 --> 00:03:16.650 will be trying to understand the dynamics of populations 00:03:16.650 --> 00:03:19.590 when their spatially extended. 00:03:19.590 --> 00:03:21.537 So this will transition naturally 00:03:21.537 --> 00:03:22.745 into the subject on Thursday. 00:03:26.070 --> 00:03:30.350 Are there any questions about where we are here? 00:03:30.350 --> 00:03:32.930 Or administrative type issues? 00:03:35.597 --> 00:03:37.930 All right, so I just want to make sure that we're first, 00:03:37.930 --> 00:03:38.830 all on the same page. 00:03:38.830 --> 00:03:40.490 We talk about logistic growth. 00:03:45.350 --> 00:03:47.350 So we can think about-- Well, what 00:03:47.350 --> 00:03:51.910 are the simplest ways that we might consider the population 00:03:51.910 --> 00:03:54.570 dynamics of just a single population. 00:03:54.570 --> 00:03:57.660 Well, the logistic growth is maybe the simplest model 00:03:57.660 --> 00:04:03.000 you might write down, where the population is at least bounded. 00:04:03.000 --> 00:04:08.520 And I might even-- Maybe I'll even just first draw 00:04:08.520 --> 00:04:10.960 exponential growth, just so that we can-- 00:04:13.820 --> 00:04:15.320 The very simplest thing you might do 00:04:15.320 --> 00:04:18.240 is you might just say that N dot might 00:04:18.240 --> 00:04:20.290 be equal to some r times n. 00:04:22.800 --> 00:04:27.760 In this case, if we plotted gamma, 00:04:27.760 --> 00:04:33.620 the per capita rate of population growth, 00:04:33.620 --> 00:04:37.280 this is just some line at r. 00:04:37.280 --> 00:04:41.130 In this case, the population grows exponentially. 00:04:41.130 --> 00:04:44.490 And that that means that if you plot 00:04:44.490 --> 00:04:46.904 say the number as a function of time, well 00:04:46.904 --> 00:04:48.320 it doesn't matter where you start, 00:04:48.320 --> 00:04:49.690 you always go to infinity. 00:04:49.690 --> 00:04:50.460 Right? 00:04:50.460 --> 00:04:58.870 Now we can compare this to the logistic growth case, where 00:04:58.870 --> 00:05:02.600 we just assume that at low density, 00:05:02.600 --> 00:05:06.190 indeed, we grow exponentially at this rate r, 00:05:06.190 --> 00:05:08.135 but then we have a linear decrease 00:05:08.135 --> 00:05:09.810 in this per capita rate of growth 00:05:09.810 --> 00:05:11.890 as a function of density. 00:05:11.890 --> 00:05:22.390 So we might write this as something that looks like this. 00:05:22.390 --> 00:05:25.756 In which case, if we write this per capita growth 00:05:25.756 --> 00:05:28.130 rate of the population, as a function of population size, 00:05:28.130 --> 00:05:32.710 it starts out at r, but then it goes to 0. 00:05:32.710 --> 00:05:34.904 And indeed you, might even if you'd like, 00:05:34.904 --> 00:05:37.070 you could say that it goes down to a negative value. 00:05:37.070 --> 00:05:39.069 So in that case, you could start at a population 00:05:39.069 --> 00:05:42.500 size above the carrying capacity k, 00:05:42.500 --> 00:05:44.395 and you still come back down it. 00:05:51.132 --> 00:05:53.340 One of things that we want to make sure we understand 00:05:53.340 --> 00:05:58.460 is what the bifurcation structure of these equations 00:05:58.460 --> 00:05:59.520 is going to look like. 00:05:59.520 --> 00:06:01.360 In particular, we can think about what 00:06:01.360 --> 00:06:04.575 happens if we add some sort of death rate to the population. 00:06:07.890 --> 00:06:11.960 Right now, in this case, a death rate 00:06:11.960 --> 00:06:16.880 corresponds to is something-- OK, we're looking 00:06:16.880 --> 00:06:20.189 at the bifurcation diagram. 00:06:20.189 --> 00:06:21.730 What we're going to do is we're going 00:06:21.730 --> 00:06:23.750 to think about what happens if you look 00:06:23.750 --> 00:06:25.850 at the size of the population. 00:06:25.850 --> 00:06:27.975 In the bifurcation diagram, what we typically do is 00:06:27.975 --> 00:06:29.920 we look at the fixed points. 00:06:29.920 --> 00:06:32.940 The fixed points in n, as a function 00:06:32.940 --> 00:06:34.430 of some external parameter. 00:06:34.430 --> 00:06:36.950 And here, we might just think about delta here 00:06:36.950 --> 00:06:38.590 which is just some death rate. 00:06:43.420 --> 00:06:45.905 The simplest way to include this here would 00:06:45.905 --> 00:06:50.210 be to have a minus delta n. 00:06:50.210 --> 00:06:52.575 So the idea is that there's this population. 00:06:52.575 --> 00:06:56.054 It grows exponentially at small sizes. 00:06:56.054 --> 00:06:57.970 As the population size grows, it might run out 00:06:57.970 --> 00:07:01.150 of nutrients or so, and that limits its overall population 00:07:01.150 --> 00:07:02.140 size. 00:07:02.140 --> 00:07:04.300 Then, we can add another term here 00:07:04.300 --> 00:07:05.959 which is corresponding to something 00:07:05.959 --> 00:07:07.500 about the quality of the environment. 00:07:07.500 --> 00:07:09.999 So this could be the amount of hunting that is taking place, 00:07:09.999 --> 00:07:13.620 or it could be a reflection of pollutants 00:07:13.620 --> 00:07:15.069 in the environment or so. 00:07:15.069 --> 00:07:16.610 And what we want to do is try to make 00:07:16.610 --> 00:07:20.220 sure we understand how the size of the population 00:07:20.220 --> 00:07:22.910 will respond to some death rate. 00:07:26.930 --> 00:07:30.980 Well maybe we'll go ahead and do a verbal vote. 00:07:30.980 --> 00:07:34.960 So in the review that you guys read, 00:07:34.960 --> 00:07:37.650 this was talking about in these early warning 00:07:37.650 --> 00:07:39.800 indicators in the context of sudden transitions 00:07:39.800 --> 00:07:41.950 and populations. 00:07:41.950 --> 00:07:43.690 So this was saying that whether you're 00:07:43.690 --> 00:07:47.830 looking at a population, or an ecosystem, or maybe even 00:07:47.830 --> 00:07:49.900 other complex system, such as climate regime 00:07:49.900 --> 00:07:52.700 shifts, epileptic seizures, and so forth, the authors were 00:07:52.700 --> 00:07:55.840 arguing that in response to a slowly 00:07:55.840 --> 00:07:57.620 changing environmental knob, the system 00:07:57.620 --> 00:08:01.730 can experience a sudden transition in the state. 00:08:01.730 --> 00:08:04.910 Now, the question is here. 00:08:04.910 --> 00:08:08.140 And the sudden transitions that they were describing, 00:08:08.140 --> 00:08:13.110 those were fold bifurcations, or saddle-node bifurcations. 00:08:13.110 --> 00:08:14.560 Where there's a sudden transition 00:08:14.560 --> 00:08:16.040 in say, the equilibrium. 00:08:16.040 --> 00:08:17.540 Size of the population is a function 00:08:17.540 --> 00:08:21.050 of some external knob describing the quality of the environment. 00:08:21.050 --> 00:08:25.530 So the question is whether this population 00:08:25.530 --> 00:08:29.212 experiences such a full bifurcation at least 00:08:29.212 --> 00:08:30.170 to a sudden transition. 00:08:34.169 --> 00:08:36.890 Do understand the question? 00:08:36.890 --> 00:08:42.085 I'll say fold bifurcation. 00:08:46.700 --> 00:08:51.812 I'll let you guys vote, just so you can use your cards. 00:08:51.812 --> 00:08:54.270 And this is a full bifurcation as a function of this delta, 00:08:54.270 --> 00:08:55.770 which is death rate. 00:08:55.770 --> 00:08:57.110 Ready? 00:08:57.110 --> 00:08:59.895 Three, two, one. 00:09:03.170 --> 00:09:05.520 We have a majority now that are saying 00:09:05.520 --> 00:09:07.510 the answer is no, this does not actually 00:09:07.510 --> 00:09:11.500 have a fold bifurcation. 00:09:11.500 --> 00:09:13.270 And the reason for that is that we 00:09:13.270 --> 00:09:17.580 can find the-- In the absence of a death rate, what 00:09:17.580 --> 00:09:20.050 is the equilibrium population size? 00:09:22.650 --> 00:09:23.650 k, right? 00:09:25.912 --> 00:09:28.370 Indeed, what's going to happen is that as we add this death 00:09:28.370 --> 00:09:33.170 rate, we can just figure out, OK well, the fixed points 00:09:33.170 --> 00:09:35.350 occur when N dots equal to zero. 00:09:35.350 --> 00:09:36.210 All of these things. 00:09:36.210 --> 00:09:38.491 There's an n here, we can divide by that. 00:09:38.491 --> 00:09:40.490 Well we could just do it, just so that we're all 00:09:40.490 --> 00:09:42.210 on the same page. 00:09:42.210 --> 00:09:43.210 So the fixed point. 00:09:46.874 --> 00:09:50.910 What we want to do is set N dot equal to zero. 00:09:50.910 --> 00:09:59.320 That tells us that n times r1 minus n over k minus delta 00:09:59.320 --> 00:10:00.740 is equal to 0. 00:10:00.740 --> 00:10:01.720 OK. 00:10:01.720 --> 00:10:04.610 How many fixed points are there going to be in the system? 00:10:04.610 --> 00:10:05.220 Two. 00:10:05.220 --> 00:10:09.540 Right, so indeed, this is N dot. 00:10:09.540 --> 00:10:14.059 So n equal to zero is going to be one fixed point, 00:10:14.059 --> 00:10:15.600 but the other one is going to be when 00:10:15.600 --> 00:10:17.660 this thing is equal to zero. 00:10:17.660 --> 00:10:20.910 And so then we end up with 1 minus n over k, 00:10:20.910 --> 00:10:23.190 is going to be equal to some delta over r. 00:10:36.890 --> 00:10:39.100 So we can see that this stable fixed point 00:10:39.100 --> 00:10:42.660 is going to decrease linearly with this death rate delta. 00:10:42.660 --> 00:10:44.510 And it's just going to go smoothly to zero. 00:10:44.510 --> 00:10:47.900 That's this key feature of this bifurcation digraph. 00:10:47.900 --> 00:10:54.190 So what happens is that we come down like this, and indeed, 00:10:54.190 --> 00:10:57.430 we could even continue this and the bifurcation occurs here. 00:10:59.940 --> 00:11:02.410 We want to draw-- This corresponds 00:11:02.410 --> 00:11:03.640 to a stable fixed point. 00:11:06.580 --> 00:11:08.215 And a dashed line is unstable. 00:11:14.580 --> 00:11:18.550 In this case, this is the stable and this is the unstable. 00:11:23.194 --> 00:11:24.610 So this thing here is what's known 00:11:24.610 --> 00:11:26.095 as a trans-critical bifurcation. 00:11:28.630 --> 00:11:31.650 And it's a bifurcation because the fixed points do something. 00:11:31.650 --> 00:11:34.030 In this case, they exchange stability. 00:11:34.030 --> 00:11:38.610 So what happens is that this thing becomes unstable, 00:11:38.610 --> 00:11:40.600 whereas the fixed point at n equals 00:11:40.600 --> 00:11:43.524 0 becomes a stable fixed point. 00:11:48.940 --> 00:11:52.630 So there's no tipping point or sudden transition 00:11:52.630 --> 00:11:57.380 in this logistic growth as you change the death rate. 00:11:57.380 --> 00:11:59.270 All right, you can think of it as a situation 00:11:59.270 --> 00:12:01.190 here where the death rate just kind of starts 00:12:01.190 --> 00:12:02.190 pulling this thing down. 00:12:06.590 --> 00:12:10.090 And this bifurcation occurs when you pull it down-- 00:12:10.090 --> 00:12:11.225 when delta is equal to r. 00:12:18.910 --> 00:12:24.050 And what this means is that if you plot say, 00:12:24.050 --> 00:12:32.400 the number as a function of time, for low delta, 00:12:32.400 --> 00:12:34.400 in particular, for delta equal to zero then this 00:12:34.400 --> 00:12:45.240 is your-- You come to this equilibrium k. 00:12:45.240 --> 00:12:48.350 Where as delta increases, eventually you just 00:12:48.350 --> 00:12:51.660 get to a situation where all of, regardless of where you start, 00:12:51.660 --> 00:12:52.300 you go extinct. 00:12:55.270 --> 00:12:58.770 Now, in many cases, we don't draw what happens down here 00:12:58.770 --> 00:13:02.620 because n being less than 0 is not a physical thing. 00:13:02.620 --> 00:13:05.185 But it's useful to think about it mathematically, 00:13:05.185 --> 00:13:07.685 because that's what tells you that that was the bifurcation. 00:13:14.310 --> 00:13:16.750 Are there any questions about where we are here? 00:13:21.530 --> 00:13:24.580 One thing that's valuable in the case of-- well, in general-- 00:13:24.580 --> 00:13:29.610 is to switch between the algebraic characterization 00:13:29.610 --> 00:13:32.094 of the dynamics a system and the graphical dynamics, 00:13:32.094 --> 00:13:34.260 because you end up seeing different things depending 00:13:34.260 --> 00:13:35.866 on how you do it. 00:13:35.866 --> 00:13:37.240 For the case of the Allee effect, 00:13:37.240 --> 00:13:40.300 we're just going to look at it and analyze it graphically, 00:13:40.300 --> 00:13:43.829 just because the equation is a little bit cumbersome, 00:13:43.829 --> 00:13:45.745 and I think don't provide very much intuition. 00:13:51.490 --> 00:13:55.530 What you can see is that for the logistic growth, 00:13:55.530 --> 00:13:57.920 other individuals in the population 00:13:57.920 --> 00:14:01.590 always decrease your happiness as an individual. 00:14:01.590 --> 00:14:05.000 So if you imagine that you are some member of this population, 00:14:05.000 --> 00:14:06.770 and having more members the population 00:14:06.770 --> 00:14:08.270 there is always kind of bad for you, 00:14:08.270 --> 00:14:10.940 because it decreases your growth rate. 00:14:10.940 --> 00:14:14.434 And to think about the happiness of an individual, what 00:14:14.434 --> 00:14:16.100 you typically want to do is divide by n, 00:14:16.100 --> 00:14:18.475 because you're thinking about the per capita growth rate. 00:14:18.475 --> 00:14:20.880 So that's telling you about the ability hat 00:14:20.880 --> 00:14:22.380 you will have to reproduce. 00:14:22.380 --> 00:14:24.440 So what you did we do is just divide by n. 00:14:24.440 --> 00:14:30.160 What you can see is that per capita division, gamma, 00:14:30.160 --> 00:14:32.910 is a monotonically decreasing function of the population 00:14:32.910 --> 00:14:33.870 size. 00:14:33.870 --> 00:14:36.550 This is saying that more individuals in the population 00:14:36.550 --> 00:14:39.410 just take up space, resources, mate, something. 00:14:39.410 --> 00:14:42.510 And so you as an individual never benefit 00:14:42.510 --> 00:14:44.970 from the presence of other individuals. 00:14:44.970 --> 00:14:47.440 Whereas, when you have the Allee effect, what that's saying 00:14:47.440 --> 00:14:50.560 is that over some range of population size or densities, 00:14:50.560 --> 00:14:52.785 there is a positive effect of other individuals 00:14:52.785 --> 00:14:53.580 in the population. 00:15:01.000 --> 00:15:07.740 This is saying that if we plot gamma, in particular, 00:15:07.740 --> 00:15:14.670 the derivative of gamma with respect to n is greater than 0 00:15:14.670 --> 00:15:19.220 for some n. 00:15:19.220 --> 00:15:24.010 This is just saying that if we plot this per capita division 00:15:24.010 --> 00:15:26.750 rate as a function of the size of the population, 00:15:26.750 --> 00:15:30.710 there's some region in which this thing has positive slope. 00:15:30.710 --> 00:15:32.870 And often, people distinguish between 00:15:32.870 --> 00:15:35.280 the so-called strong Allee effect and the weak Allee 00:15:35.280 --> 00:15:36.500 effect. 00:15:36.500 --> 00:15:38.700 The strong Allee effect corresponds to the situation 00:15:38.700 --> 00:15:43.740 where at n equal to zero, you start out with negative 00:15:43.740 --> 00:15:44.810 per capita growth rates. 00:15:47.900 --> 00:15:50.140 So this is just some generic curve here, 00:15:50.140 --> 00:15:51.530 that describes the Allee effect. 00:15:51.530 --> 00:15:56.230 The important thing is that it's coming up here at low n. 00:15:56.230 --> 00:15:59.710 And what this tells us is that we 00:15:59.710 --> 00:16:01.860 will have some sort of different dynamics 00:16:01.860 --> 00:16:05.780 where this is going to be the stable size of the population 00:16:05.780 --> 00:16:07.380 here. 00:16:07.380 --> 00:16:10.140 But there's also a minimal size required for survival. 00:16:10.140 --> 00:16:13.310 So if you start out below this n, then you come down. 00:16:17.695 --> 00:16:19.320 If you want you want, you could call it 00:16:19.320 --> 00:16:27.960 some-- This could be some k, and this could be n-min. 00:16:27.960 --> 00:16:32.450 This is saying if we plot n as a function of time 00:16:32.450 --> 00:16:36.980 starting with different sizes of the population, 00:16:36.980 --> 00:16:40.960 then indeed, this thing is a stable state. 00:16:40.960 --> 00:16:44.904 But the key thing is that there's this other minimal size 00:16:44.904 --> 00:16:45.820 required for survival. 00:16:45.820 --> 00:16:49.360 So if you start out around here, then just above it you come up, 00:16:49.360 --> 00:16:51.382 but just below it, you come down. 00:16:57.154 --> 00:16:57.820 What do you say? 00:16:57.820 --> 00:16:59.760 What you see is that this population 00:16:59.760 --> 00:17:02.130 that experiences the strong Allee effect 00:17:02.130 --> 00:17:03.670 has bistable fates. 00:17:06.290 --> 00:17:07.190 So it's bistable. 00:17:07.190 --> 00:17:09.730 This is bistable depending on the starting size 00:17:09.730 --> 00:17:10.564 n of the population. 00:17:14.359 --> 00:17:20.310 Now, can somebody suggest possible explanations 00:17:20.310 --> 00:17:22.240 for why there might an Allee effect? 00:17:41.960 --> 00:17:43.789 AUDIENCE: Sexual reproduction? 00:17:43.789 --> 00:17:45.080 PROFESSOR: Sexual reproduction. 00:17:45.080 --> 00:17:47.583 And can you explain that a little bit more? 00:17:47.583 --> 00:17:48.458 AUDIENCE: [INAUDIBLE] 00:17:55.920 --> 00:17:56.920 PROFESSOR: That's right. 00:17:56.920 --> 00:17:59.360 And basically, just the need to find mates. 00:18:10.440 --> 00:18:15.600 So this implies that at some density, 00:18:15.600 --> 00:18:17.287 sexual reproducing species are expected 00:18:17.287 --> 00:18:18.370 to have this Allee effect. 00:18:18.370 --> 00:18:20.510 It doesn't mean that the Allee fact 00:18:20.510 --> 00:18:23.980 is super strong at moderate sizes, right? 00:18:23.980 --> 00:18:25.940 I think that this statement is true, 00:18:25.940 --> 00:18:29.679 but it might lead us to conclude that these extreme dynamics 00:18:29.679 --> 00:18:32.220 that we're talking about here are relevant for every sexually 00:18:32.220 --> 00:18:33.053 reproducing species. 00:18:33.053 --> 00:18:38.670 But ultimately, the question is maybe how big is this n-min? 00:18:38.670 --> 00:18:43.850 If n-min is two, then it's not a severe constraint 00:18:43.850 --> 00:18:45.700 on the population. 00:18:45.700 --> 00:18:50.850 Whereas if n-min is-- if you need to have 100 California 00:18:50.850 --> 00:18:53.110 condors in order to have a viable population, 00:18:53.110 --> 00:18:54.580 then that's a problem. 00:18:54.580 --> 00:18:56.990 So there really is a quantitative question 00:18:56.990 --> 00:19:01.790 of how big is this minimal size. 00:19:01.790 --> 00:19:05.474 Other suggestions of why it is you might have an Allee effect? 00:19:05.474 --> 00:19:08.210 AUDIENCE: The animal species in groups? 00:19:08.210 --> 00:19:09.210 PROFESSOR: That's right. 00:19:09.210 --> 00:19:10.350 Group hunting. 00:19:10.350 --> 00:19:15.106 And are you aware of any cases that have group hunting ? 00:19:15.106 --> 00:19:17.299 AUDIENCE: Wolves? 00:19:17.299 --> 00:19:18.840 PROFESSOR: That's right, so, exactly. 00:19:21.670 --> 00:19:24.751 Right, so wolves for example, can take down 00:19:24.751 --> 00:19:26.750 like bison, of large animals they would never be 00:19:26.750 --> 00:19:28.980 able to take down on their own. 00:19:28.980 --> 00:19:33.920 So this could be this could be wolves, it could be primates. 00:19:33.920 --> 00:19:37.350 We historically did-- I guess we still have some groups. 00:19:37.350 --> 00:19:41.970 Societies will have things like this. 00:19:41.970 --> 00:19:44.240 But even at the microscopic realm, 00:19:44.240 --> 00:19:47.500 if you look at just bacteria or yeast and so forth, 00:19:47.500 --> 00:19:51.480 there are often cases where food is generated or broken 00:19:51.480 --> 00:19:53.230 down outside of the cell. 00:19:53.230 --> 00:19:54.810 So any of these cases where you have 00:19:54.810 --> 00:19:57.000 secreted enzymatic breakdown. 00:20:07.190 --> 00:20:07.840 That was an i. 00:20:12.440 --> 00:20:13.100 Other thoughts? 00:20:20.777 --> 00:20:21.277 Yes? 00:20:21.277 --> 00:20:22.777 AUDIENCE: Protection from predators? 00:20:22.777 --> 00:20:24.000 PROFESSOR: Right, protection. 00:20:24.000 --> 00:20:26.050 And so this is the flip side of this, 00:20:26.050 --> 00:20:28.300 so this would be predator avoidance. 00:20:28.300 --> 00:20:32.490 Or an example of that would be what? 00:20:32.490 --> 00:20:34.430 AUDIENCE: Antelopes. 00:20:34.430 --> 00:20:37.093 PROFESSOR: What do antelopes do? 00:20:37.093 --> 00:20:38.980 AUDIENCE: They run in groups. 00:20:38.980 --> 00:20:39.980 PROFESSOR: That's right. 00:20:39.980 --> 00:20:41.880 So the standard example here is kind 00:20:41.880 --> 00:20:45.130 of what herding behavior of land animals, schooling 00:20:45.130 --> 00:20:47.450 behavior of fish. 00:20:47.450 --> 00:20:51.370 Of course, researchers argue about to what degree 00:20:51.370 --> 00:20:54.750 the benefits are primarily this, versus something else. 00:20:54.750 --> 00:20:56.940 I am not going to weigh in on that debate. 00:20:56.940 --> 00:21:04.680 But, I'd be surprised if this were never true. 00:21:04.680 --> 00:21:09.560 In all these examples, there's clearly something cooperative 00:21:09.560 --> 00:21:10.779 about the dynamics. 00:21:10.779 --> 00:21:12.320 In that sense, it kind of makes sense 00:21:12.320 --> 00:21:14.920 to think there's some element of cooperative growth, 00:21:14.920 --> 00:21:17.750 whether it's on the hunting side, or the protection side. 00:21:20.560 --> 00:21:24.440 So if you look at all these cases 00:21:24.440 --> 00:21:26.424 these would all be very naturally described 00:21:26.424 --> 00:21:27.590 as some sort of cooperation. 00:21:30.057 --> 00:21:32.390 I do want to highlight though, that the Allee effect can 00:21:32.390 --> 00:21:34.640 be caused by things that lead to what you might 00:21:34.640 --> 00:21:36.100 call effective cooperation. 00:21:36.100 --> 00:21:38.145 So it's not like obviously cooperative, 00:21:38.145 --> 00:21:39.520 or not intentionally cooperative, 00:21:39.520 --> 00:21:42.220 but still has the same effect. 00:21:42.220 --> 00:21:45.075 A nice example of this is what we call predator satiation. 00:21:51.130 --> 00:21:53.692 Can somebody guess what this might mean? 00:21:53.692 --> 00:21:54.900 Somebody's laughing so it's-- 00:21:57.265 --> 00:21:59.139 AUDIENCE: If there are enough animals around, 00:21:59.139 --> 00:22:00.361 the predator will be fed. 00:22:00.361 --> 00:22:01.360 PROFESSOR: That's right. 00:22:01.360 --> 00:22:03.630 If the predators gets full, then you 00:22:03.630 --> 00:22:05.660 can end up with something that's like this. 00:22:05.660 --> 00:22:09.620 And the way I think about this is just in a group like this, 00:22:09.620 --> 00:22:15.180 if 20 bears come in and eat 20 of us, 00:22:15.180 --> 00:22:18.350 then we hope that there are 20 other people that 00:22:18.350 --> 00:22:21.070 will get eaten first, so that you know that the leftovers can 00:22:21.070 --> 00:22:21.570 run off. 00:22:21.570 --> 00:22:22.070 Right? 00:22:25.180 --> 00:22:28.170 This is embodied in that joke that people tell when they're 00:22:28.170 --> 00:22:32.004 the two hikers in the woods, and they see the mother bear that's 00:22:32.004 --> 00:22:33.670 angry and one of the hikers reaches down 00:22:33.670 --> 00:22:36.350 and starts tying his shoelaces. 00:22:36.350 --> 00:22:38.990 And the other thing the hiking partner says why are you 00:22:38.990 --> 00:22:40.230 tying your shoelaces? 00:22:40.230 --> 00:22:41.632 You can't outrun a bear, right? 00:22:41.632 --> 00:22:43.340 And the guys says, well yeah I know that, 00:22:43.340 --> 00:22:46.370 but I only have to outrun you. 00:22:46.370 --> 00:22:48.680 That's predator satiation. 00:22:48.680 --> 00:22:52.500 So if the predator gets full, then you 00:22:52.500 --> 00:22:54.670 can get something that looks like this. 00:22:54.670 --> 00:22:57.350 It's below some size, below this n-min size, 00:22:57.350 --> 00:22:59.100 they all get eaten by that bear. 00:22:59.100 --> 00:23:03.960 And above it, the population may be able to grow. 00:23:03.960 --> 00:23:06.425 So this is a good example of just 00:23:06.425 --> 00:23:08.240 that it's an effective form of cooperation. 00:23:14.580 --> 00:23:21.140 Now can somebody explain-- Well, first of all, 00:23:21.140 --> 00:23:23.530 does this lead to a full bifurcation 00:23:23.530 --> 00:23:24.730 if we add a death rate? 00:23:27.424 --> 00:23:29.590 We're going to think about this first seven seconds, 00:23:29.590 --> 00:23:30.760 then we'll vote. 00:23:30.760 --> 00:23:34.430 The question: if we add a death rate to this, 00:23:34.430 --> 00:23:36.645 does that lead to a fold or saddle-node bifurcation? 00:23:42.497 --> 00:23:44.830 I'm going to give you more than seven seconds, because I 00:23:44.830 --> 00:23:46.620 see a lot of brain activity. 00:23:53.310 --> 00:23:55.018 Does this one lead to a full bifurcation? 00:24:04.000 --> 00:24:04.690 Ready? 00:24:04.690 --> 00:24:07.040 Three, two, one. 00:24:13.980 --> 00:24:15.420 The answer is, in this case, yes. 00:24:19.180 --> 00:24:23.384 So we'll say Allee effect is in general the kind of thing that 00:24:23.384 --> 00:24:24.550 leads to a full bifurcation. 00:24:24.550 --> 00:24:26.010 In this case, it definitely does. 00:24:28.920 --> 00:24:31.100 So it's really very important to be 00:24:31.100 --> 00:24:34.040 able to take curves like this, and figure out 00:24:34.040 --> 00:24:37.750 what the bifurcation diagram looks like. 00:24:37.750 --> 00:24:41.380 And just remember that the solid and dashed lines here 00:24:41.380 --> 00:24:43.640 are very useful for getting intuition 00:24:43.640 --> 00:24:45.950 on what's going on, because stable is telling us 00:24:45.950 --> 00:24:49.932 that any perturbation away from this point, it goes away. 00:24:49.932 --> 00:24:51.390 So this is why we draw arrows here. 00:24:54.590 --> 00:24:57.790 Arrows come here. 00:24:57.790 --> 00:25:04.325 So this is saying that starting at 0, from a differential 00:25:04.325 --> 00:25:05.700 equation standpoint at least, you 00:25:05.700 --> 00:25:09.895 can add a single individual n, and that individual 00:25:09.895 --> 00:25:12.320 will reproduce and gets to whatever the effective carrying 00:25:12.320 --> 00:25:14.050 capacity is at that delta. 00:25:19.080 --> 00:25:21.370 So what we can do is we can draw the same bifurcation 00:25:21.370 --> 00:25:23.351 diagram here. 00:25:23.351 --> 00:25:29.440 Where we look at the population size as a function of delta. 00:25:29.440 --> 00:25:31.810 And now it's actually somewhat more interesting looking, 00:25:31.810 --> 00:25:35.560 because in the absence of this death rate, 00:25:35.560 --> 00:25:37.109 we have it start at k. 00:25:37.109 --> 00:25:39.525 So what's going to happen is it's going to come like this. 00:25:43.897 --> 00:25:45.730 And the reason we call it a full bifurcation 00:25:45.730 --> 00:25:50.860 is because these fixed points fold over on themselves. 00:25:50.860 --> 00:25:53.980 Now this is a bifurcation where as a function is a control 00:25:53.980 --> 00:25:56.650 parameter, it's not that these fixed points exchange 00:25:56.650 --> 00:25:59.070 stability, but rather that the fixed points collide 00:25:59.070 --> 00:26:00.820 and annihilate. 00:26:00.820 --> 00:26:03.710 So in general, a bifurcation is where something qualitative 00:26:03.710 --> 00:26:06.670 happens to the fixed points. 00:26:06.670 --> 00:26:08.550 In this case, the number of fixed points 00:26:08.550 --> 00:26:11.200 did not change as a function of the control parameter. 00:26:11.200 --> 00:26:14.400 Whereas in this case, it does change. 00:26:14.400 --> 00:26:17.600 But there are only certain characteristic ways 00:26:17.600 --> 00:26:19.610 in which these six points are allowed to change. 00:26:28.520 --> 00:26:30.670 So from this, we can draw our arrows. 00:26:34.610 --> 00:26:38.510 n equal to 0 is still a stable point down here. 00:26:38.510 --> 00:26:41.547 So we can draw. 00:26:41.547 --> 00:26:43.380 This is where colored chalk would be useful. 00:26:51.602 --> 00:26:52.560 We can draw our arrows. 00:27:01.804 --> 00:27:02.720 What do I do out here? 00:27:07.430 --> 00:27:11.690 Now, the reason that we say that this population experiences 00:27:11.690 --> 00:27:14.170 a sudden transition, or a tipping point, 00:27:14.170 --> 00:27:16.920 is because you can imagine that this death rate being 00:27:16.920 --> 00:27:18.794 a slowly changing parameter. 00:27:18.794 --> 00:27:20.210 So you could imagine that it could 00:27:20.210 --> 00:27:22.470 be a slow acidification of the oceans, 00:27:22.470 --> 00:27:25.560 or a slow increase in the amount of fishing 00:27:25.560 --> 00:27:26.790 that takes place here. 00:27:26.790 --> 00:27:29.730 So then what can happen is that the size of the population, 00:27:29.730 --> 00:27:31.550 it's very healthy here. 00:27:31.550 --> 00:27:32.940 It's pretty happy, happy. 00:27:32.940 --> 00:27:36.020 Here it's maybe decreasing some, but you might not 00:27:36.020 --> 00:27:36.810 sound an alarm. 00:27:36.810 --> 00:27:39.950 But what can happen is that at some critical level 00:27:39.950 --> 00:27:42.540 of environmental conditions, some critical level of fishing, 00:27:42.540 --> 00:27:45.380 for example, you can get this catastrophic collapse 00:27:45.380 --> 00:27:46.340 of the population. 00:27:46.340 --> 00:27:50.162 Where over short periods of time, it can suddenly collapse. 00:27:50.162 --> 00:27:51.620 Indeed, this sort of thing has been 00:27:51.620 --> 00:27:54.560 observed in a number of fisheries around the world. 00:27:54.560 --> 00:27:57.480 So if you go to, for example, the Monterey Bay Aquarium 00:27:57.480 --> 00:28:03.110 in Monterey, California, they have a very nice display 00:28:03.110 --> 00:28:06.050 explaining how that space ended up becoming and aquarium. 00:28:06.050 --> 00:28:08.380 Because it used to be a sardine fishery, 00:28:08.380 --> 00:28:12.230 and was a very productive fishery in the years leading up 00:28:12.230 --> 00:28:13.590 to World War II. 00:28:13.590 --> 00:28:15.815 But if you look at the number of fish that 00:28:15.815 --> 00:28:18.010 were caught as a function of time, it goes up, 00:28:18.010 --> 00:28:19.970 up, because there's fishing more and more, then 00:28:19.970 --> 00:28:22.200 all of the sudden, they just presumably fish 00:28:22.200 --> 00:28:25.369 too much and over a time span of a few years, 00:28:25.369 --> 00:28:26.910 the fishery collapsed, and there were 00:28:26.910 --> 00:28:30.570 just no more sardines to catch. 00:28:30.570 --> 00:28:33.630 That whole street there used to be canneries. 00:28:33.630 --> 00:28:35.330 But then it all went out of business. 00:28:35.330 --> 00:28:38.890 And those buildings were largely unused for many years. 00:28:38.890 --> 00:28:41.107 But then eventually, the aquarium 00:28:41.107 --> 00:28:43.190 went and bought some of them and refurbished them, 00:28:43.190 --> 00:28:44.606 and now it's a beautiful aquarium. 00:28:44.606 --> 00:28:48.250 So I encourage you to check it out. 00:28:48.250 --> 00:28:50.370 But that being said, the collapse 00:28:50.370 --> 00:28:52.740 is probably still bad, even though it 00:28:52.740 --> 00:28:54.750 led to this nice outcome eventually. 00:28:54.750 --> 00:28:59.100 And what you can see is that not only are these tipping points 00:28:59.100 --> 00:29:03.020 or the sudden transitions maybe undesirable in the case 00:29:03.020 --> 00:29:06.380 of fishery, but it also would be very difficult to reverse. 00:29:06.380 --> 00:29:08.590 Because it's not that-- You couldn't imagined 00:29:08.590 --> 00:29:10.750 that the bifurcation diagram look rather different. 00:29:10.750 --> 00:29:12.041 It could have looked like this. 00:29:16.605 --> 00:29:17.980 What kind of bifurcation is this? 00:29:21.640 --> 00:29:23.290 So this is still a transcritical. 00:29:23.290 --> 00:29:25.475 It doesn't have to be a line throughout, 00:29:25.475 --> 00:29:29.390 because in principle there was this unstable thing down here. 00:29:29.390 --> 00:29:31.960 This is again, and, as a function of some parameter, 00:29:31.960 --> 00:29:33.610 we'll say delta. 00:29:33.610 --> 00:29:37.400 And so you'd say, oh this is a rather sudden transition. 00:29:37.400 --> 00:29:39.590 In the sense that a modest change in delta 00:29:39.590 --> 00:29:43.170 will lead to a dramatic change in the size of the population. 00:29:43.170 --> 00:29:45.622 But there's something that's very qualitatively different 00:29:45.622 --> 00:29:46.580 about these situations. 00:29:46.580 --> 00:29:49.430 Which is that, here you can imagine that if you improve 00:29:49.430 --> 00:29:52.130 the quality of the environment, and then you reintroduce fish, 00:29:52.130 --> 00:29:54.730 or maybe you don't even have to introduce fish, 00:29:54.730 --> 00:29:56.719 they can come from a different area. 00:29:56.719 --> 00:29:59.010 Then, just by improving the quality of the environment, 00:29:59.010 --> 00:30:01.180 you get recovery. 00:30:01.180 --> 00:30:03.720 Whereas here, that's very difficult to do, 00:30:03.720 --> 00:30:06.135 because the system is hysteretic, or has memory. 00:30:09.012 --> 00:30:10.470 That means that even if you improve 00:30:10.470 --> 00:30:11.969 the quality of the environment here, 00:30:11.969 --> 00:30:15.302 it may be very difficult to get back to your previous state. 00:30:15.302 --> 00:30:16.760 People think that they've seen this 00:30:16.760 --> 00:30:20.040 in the context of these transitions 00:30:20.040 --> 00:30:23.520 in lake ecosystems, so-called eutrophication transition, 00:30:23.520 --> 00:30:26.570 where you get as a result of maybe 00:30:26.570 --> 00:30:29.880 runoff from agricultural uses. 00:30:29.880 --> 00:30:32.370 You can get the sun transitions of not 00:30:32.370 --> 00:30:34.990 just a single population, but of an entire ecosystem. 00:30:34.990 --> 00:30:36.700 So the lake can go from this thing 00:30:36.700 --> 00:30:40.330 where it's nice and clear and beautiful summer vacations 00:30:40.330 --> 00:30:45.580 spot, to this really green kind of algal take over. 00:30:45.580 --> 00:30:49.060 And it can be very difficult to get recovery. 00:30:49.060 --> 00:30:50.716 There's was a question? 00:30:50.716 --> 00:30:51.216 No? 00:30:58.960 --> 00:31:02.430 And so the basic-- We'll say something 00:31:02.430 --> 00:31:04.450 about the early warning indicators. 00:31:04.450 --> 00:31:06.620 Well maybe I'll say it now. 00:31:06.620 --> 00:31:08.660 So based on the review that you read, 00:31:08.660 --> 00:31:11.270 there's this phenomenon of critical slowing down 00:31:11.270 --> 00:31:16.220 that tells us that in principle, you 00:31:16.220 --> 00:31:19.050 can anticipate the transitions about to take place. 00:31:22.350 --> 00:31:25.590 And the statement was that the dominant eigenvalue 00:31:25.590 --> 00:31:27.120 described in dynamics, a system went 00:31:27.120 --> 00:31:29.150 to 0 at one of these points. 00:31:35.690 --> 00:31:37.450 Now, the systems they were talking about 00:31:37.450 --> 00:31:40.690 were presumably systems like this. 00:31:40.690 --> 00:31:44.000 This is a so-called zero eigenvalue bifurcation, 00:31:44.000 --> 00:31:49.300 were the eigenvalue goes to 0 at this point right here. 00:31:54.110 --> 00:31:55.420 So here's a question for you. 00:31:57.950 --> 00:32:00.630 Does the eigenvalue describe the dynamics of a system 00:32:00.630 --> 00:32:02.470 go to 0 at this point? 00:32:12.030 --> 00:32:13.090 At transcritical. 00:32:15.675 --> 00:32:17.130 In a transcritical bifurcation. 00:32:22.449 --> 00:32:23.740 Do you understand the question? 00:32:30.709 --> 00:32:32.500 I'll let you think about it for 20 seconds, 00:32:32.500 --> 00:32:35.180 because this is maybe not totally obvious. 00:32:55.580 --> 00:32:57.350 Do you need more time? 00:32:57.350 --> 00:32:59.080 I'm wondering. 00:32:59.080 --> 00:33:02.500 Maybe people are sufficiently lost or confused. 00:33:02.500 --> 00:33:10.580 They're not sure what-- Nod if you want more time. 00:33:10.580 --> 00:33:11.170 OK. 00:33:11.170 --> 00:33:11.670 Ready? 00:33:14.410 --> 00:33:16.305 Three, two, one. 00:33:20.200 --> 00:33:22.590 I'd say it's a kind of a 50-50 thing. 00:33:22.590 --> 00:33:24.120 So go ahead and turn to you neighbor 00:33:24.120 --> 00:33:27.460 and try to figure out, using some combination of math 00:33:27.460 --> 00:33:30.300 and graphical analysis, whether the bifurcate-- 00:33:30.300 --> 00:33:32.870 Whether at this transcritical bifurcation 00:33:32.870 --> 00:33:34.357 the eigenvalue goes to 0. 00:33:37.339 --> 00:34:50.830 [SIDE CONVERSATIONS] 00:34:50.830 --> 00:34:53.730 PROFESSOR: It sounds like the discussion is maybe 00:34:53.730 --> 00:34:56.830 gone to completion. 00:34:56.830 --> 00:34:58.220 Let's go ahead and vote again. 00:34:58.220 --> 00:35:00.930 The question is, does the dominant eigenvalue 00:35:00.930 --> 00:35:03.140 of this system go to 0 at this point here? 00:35:03.140 --> 00:35:04.940 The trans critical bifurcation. 00:35:04.940 --> 00:35:05.700 Ready? 00:35:05.700 --> 00:35:07.885 Three, two, one. 00:35:10.450 --> 00:35:10.950 Alright. 00:35:13.440 --> 00:35:15.440 It doesn't seem like it's had much of an effect. 00:35:15.440 --> 00:35:18.194 I see that one person has been convinced of something. 00:35:18.194 --> 00:35:19.860 I don't know if he's actually convinced, 00:35:19.860 --> 00:35:21.805 of if he was just surrounded and bullied. 00:35:25.500 --> 00:35:27.530 All right, so what are some different ways 00:35:27.530 --> 00:35:28.780 of thinking about this? 00:35:32.410 --> 00:35:33.210 Yes? 00:35:33.210 --> 00:35:37.674 AUDIENCE: Well, one of my neighbors 00:35:37.674 --> 00:35:40.154 decided to actually just lineralize 00:35:40.154 --> 00:35:41.445 that equation at that point. 00:35:41.445 --> 00:35:42.320 PROFESSOR: All right. 00:35:46.040 --> 00:35:48.920 AUDIENCE: If you were completely into you intuition, 00:35:48.920 --> 00:35:50.357 this would be a way to check. 00:35:50.357 --> 00:35:51.190 PROFESSOR: Right OK. 00:35:51.190 --> 00:35:54.240 And now in general, in life, I'm a big believer 00:35:54.240 --> 00:35:55.990 when you're confronted with some question, 00:35:55.990 --> 00:35:59.930 you should first think about it, what 00:35:59.930 --> 00:36:02.390 you should guess based on whatever intuition 00:36:02.390 --> 00:36:04.280 might be available to you. 00:36:04.280 --> 00:36:07.080 And then after you have made a guess based on intuition, 00:36:07.080 --> 00:36:09.110 then you go and you do the calculation. 00:36:09.110 --> 00:36:11.980 Which in this case, linearizing around. 00:36:11.980 --> 00:36:13.230 And where do linearize around? 00:36:16.640 --> 00:36:17.620 Around the fixed point. 00:36:17.620 --> 00:36:20.380 I would say that you really want to linearize 00:36:20.380 --> 00:36:24.069 maybe not around-- You get the same answer if you lineralize 00:36:24.069 --> 00:36:26.360 around 0, but you really want to, I think conceptually, 00:36:26.360 --> 00:36:30.611 you want to linearize around this stable fixed point. 00:36:30.611 --> 00:36:31.110 Right? 00:36:31.110 --> 00:36:33.680 Because, in some ways, it's the stable fixed point that 00:36:33.680 --> 00:36:36.340 the system is sitting in, and you want to know how is it that 00:36:36.340 --> 00:36:38.673 you're-- How is it is you're going to react when you get 00:36:38.673 --> 00:36:40.650 perturbed away from that fixed point? 00:36:40.650 --> 00:36:44.520 And you do get the same answer. 00:36:44.520 --> 00:36:48.430 The thing is that it's true that the eigenvalue describing 00:36:48.430 --> 00:36:50.560 the dynamics around this unstable fixed point 00:36:50.560 --> 00:36:53.030 also goes to zero. 00:36:53.030 --> 00:36:55.660 But that's because, I think the rules of mathematics 00:36:55.660 --> 00:36:57.785 somehow say that the fixed points are going to have 00:36:57.785 --> 00:36:59.180 to do something similar. 00:36:59.180 --> 00:37:02.185 So I think you do also find the eigenvalue here 00:37:02.185 --> 00:37:03.560 as it goes to 0, but conceptually 00:37:03.560 --> 00:37:05.351 that's not the eigenvalue that you actually 00:37:05.351 --> 00:37:06.791 want to know about. 00:37:06.791 --> 00:37:08.280 Do you see why I'm saying that? 00:37:11.290 --> 00:37:14.860 Again, the eigenvalue here is going to 0. 00:37:14.860 --> 00:37:16.660 We can try to explore this a little bit, 00:37:16.660 --> 00:37:20.950 but I just want to kind of tie this together and say-- 00:37:20.950 --> 00:37:23.980 These local bifurcations that you have maybe 00:37:23.980 --> 00:37:27.070 heard about, which is that this fold bifurcation, 00:37:27.070 --> 00:37:31.250 the transcritical bifurcation, and also the Hopf bifurcation. 00:37:31.250 --> 00:37:34.020 So the Hopf is one that looks where 00:37:34.020 --> 00:37:35.820 you have a stable fixed point that goes-- 00:37:35.820 --> 00:37:37.910 And in terms of an unstable line, 00:37:37.910 --> 00:37:39.394 you end up getting oscillations. 00:37:39.394 --> 00:37:40.810 So that's where you have something 00:37:40.810 --> 00:37:49.340 that that's going to look like-- but now that I'm drawing this, 00:37:49.340 --> 00:37:51.970 I'm a little bit-- Is there a difference 00:37:51.970 --> 00:37:53.809 between a Hopf and a pitchfork bifurcation? 00:37:53.809 --> 00:37:55.975 AUDIENCE: It's whether you have oscillations or not. 00:37:55.975 --> 00:37:57.597 You can't really show that. 00:37:57.597 --> 00:37:58.180 PROFESSOR: OK. 00:37:58.180 --> 00:38:02.912 So as drawn, it's either or both, or? 00:38:02.912 --> 00:38:04.060 AUDIENCE: It's both. 00:38:04.060 --> 00:38:05.870 PROFESSOR: OK well. 00:38:05.870 --> 00:38:10.670 I'll say that this thing also the eigenvalue goes to 0 here. 00:38:10.670 --> 00:38:13.610 Because I guess, in this-- You don't 00:38:13.610 --> 00:38:15.690 have to have oscillations in order-- 00:38:15.690 --> 00:38:18.400 And it depends on the higher dynamic systems. 00:38:18.400 --> 00:38:20.985 But again, this is another 0 eigenvalue bifurcation. 00:38:24.160 --> 00:38:27.740 And so the statement is that in principle, you 00:38:27.740 --> 00:38:30.500 can use this fact that the eigenvalue goes to 0, 00:38:30.500 --> 00:38:32.540 this thing called critical slowing down, 00:38:32.540 --> 00:38:36.840 to measure some changes in the dynamics of the system 00:38:36.840 --> 00:38:39.230 before, in this case, you get collapse, 00:38:39.230 --> 00:38:43.120 or before, in this case, you get extinction. 00:38:43.120 --> 00:38:47.344 The issue maybe is that in this transition, 00:38:47.344 --> 00:38:48.760 there really is something big that 00:38:48.760 --> 00:38:50.359 happened at the bifurcation. 00:38:50.359 --> 00:38:52.150 So you really want to have an early warning 00:38:52.150 --> 00:38:55.450 signal because there's a dramatic change, 00:38:55.450 --> 00:38:56.920 and it may be difficult to reverse. 00:38:56.920 --> 00:39:00.830 Whereas here, the population size is smoothly going to 0. 00:39:00.830 --> 00:39:04.347 And once you get down to your last two California condors, 00:39:04.347 --> 00:39:05.680 you know that you're in trouble. 00:39:05.680 --> 00:39:08.510 You don't need any sort of fancy early warning 00:39:08.510 --> 00:39:11.420 indicator based on fluctuations or return time, and so forth. 00:39:11.420 --> 00:39:15.400 Instead, just the number is maybe 00:39:15.400 --> 00:39:20.640 your best measure for the health of the population. 00:39:20.640 --> 00:39:24.140 So I think that yeah, maybe I won't do the linearization, 00:39:24.140 --> 00:39:26.410 but I encourage you to do it. 00:39:26.410 --> 00:39:30.270 Because, that's what's going to be relevant. 00:39:30.270 --> 00:39:31.810 In particular, what you want to know 00:39:31.810 --> 00:39:36.280 is if there's some-- If you linearize around there some, 00:39:36.280 --> 00:39:41.330 just be clear, there's some n equilibrium. 00:39:41.330 --> 00:39:44.250 OK there's some equilibrium, and then you want to ask, 00:39:44.250 --> 00:39:49.240 well OK now you want to say if your n is equal to some n 00:39:49.240 --> 00:39:52.330 equilibrium, plus some epsilon. 00:39:52.330 --> 00:39:56.910 Then, if you plug it into the N dot function, 00:39:56.910 --> 00:39:58.730 you should get an equation that is 00:39:58.730 --> 00:40:01.319 something where this epsilon dot is 00:40:01.319 --> 00:40:02.610 going to be equal to something. 00:40:02.610 --> 00:40:03.735 What should it be equal to? 00:40:07.580 --> 00:40:10.240 Something times epsilon. 00:40:10.240 --> 00:40:11.270 And what something? 00:40:14.002 --> 00:40:15.710 AUDIENCE: What we've been calling lambda? 00:40:15.710 --> 00:40:18.400 PROFESSOR: What we've been calling labmda. 00:40:18.400 --> 00:40:21.210 So this is saying that if you go a little bit away 00:40:21.210 --> 00:40:25.351 from the equilibrium, so small epsilon here, 00:40:25.351 --> 00:40:27.600 that epsilon-- The solution to this is an exponential, 00:40:27.600 --> 00:40:29.340 it's going to get exponential decay, 00:40:29.340 --> 00:40:32.120 assuming that lambda is what? 00:40:32.120 --> 00:40:32.750 Negative. 00:40:32.750 --> 00:40:37.512 And that's the definition of-- is 00:40:37.512 --> 00:40:39.220 When we say this is a stable fixed point, 00:40:39.220 --> 00:40:40.760 this is unstable, that's equivalent to saying 00:40:40.760 --> 00:40:42.980 that the lambda here is negative and the lambda here 00:40:42.980 --> 00:40:43.480 is positive. 00:40:49.020 --> 00:40:52.440 AUDIENCE: That's a very easy argument. 00:40:52.440 --> 00:40:55.250 They exchange stability. 00:40:55.250 --> 00:40:57.270 So you know that lambda has to be completely 0. 00:40:57.270 --> 00:40:59.410 PROFESSOR: Yes, this is the point. 00:40:59.410 --> 00:41:03.679 So we know that lambda here is positive lambda-- I'm sorry, 00:41:03.679 --> 00:41:05.720 lambda here is negative, lambda here is positive. 00:41:05.720 --> 00:41:07.099 These fixed points at this point, 00:41:07.099 --> 00:41:08.890 they're going to exchange stability so that 00:41:08.890 --> 00:41:10.989 means they had to be equal. 00:41:10.989 --> 00:41:12.780 And where they exchange is when it crosses. 00:41:12.780 --> 00:41:16.520 So they both-- Well, I guess independently this 00:41:16.520 --> 00:41:18.790 went-- We should be able to draw this, 00:41:18.790 --> 00:41:20.600 although I'm a little bit worried 00:41:20.600 --> 00:41:22.270 that once I try to draw something 00:41:22.270 --> 00:41:23.353 I'm going to get confused. 00:41:23.353 --> 00:41:25.355 But this is a useful exercise. 00:41:29.240 --> 00:41:34.360 In particular, if we plot as a function of delta. 00:41:34.360 --> 00:41:38.890 So here, we have the lambdas, and we have this delta. 00:41:38.890 --> 00:41:42.350 and a delta equal to r what's going to happen 00:41:42.350 --> 00:41:45.974 is that we're going to get the one 00:41:45.974 --> 00:41:47.140 guy's going to go like this. 00:41:55.220 --> 00:41:58.080 Something like that. 00:41:58.080 --> 00:42:01.800 Given, that the fixed points went from stable to unstable, 00:42:01.800 --> 00:42:03.757 unstable to stable, so they had to cross 0. 00:42:03.757 --> 00:42:05.340 So if they both cross 0 at that point, 00:42:05.340 --> 00:42:09.970 they're going to be equal n zero. 00:42:09.970 --> 00:42:11.866 You guys agree? 00:42:11.866 --> 00:42:15.642 AUDIENCE: Do we know that the eigenvalues are going to change 00:42:15.642 --> 00:42:18.207 linearly with the death rate? 00:42:18.207 --> 00:42:18.790 PROFESSOR: No. 00:42:18.790 --> 00:42:20.130 So in general, they don't. 00:42:25.670 --> 00:42:29.970 I'm trying to think if close to the bifurcation, 00:42:29.970 --> 00:42:33.115 whether I can say. 00:42:33.115 --> 00:42:35.440 The mathematicians I'm sure can say something. 00:42:35.440 --> 00:42:38.934 I'm not going to. 00:42:38.934 --> 00:42:40.725 AUDIENCE: Does this also mean that whenever 00:42:40.725 --> 00:42:44.250 we're on a steady state, approaching bifurcation, 00:42:44.250 --> 00:42:48.800 we'll always see the critical slowing down? 00:42:48.800 --> 00:42:53.250 If it's a stable fixed point at a critical value. 00:42:53.250 --> 00:42:58.050 PROFESSOR: I think that the statement is that in principle, 00:42:58.050 --> 00:42:59.974 critical slowing down occurs. 00:42:59.974 --> 00:43:01.890 But that does not mean that you'll necessarily 00:43:01.890 --> 00:43:03.000 be able to see it. 00:43:03.000 --> 00:43:05.416 Because it could be, there's just too much noise, or this, 00:43:05.416 --> 00:43:07.620 or that. 00:43:07.620 --> 00:43:11.540 AUDIENCE: So something that I was thinking about related 00:43:11.540 --> 00:43:15.210 to that is-- So in this case, it seems like everything 00:43:15.210 --> 00:43:20.970 about the system, the fluctuations, or anything, 00:43:20.970 --> 00:43:26.282 should be getting very small, as you approach that. 00:43:26.282 --> 00:43:27.990 PROFESSOR: In principle, the fluctuations 00:43:27.990 --> 00:43:29.710 are supposed to grow. 00:43:29.710 --> 00:43:32.510 Although this is assuming that the strength of the noise 00:43:32.510 --> 00:43:33.950 is constant. 00:43:33.950 --> 00:43:36.310 And in this case, the strength of the noise 00:43:36.310 --> 00:43:39.220 may not be constant, because if it's demographic noise, 00:43:39.220 --> 00:43:41.620 and the size of your population is going down. 00:43:41.620 --> 00:43:43.850 This is actually subtle then, I think. 00:43:47.260 --> 00:43:49.820 And when we typically talk about these dynamics, 00:43:49.820 --> 00:43:52.480 we're assuming that there's an added noise source that 00:43:52.480 --> 00:43:56.030 is independent of the variable. 00:43:56.030 --> 00:43:57.790 But if it's demographic fluctuations 00:43:57.790 --> 00:43:59.706 you're talking about, that that won't be true. 00:44:01.970 --> 00:44:04.190 I think that it's also useful to draw, 00:44:04.190 --> 00:44:06.060 to get some intuition around this, 00:44:06.060 --> 00:44:08.840 based on this idea of an effective potential. 00:44:08.840 --> 00:44:10.810 So these are one-dimensional systems, 00:44:10.810 --> 00:44:14.920 which means you can always write down an effective potential. 00:44:14.920 --> 00:44:21.020 And in this case, what it's going to look like is-- 00:44:21.020 --> 00:44:24.930 This is some u effective as a function of the population 00:44:24.930 --> 00:44:26.150 size. 00:44:26.150 --> 00:44:32.080 And up here, we have some stable state here 00:44:32.080 --> 00:44:38.250 that corresponds to something that looks like this. 00:44:38.250 --> 00:44:40.070 So it starts out maybe at k. 00:44:40.070 --> 00:44:42.720 What happens is, as the death rate increases, 00:44:42.720 --> 00:44:53.110 this thing kind of turns-- Or, maybe I didn't quite make it. 00:44:53.110 --> 00:44:57.120 So what you see is that as the death rate 00:44:57.120 --> 00:45:00.920 here-- So, as we go down the death rate is going up. 00:45:04.190 --> 00:45:06.670 So that what happens is the effective potential, 00:45:06.670 --> 00:45:10.110 describing the dynamics of the population near the equilibrium 00:45:10.110 --> 00:45:12.702 is broadening out. 00:45:12.702 --> 00:45:14.160 So you can think about the dynamics 00:45:14.160 --> 00:45:17.930 as being equivalent to an overdamped particle 00:45:17.930 --> 00:45:19.710 in effective potential. 00:45:19.710 --> 00:45:22.700 So for example, for those of you that have studied single 00:45:22.700 --> 00:45:24.530 molecule biophysics kinds of things, 00:45:24.530 --> 00:45:28.760 this could be thought of as a [? poly-centered ?] bead 00:45:28.760 --> 00:45:30.860 that's trapped in a the laser trap. 00:45:30.860 --> 00:45:33.500 And as you turn down the power of your laser, 00:45:33.500 --> 00:45:36.660 then the spring constant describing the dynamics 00:45:36.660 --> 00:45:40.290 of that B in the trap, it gets weaker and weaker. 00:45:40.290 --> 00:45:42.160 So the potential broadens, and that 00:45:42.160 --> 00:45:45.680 means that for fixed injection noise-- fixed temperature-- you 00:45:45.680 --> 00:45:50.000 get these effects, where the fluctuations will increase 00:45:50.000 --> 00:45:52.610 in magnitude, and you get an increase 00:45:52.610 --> 00:45:54.610 in this autocorrelation time. 00:45:54.610 --> 00:45:56.660 Because for example, the bead in the trap, 00:45:56.660 --> 00:45:58.180 the autocorrelation time is indeed 00:45:58.180 --> 00:46:06.360 just equal to the relaxation time of the bead in the trap. 00:46:06.360 --> 00:46:09.060 And the bifurcation occurs right at this point 00:46:09.060 --> 00:46:13.230 here, where you see that this local potential is, 00:46:13.230 --> 00:46:15.580 or the local minimum, is disappearing. 00:46:15.580 --> 00:46:19.410 That's when this stable fixed point is gone. 00:46:19.410 --> 00:46:21.440 And then you just kind of fall off here. 00:46:21.440 --> 00:46:26.970 And that's what leads to this collapse, which 00:46:26.970 --> 00:46:27.865 I have covered up. 00:46:27.865 --> 00:46:29.115 That leads to the bifurcation. 00:46:35.340 --> 00:46:38.320 And you should, for example, be able to draw 00:46:38.320 --> 00:46:40.095 an effective potential here as well. 00:46:42.890 --> 00:46:45.900 But I'll let you play with it. 00:46:49.496 --> 00:46:51.620 But you can see the location of this unstable fixed 00:46:51.620 --> 00:46:52.700 point shifting as well. 00:46:56.856 --> 00:47:00.780 I'm not sure how good my drawing is. 00:47:00.780 --> 00:47:03.020 Because in principal, this unstable fixed point 00:47:03.020 --> 00:47:05.661 should have to come together. 00:47:05.661 --> 00:47:07.660 I'm not sure if that's very clear in my drawing. 00:47:11.950 --> 00:47:15.900 Just to summarize this discussion, 00:47:15.900 --> 00:47:18.800 I think there are a few ways that you 00:47:18.800 --> 00:47:21.480 can think of these early warning indicators. 00:47:21.480 --> 00:47:24.700 And there's a diagram that I like to make, 00:47:24.700 --> 00:47:27.640 that I think makes things more clear, for me, at least. 00:47:27.640 --> 00:47:34.980 Which is that if you look at this population 00:47:34.980 --> 00:47:39.030 as a function of time, it goes. 00:47:39.030 --> 00:47:44.390 And if there's an environment quality as a function of time. 00:47:44.390 --> 00:47:47.950 If the environment has a perturbation, 00:47:47.950 --> 00:47:52.180 then the population will shrink, and then you'll get recovery. 00:47:52.180 --> 00:47:55.822 And this thing here tells you about the time for recovery. 00:47:55.822 --> 00:47:57.780 And this basic phenomenon critical slowing down 00:47:57.780 --> 00:48:00.910 tells you that as the tipping point approaches, 00:48:00.910 --> 00:48:03.176 as you approach the bifurcation, that time 00:48:03.176 --> 00:48:05.300 to recover from this perturbation is going to grow. 00:48:08.660 --> 00:48:10.470 Of course, the other thing that you can say 00:48:10.470 --> 00:48:14.370 is that even in the absence of a defied perturbation, even 00:48:14.370 --> 00:48:16.760 if the environment is constant over time, 00:48:16.760 --> 00:48:20.050 there could just a natural noise in the system that 00:48:20.050 --> 00:48:21.780 will fluctuate, just like temperature, 00:48:21.780 --> 00:48:23.100 for the bead in the trap. 00:48:23.100 --> 00:48:25.050 And principle, then you can measure 00:48:25.050 --> 00:48:27.870 the size of the fluctuations, the variance, 00:48:27.870 --> 00:48:31.010 as well as the autocorrelation time 00:48:31.010 --> 00:48:34.540 tau, which is, for civil systems, equal to t. 00:48:34.540 --> 00:48:39.323 And this also grows as you approach the bifurcation 00:48:39.323 --> 00:48:42.269 AUDIENCE: Is it [? clear ?] that-- Because the perturbation 00:48:42.269 --> 00:48:46.540 [INAUDIBLE] that you're talking about is I think n [INAUDIBLE] 00:48:46.540 --> 00:48:49.160 perturbed n into epsilon, and then it 00:48:49.160 --> 00:48:50.360 relaxes the steady state. 00:48:50.360 --> 00:48:53.002 Is it [INAUDIBLE] to perturb the environment? 00:48:53.002 --> 00:48:53.960 PROFESSOR: Yeah, right. 00:48:53.960 --> 00:48:55.540 So that's a good question. 00:48:55.540 --> 00:48:57.680 What I've been talking about is a situation 00:48:57.680 --> 00:49:00.457 where you really literally pull it away, and then you let go. 00:49:00.457 --> 00:49:02.540 And that's equivalent for in the bead in the trap, 00:49:02.540 --> 00:49:04.581 that you pull the bead away, and then you let go, 00:49:04.581 --> 00:49:05.717 and you watch it come back. 00:49:05.717 --> 00:49:07.550 The situation in the case of the environment 00:49:07.550 --> 00:49:11.270 is that, if it's a kind of a sudden shift-- It could 00:49:11.270 --> 00:49:15.890 be anything, it could be a brief change in-- Well it could just 00:49:15.890 --> 00:49:16.490 be delta. 00:49:16.490 --> 00:49:19.249 So it could be that over some period of time, 00:49:19.249 --> 00:49:21.290 the death rate increases, or some period of time, 00:49:21.290 --> 00:49:23.430 the growth rate decreases. 00:49:23.430 --> 00:49:26.105 Or it could be that for-- The perturbations don't 00:49:26.105 --> 00:49:27.860 have to be bad, they can also be good, 00:49:27.860 --> 00:49:29.450 and you actually get the same. 00:49:29.450 --> 00:49:31.420 The principal for small perturbations to go, 00:49:31.420 --> 00:49:32.545 it's the same thing anyway. 00:49:37.680 --> 00:49:39.790 So these are early warning indicators 00:49:39.790 --> 00:49:41.990 of an impending transition that are 00:49:41.990 --> 00:49:43.205 based on temporal indicators. 00:49:45.950 --> 00:49:48.990 And one of the things that we've been excited about in my group 00:49:48.990 --> 00:49:51.080 is actually just trying to measure these things 00:49:51.080 --> 00:49:53.300 in a well controlled laboratory population. 00:49:53.300 --> 00:49:55.097 Since in our case, we're using yeast 00:49:55.097 --> 00:49:57.430 that are engaging in what you might call a group hunting 00:49:57.430 --> 00:50:00.060 behavior, where they secrete an enzyme that breaks down sugar, 00:50:00.060 --> 00:50:02.143 and that kind of leads to this cooperative growth. 00:50:02.143 --> 00:50:04.270 And at least in that case, we can experimentally 00:50:04.270 --> 00:50:06.230 measure an increase in all of these things 00:50:06.230 --> 00:50:07.563 as we approach this bifurcation. 00:50:10.549 --> 00:50:12.090 The other thing you might think about 00:50:12.090 --> 00:50:14.620 is what happens for spatially extended populations? 00:50:14.620 --> 00:50:17.640 We'll talk more about spatial populations on Thursday, 00:50:17.640 --> 00:50:20.180 but just while we're here, it's useful to think about it. 00:50:20.180 --> 00:50:22.596 So instead of thinking about just n as a function of time, 00:50:22.596 --> 00:50:24.690 now you want to think about density of population. 00:50:24.690 --> 00:50:26.240 So one thing that people talked about 00:50:26.240 --> 00:50:29.300 is that-- Sorry, this is not a function of time, 00:50:29.300 --> 00:50:31.230 it's now a function of position x. 00:50:31.230 --> 00:50:33.259 Density is a function of position. 00:50:33.259 --> 00:50:35.050 Now, environment is a function of position. 00:50:35.050 --> 00:50:37.980 If you have a uniform environment over position 00:50:37.980 --> 00:50:39.442 or space, then in principle you can 00:50:39.442 --> 00:50:40.650 look at density fluctuations. 00:50:40.650 --> 00:50:43.400 And this is something that people have talked about. 00:50:43.400 --> 00:50:45.850 One of things that we have argued 00:50:45.850 --> 00:50:49.680 is that there should be a spatial analog to this recovery 00:50:49.680 --> 00:50:50.614 time. 00:50:50.614 --> 00:50:52.030 So that corresponds to a situation 00:50:52.030 --> 00:50:54.239 where you have environment as a function of position. 00:50:54.239 --> 00:50:56.363 Like for example, we have a sharp boundary, or just 00:50:56.363 --> 00:50:57.460 a region of poor quality. 00:50:57.460 --> 00:51:01.120 And in that case, you can look at the density 00:51:01.120 --> 00:51:05.240 as a function of position, and you get some linked scale here. 00:51:05.240 --> 00:51:07.862 So the statement here is that, just because you're 00:51:07.862 --> 00:51:09.320 in a region of high quality doesn't 00:51:09.320 --> 00:51:10.990 mean that you're at your equilibrium density, 00:51:10.990 --> 00:51:12.698 because if you're close to a poor region, 00:51:12.698 --> 00:51:14.480 so if you're close to hunting grounds, 00:51:14.480 --> 00:51:17.724 then you'll get local depletion of the population. 00:51:17.724 --> 00:51:19.890 You have to get some distance away from a bad region 00:51:19.890 --> 00:51:21.223 before you get your equilibrium. 00:51:21.223 --> 00:51:23.960 That distance or recovery length tells you 00:51:23.960 --> 00:51:26.500 about the quality of the environment. 00:51:26.500 --> 00:51:29.120 And it's the quality of the environment at this region, 00:51:29.120 --> 00:51:30.430 the good region. 00:51:30.430 --> 00:51:31.632 Where you are. 00:51:31.632 --> 00:51:33.090 At least in the laboratory, this is 00:51:33.090 --> 00:51:34.120 something that's actually much easier 00:51:34.120 --> 00:51:35.350 to measure than other things. 00:51:35.350 --> 00:51:38.670 Because, this is a deterministic phenomenon. 00:51:38.670 --> 00:51:41.670 You don't have to measure fluctuations over time 00:51:41.670 --> 00:51:43.110 with a high quality time series. 00:51:43.110 --> 00:51:45.200 You don't have to wait for a perturbation in time, 00:51:45.200 --> 00:51:46.870 like a drought, and look at recovery. 00:51:46.870 --> 00:51:49.870 Instead, you just take advantage of natural variations 00:51:49.870 --> 00:51:53.240 in quality of the environment over space, position, and then 00:51:53.240 --> 00:51:54.860 you measure profiles. 00:51:54.860 --> 00:51:58.370 We have a collaboration with a professor 00:51:58.370 --> 00:52:01.270 at the University of Pisa who does field ecology experiments 00:52:01.270 --> 00:52:03.830 with these algal mats on intertidal communities, 00:52:03.830 --> 00:52:05.265 on islands in the Mediterranean. 00:52:08.490 --> 00:52:10.410 And we have a [? pronates ?] that we 00:52:10.410 --> 00:52:13.540 can measure this in those island communities as well. 00:52:17.094 --> 00:52:18.760 Are there any questions about this stuff 00:52:18.760 --> 00:52:20.652 before we switch gears? 00:52:20.652 --> 00:52:21.152 Yes? 00:52:24.068 --> 00:52:27.660 AUDIENCE: In that example, is it-- How much control can you 00:52:27.660 --> 00:52:31.224 have over actually [? saying ?] the quality to the environment? 00:52:33.950 --> 00:52:37.190 PROFESSOR: So in that manipulation, what they did was 00:52:37.190 --> 00:52:41.870 they basically go in and they-- So there these, 00:52:41.870 --> 00:52:43.820 they're like miniature forest somehow. 00:52:43.820 --> 00:52:47.460 So they're little-- They have alternative stable states. 00:52:47.460 --> 00:52:50.300 What they actually do, they go and they like chip away 00:52:50.300 --> 00:52:52.390 at the rock to remove the things. 00:52:52.390 --> 00:52:54.950 And they do it over some range. 00:52:54.950 --> 00:52:57.670 So they experimentally basically make it a 00:52:57.670 --> 00:53:00.360 challenging environment. 00:53:00.360 --> 00:53:03.380 And apparently, they have permission to do this. 00:53:03.380 --> 00:53:06.690 So don't try that at home without asking 00:53:06.690 --> 00:53:08.940 the proper authorities. 00:53:08.940 --> 00:53:11.850 Any other questions about this? 00:53:16.350 --> 00:53:19.030 So I think that the nice thing about studying 00:53:19.030 --> 00:53:21.350 all these dynamics just for a single population, 00:53:21.350 --> 00:53:25.350 is that it really clarifies the essential ingredients, in order 00:53:25.350 --> 00:53:27.280 to get things like sudden transitions. 00:53:27.280 --> 00:53:29.950 Of course, in natural populations, 00:53:29.950 --> 00:53:31.660 we have many, many species and tracking 00:53:31.660 --> 00:53:32.826 in lots of complicated ways. 00:53:32.826 --> 00:53:35.030 But we would like to understand those things, 00:53:35.030 --> 00:53:36.710 but I think since it's so complicated, 00:53:36.710 --> 00:53:38.830 you kind of like, oh anything can happen, 00:53:38.830 --> 00:53:40.790 and you get a little bit discouraged 00:53:40.790 --> 00:53:42.337 from thinking deeply about it. 00:53:42.337 --> 00:53:44.670 Because, you just think it's going to be too complicated 00:53:44.670 --> 00:53:45.760 to try to understand. 00:53:45.760 --> 00:53:48.920 I very much like the idea of trying to really hone 00:53:48.920 --> 00:53:51.480 your intuition on the simplest possible situation like this, 00:53:51.480 --> 00:53:56.510 and then bringing that intuition to more complicated or complex 00:53:56.510 --> 00:53:57.460 situations. 00:53:57.460 --> 00:53:58.140 Yeah? 00:53:58.140 --> 00:53:59.806 AUDIENCE: Can you partially [? adjust ?] 00:53:59.806 --> 00:54:01.900 But what I want to ask you is whether any of this 00:54:01.900 --> 00:54:04.250 can be applied to human society? 00:54:04.250 --> 00:54:05.620 PROFESSOR: Oh, yeah. 00:54:05.620 --> 00:54:08.270 So whether this could be applied to human society. 00:54:08.270 --> 00:54:09.660 It's a good question. 00:54:09.660 --> 00:54:11.360 People certainly try to. 00:54:11.360 --> 00:54:16.330 I think you'll always have to decide what we mean by apply. 00:54:16.330 --> 00:54:18.570 I would say that this basic idea that there 00:54:18.570 --> 00:54:20.280 could be these feedback loops that 00:54:20.280 --> 00:54:22.630 can lead to sudden transitions in complex systems. 00:54:22.630 --> 00:54:26.044 I think this is a very robust phenomenon, in the sense 00:54:26.044 --> 00:54:28.210 that I think if you have strong enough interactions, 00:54:28.210 --> 00:54:31.750 then I think you kind of expect it to be true. 00:54:31.750 --> 00:54:36.780 Another question is whether you could quantitatively predict 00:54:36.780 --> 00:54:38.960 when that's going to happen. 00:54:38.960 --> 00:54:42.040 There was a recent article written in science or nature 00:54:42.040 --> 00:54:47.540 about potential tipping points in human society 00:54:47.540 --> 00:54:49.000 on a global scale. 00:54:49.000 --> 00:54:51.620 Things in terms of productivity of crops. 00:54:51.620 --> 00:54:54.160 And so they have a question mark about, 00:54:54.160 --> 00:55:00.150 they say 2050, question mark, collapse maybe. 00:55:00.150 --> 00:55:03.770 I say it's very important to be thinking about these things in. 00:55:03.770 --> 00:55:07.710 But the question is what to do, whether given the uncertainties 00:55:07.710 --> 00:55:10.050 and your knowledge of where these tipping points might 00:55:10.050 --> 00:55:12.700 occur, it's always hard to know whether you could make 00:55:12.700 --> 00:55:16.330 a strong argument saying, oh we have to stop fishing here 00:55:16.330 --> 00:55:17.780 because it's going to collapse. 00:55:20.520 --> 00:55:22.880 There's always uncertainty in your decision making. 00:55:22.880 --> 00:55:25.130 But certainly in the context of climate regime shifts, 00:55:25.130 --> 00:55:26.730 people are worried about these sorts 00:55:26.730 --> 00:55:29.399 of feedback loops and the North Atlantic Oscillation 00:55:29.399 --> 00:55:29.940 and so forth. 00:55:29.940 --> 00:55:31.523 And I'm not at all an expert for that, 00:55:31.523 --> 00:55:33.710 so I don't know whether we should be worried. 00:55:33.710 --> 00:55:38.390 But it's at least good to remember 00:55:38.390 --> 00:55:44.750 that systems can respond in dramatic ways to small changes. 00:55:44.750 --> 00:55:47.650 And then, you have to decide what to do with that knowledge, 00:55:47.650 --> 00:55:50.034 and I think that's more of a judgement call. 00:55:52.998 --> 00:55:56.950 AUDIENCE: In the time example, you had death, 00:55:56.950 --> 00:55:59.895 it was a death rate. 00:55:59.895 --> 00:56:02.557 Is that in a [INAUDIBLE] space? 00:56:02.557 --> 00:56:03.140 PROFESSOR: OK. 00:56:03.140 --> 00:56:05.280 So I want to be a little bit maybe more clear. 00:56:05.280 --> 00:56:09.390 The claim is that-- Like the death, the delta there 00:56:09.390 --> 00:56:11.380 that we're thinking about, that didn't 00:56:11.380 --> 00:56:12.870 have to be the perturbation. 00:56:12.870 --> 00:56:15.467 It could be, but it didn't have to be. 00:56:15.467 --> 00:56:17.300 When we're plotting the bifurcation diagram, 00:56:17.300 --> 00:56:19.216 we're assuming implicitly somehow that there's 00:56:19.216 --> 00:56:21.614 a separation of time scales, such that this delta might 00:56:21.614 --> 00:56:23.530 be changing very slowly, and then other things 00:56:23.530 --> 00:56:25.480 are changing more rapidly. 00:56:25.480 --> 00:56:27.050 So this bifurcation diagram, it's 00:56:27.050 --> 00:56:29.630 really that you're tracking it. 00:56:29.630 --> 00:56:33.380 So in the context of this n as a function of delta, 00:56:33.380 --> 00:56:34.620 this thing goes like this. 00:56:34.620 --> 00:56:37.290 And the idea is that oh, slowly you're 00:56:37.290 --> 00:56:39.200 getting more and more agricultural runoff, 00:56:39.200 --> 00:56:40.920 or the temperature is increasing. 00:56:40.920 --> 00:56:42.794 Doesn't have to be human induced, by the way, 00:56:42.794 --> 00:56:45.070 it could be something else. 00:56:45.070 --> 00:56:47.810 So the idea is that the population is slowly 00:56:47.810 --> 00:56:49.200 changing like this. 00:56:49.200 --> 00:56:53.480 And then eventually maybe you get this collapse. 00:56:53.480 --> 00:56:57.100 Now, a perturbation could be something that could just 00:56:57.100 --> 00:57:00.290 be a drought, or something that is independent of this delta. 00:57:00.290 --> 00:57:05.367 But the statement is that, over here-- Now I'm 00:57:05.367 --> 00:57:06.450 going to be mixing things. 00:57:09.570 --> 00:57:13.170 At low delta, the claim is that you should get a dip and then 00:57:13.170 --> 00:57:14.910 rapid recovery. 00:57:14.910 --> 00:57:17.252 As the delta gets bigger, you get maybe even 00:57:17.252 --> 00:57:19.710 a larger-- The same perturbation could [? principally ?] do 00:57:19.710 --> 00:57:22.650 a larger dip, and it takes longer to get recovery. 00:57:32.357 --> 00:57:33.940 This actually brings up another point, 00:57:33.940 --> 00:57:39.030 which is that these could all be the same perturbations. 00:57:39.030 --> 00:57:45.810 So this is a case where delta is increasing as we go down. 00:57:45.810 --> 00:57:48.420 So they could be the same perturbation here. 00:57:48.420 --> 00:57:50.830 Here you survive, here you survive. 00:57:50.830 --> 00:57:53.130 But then that same perturbation that was survived here 00:57:53.130 --> 00:57:56.290 could be the same magnitude drought, just as 00:57:56.290 --> 00:57:59.351 delta increases you expect this-- 00:57:59.351 --> 00:58:01.720 This may be able to push the system 00:58:01.720 --> 00:58:03.530 past the unstable fixed point, because 00:58:03.530 --> 00:58:04.956 of a loss of resilience. 00:58:04.956 --> 00:58:06.330 This is something that we've seen 00:58:06.330 --> 00:58:07.788 in a variety of different contexts. 00:58:07.788 --> 00:58:10.280 I think it's a very general phenomenon. 00:58:10.280 --> 00:58:11.759 Just because the separation here is 00:58:11.759 --> 00:58:13.800 some measure of the resilience of the population, 00:58:13.800 --> 00:58:16.670 or the ability of the population to withstand perturbations. 00:58:16.670 --> 00:58:19.980 We can see that the resilience shrinks as we 00:58:19.980 --> 00:58:21.230 get close to this bifurcation. 00:58:30.380 --> 00:58:32.090 Did that answer your question? 00:58:32.090 --> 00:58:32.820 Sort of? 00:58:32.820 --> 00:58:33.640 Not really. 00:58:33.640 --> 00:58:35.000 AUDIENCE: No, but it answered another question. 00:58:35.000 --> 00:58:35.550 PROFESSOR: It answered another question? 00:58:35.550 --> 00:58:36.442 Oh good. 00:58:36.442 --> 00:58:37.900 I'm glad that I answered something. 00:58:37.900 --> 00:58:40.460 But what was your-- Because you were saying 00:58:40.460 --> 00:58:43.955 is it the death rate that is causing this perturbation? 00:58:43.955 --> 00:58:44.945 AUDIENCE: Well, no. 00:58:44.945 --> 00:58:47.327 I was wondering for [? per ?] space. 00:58:47.327 --> 00:58:47.910 PROFESSOR: OK. 00:58:47.910 --> 00:58:49.180 Oh yes, I forgot. 00:58:49.180 --> 00:58:53.100 Yes, I do remember that you did ask that. 00:58:53.100 --> 00:58:58.570 So the idea here is that this could be a region that we fish, 00:58:58.570 --> 00:59:01.680 and this could be region that we don't fish, for example. 00:59:01.680 --> 00:59:04.980 Or it could be that, here there's 00:59:04.980 --> 00:59:07.860 just not as much food for the organism as there are here. 00:59:07.860 --> 00:59:09.730 So just the idea is that you would 00:59:09.730 --> 00:59:12.870 need kind of a sudden sharp boundary between regions 00:59:12.870 --> 00:59:13.760 of different quality. 00:59:13.760 --> 00:59:15.343 And that's kind of the situation where 00:59:15.343 --> 00:59:18.315 you would be able to measure this recovery length. 00:59:18.315 --> 00:59:19.690 And the basic reason for that, is 00:59:19.690 --> 00:59:22.510 that if the environmental quality is something 00:59:22.510 --> 00:59:26.370 that changes very slowly, then the population density 00:59:26.370 --> 00:59:29.060 will just track that. 00:59:29.060 --> 00:59:33.680 So then you can't use that to measure this recovery length. 00:59:33.680 --> 00:59:36.390 Of course, the same statement is true here. 00:59:36.390 --> 00:59:39.120 That if you have environmental perturbations that 00:59:39.120 --> 00:59:40.870 are changing slowly over time, then 00:59:40.870 --> 00:59:42.681 that also can complicate this picture. 00:59:48.940 --> 00:59:51.630 Any other questions about that before we 00:59:51.630 --> 00:59:53.890 move towards multispecies ecosystems? 00:59:53.890 --> 00:59:55.970 We're going to start with two species. 01:00:07.265 --> 01:00:08.640 So what I'm going to do is I want 01:00:08.640 --> 01:00:10.850 to spend the rest of the class talking 01:00:10.850 --> 01:00:12.910 about Lotka-Volterra populations. 01:00:12.910 --> 01:00:15.400 And we're going to move the rock, paper, 01:00:15.400 --> 01:00:18.120 scissors discussion to Thursday, because it's 01:00:18.120 --> 01:00:20.570 a nice spatial example anyways. 01:00:20.570 --> 01:00:23.810 So let's think about Lotka-Volterra. 01:00:23.810 --> 01:00:27.380 Now, there's a lot. 01:00:27.380 --> 01:00:28.980 Lotka-Volterra competition model. 01:00:33.329 --> 01:00:34.870 So the assumption is that we're going 01:00:34.870 --> 01:00:38.880 to make here is that we have these two species that 01:00:38.880 --> 01:00:40.950 going to be interacting, but they're 01:00:40.950 --> 01:00:44.722 going to interact in kind of the simplest way 01:00:44.722 --> 01:00:45.680 that you might imagine. 01:00:45.680 --> 01:00:50.614 Which is that, if we plot the derivative of population size 01:00:50.614 --> 01:00:53.280 n1 as a function of time, we get something that looks like this. 01:01:12.846 --> 01:01:14.220 Now the first thing that you want 01:01:14.220 --> 01:01:16.470 to do when you see something like this is just to make 01:01:16.470 --> 01:01:17.970 sense the basic equation. 01:01:26.170 --> 01:01:33.780 In particular, in the absence of the other species, 01:01:33.780 --> 01:01:36.235 what happens to these populations? 01:01:38.810 --> 01:01:45.670 We're going to assume here maybe first is that r1 and r2 are 01:01:45.670 --> 01:01:46.670 both greater than 0. 01:01:49.752 --> 01:01:51.460 It's in the absence of the other species. 01:01:56.100 --> 01:02:00.260 Do these populations survive or not? 01:02:00.260 --> 01:02:00.990 Ready? 01:02:00.990 --> 01:02:06.210 Three, we're going do it A, yes, B, no, our typical. 01:02:06.210 --> 01:02:07.710 All right, absence of a species. 01:02:11.570 --> 01:02:12.070 ready? 01:02:12.070 --> 01:02:14.580 Three, two, one, survival? 01:02:14.580 --> 01:02:15.550 Yeah, they survive. 01:02:15.550 --> 01:02:18.490 For sure, because I told you that these guys are 01:02:18.490 --> 01:02:21.360 greater than 0, that means-- We can think about n1. 01:02:21.360 --> 01:02:27.100 In the absence n2, what is this equation? 01:02:27.100 --> 01:02:28.920 Logistic. 01:02:28.920 --> 01:02:33.210 Now, if we wanted to think about these as describing 01:02:33.210 --> 01:02:36.720 competitive interactions, what does that tell us 01:02:36.720 --> 01:02:40.590 about the sign of the betas? 01:02:40.590 --> 01:02:45.940 Are the betas greater than 0, or are they less than 0? 01:02:48.842 --> 01:02:50.300 I'll think about it for 10 seconds. 01:03:02.990 --> 01:03:04.998 All right. 01:03:04.998 --> 01:03:07.230 Do you need more time? 01:03:07.230 --> 01:03:07.760 Ready? 01:03:07.760 --> 01:03:12.080 Three, two, one. 01:03:12.080 --> 01:03:16.330 Competitive means that the betas are greater than 0. 01:03:16.330 --> 01:03:20.420 And what is the assumption somehow that's going into this? 01:03:23.940 --> 01:03:28.630 And we should remember that this is a minus everything up here. 01:03:32.039 --> 01:03:34.961 AUDIENCE: I guess that somehow these two things 01:03:34.961 --> 01:03:39.344 are-- there's a fixed amount of some resource, 01:03:39.344 --> 01:03:41.497 that both of these things need. 01:03:41.497 --> 01:03:43.705 PROFESSOR: OK, so there could be some fixed resource. 01:03:50.157 --> 01:03:54.450 AUDIENCE: It's basically like an effective carrying capacity. 01:03:54.450 --> 01:03:56.769 PROFESSOR: That's right, so somehow it's 01:03:56.769 --> 01:03:58.560 modulating the effect of carrying capacity. 01:04:01.210 --> 01:04:05.430 And how is it that you would describe these betas? 01:04:09.010 --> 01:04:12.970 AUDIENCE: Sort of seems like you expect that the beta should 01:04:12.970 --> 01:04:16.373 be probably less than one. 01:04:16.373 --> 01:04:21.100 [INAUDIBLE] I guess it doesn't have to be less than one. 01:04:23.660 --> 01:04:29.350 Compared to how well n1-- and individual of n1 01:04:29.350 --> 01:04:31.780 takes up some of this. 01:04:31.780 --> 01:04:35.250 This is how much [INAUDIBLE] 01:04:35.250 --> 01:04:38.230 PROFESSOR: That's right. 01:04:38.230 --> 01:04:39.200 I think that's right. 01:04:39.200 --> 01:04:41.950 Now I think we-- There's a question 01:04:41.950 --> 01:04:44.690 about how explicitly to be thinking about these resources. 01:04:44.690 --> 01:04:47.040 Because it could be, it doesn't have to be one resource. 01:04:47.040 --> 01:04:50.530 And this is of course, a very phenomenological model. 01:04:50.530 --> 01:04:52.700 It's lumping all the interact-- All of the ways 01:04:52.700 --> 01:04:57.340 in which a species interacts in just a single parameter beta. 01:04:57.340 --> 01:04:59.820 But it's somehow, the betas are telling us something 01:04:59.820 --> 01:05:04.920 about how much does a member of the other species 01:05:04.920 --> 01:05:08.820 inhibit my growth, as compared to a member of my own species? 01:05:08.820 --> 01:05:13.950 Because we have this n1 here, and just this species one 01:05:13.950 --> 01:05:16.330 will naturally lead to a carrying capacity k1, 01:05:16.330 --> 01:05:17.910 you only have species one. 01:05:17.910 --> 01:05:21.190 Now, the betas are telling you something about 01:05:21.190 --> 01:05:23.280 how much overlap they have in terms of maybe 01:05:23.280 --> 01:05:24.870 [? niche ?] or so. 01:05:24.870 --> 01:05:27.340 Now, if the betas are small, it means 01:05:27.340 --> 01:05:31.270 that they're not competing with each other very much. 01:05:31.270 --> 01:05:33.890 The betas could be larger than one, and what that's saying 01:05:33.890 --> 01:05:36.360 is that a member of this other species 01:05:36.360 --> 01:05:38.720 is inhibiting my growth more than a member 01:05:38.720 --> 01:05:39.485 of my own species. 01:05:51.170 --> 01:05:54.640 So the standard way to analyze these equations 01:05:54.640 --> 01:05:57.630 is to look at these isoclines. 01:05:57.630 --> 01:06:01.960 So basically, if you say, OK well n1 dot is equal to zero, 01:06:01.960 --> 01:06:03.950 what does that mean? 01:06:03.950 --> 01:06:07.060 Well, we can just see that that's equivalent to saying 01:06:07.060 --> 01:06:17.500 that it's n1 plus beta1,2 n2 is equal to k1. 01:06:21.850 --> 01:06:27.430 So this is giving us a relationship between n1 and n2. 01:06:27.430 --> 01:06:33.210 What do these look like on a plane, n1 versus n2 drawings? 01:06:42.220 --> 01:06:47.220 So the parabola, it's a line. 01:06:47.220 --> 01:06:49.810 And indeed, we can think about what happens when each of these 01:06:49.810 --> 01:06:52.350 is equal to one thing or another. 01:06:52.350 --> 01:06:57.010 So if n1-- If n2 is equal to 0, this thing 01:06:57.010 --> 01:06:58.060 is going across at k1. 01:07:02.820 --> 01:07:05.340 And that makes sense, because we already 01:07:05.340 --> 01:07:07.730 decided that in the absence of n2, 01:07:07.730 --> 01:07:09.700 this thing is just following logistic growth, 01:07:09.700 --> 01:07:13.550 where the species one will grow and go to k1. 01:07:13.550 --> 01:07:16.800 And that's a stable fixed point that n1 dot 01:07:16.800 --> 01:07:18.185 has to be equal to zero. 01:07:18.185 --> 01:07:21.019 It's a fixed point, so n1 dot has to be equal to 0. 01:07:21.019 --> 01:07:22.810 And it's going to cross at this other point 01:07:22.810 --> 01:07:33.479 here where n2 is going to be some k1 over beta 1, 2. 01:07:33.479 --> 01:07:34.770 And then we end up with a line. 01:07:43.690 --> 01:07:47.540 Now if we go, and we ask what n2 dot is equal to, 01:07:47.540 --> 01:07:49.820 and this is equal to 0, we're going 01:07:49.820 --> 01:07:52.390 to end up with something that looks very similar. 01:07:52.390 --> 01:07:58.767 But now it's going to be n2 plus beta 2, 1, n1 is equal to k2. 01:07:58.767 --> 01:08:00.100 This is also going to be a line. 01:08:04.620 --> 01:08:10.030 Given what I have said, how many different qualitative outcomes 01:08:10.030 --> 01:08:12.460 can you get in this model? 01:08:12.460 --> 01:08:15.414 Where we just assume-- Where we have beta-- Overall 01:08:15.414 --> 01:08:17.664 I think we're thinking about competitive interactions. 01:08:23.420 --> 01:08:27.029 How many cases do we need to consider? 01:08:33.990 --> 01:08:36.914 Alright, it's going to be a, 0. 01:08:49.385 --> 01:08:52.140 Do you guys understand what I mean by cases? 01:08:58.720 --> 01:09:00.479 No? 01:09:00.479 --> 01:09:03.069 I mean, how many different kind of qualitative outcomes 01:09:03.069 --> 01:09:12.660 can there be in terms of species one or species two winning, 01:09:12.660 --> 01:09:14.420 or can you get coexistence? 01:09:14.420 --> 01:09:17.890 Or how many different kind of qualitative outcomes 01:09:17.890 --> 01:09:18.834 can there be? 01:09:23.180 --> 01:09:26.890 And if you're confused, you can not vote, or give me 01:09:26.890 --> 01:09:27.920 an unhappy face. 01:09:27.920 --> 01:09:29.040 OK, ready? 01:09:29.040 --> 01:09:30.940 Three, two, one. 01:09:33.050 --> 01:09:33.550 So 01:09:33.550 --> 01:09:35.454 We have many Es. 01:09:38.060 --> 01:09:40.640 And can somebody explain why it that this 01:09:40.640 --> 01:09:45.071 is the case, without invoking that you've already studied 01:09:45.071 --> 01:09:46.279 this and you know the answer? 01:09:51.019 --> 01:09:55.470 AUDIENCE: If you add another line, completely on top, 01:09:55.470 --> 01:09:57.920 or under, or it can cross too. 01:09:57.920 --> 01:09:59.730 PROFESSOR: That's right, that's right. 01:09:59.730 --> 01:10:04.320 So the next thing I was about to do is draw another line. 01:10:04.320 --> 01:10:07.190 And the reason I stopped is because there are actually 01:10:07.190 --> 01:10:11.720 four different ways in which this other line can be drawn. 01:10:11.720 --> 01:10:17.560 And basically, you can have another line that does one. 01:10:17.560 --> 01:10:19.306 You can have two. 01:10:19.306 --> 01:10:21.910 You can have three. 01:10:21.910 --> 01:10:23.470 And you can have four. 01:10:26.630 --> 01:10:27.130 OK? 01:10:30.100 --> 01:10:33.875 These two cases-- So this is we'll say one. 01:10:33.875 --> 01:10:35.250 I don't know what order i did it. 01:10:35.250 --> 01:10:39.190 Two, three, and four. 01:10:39.190 --> 01:10:44.280 Two and three, in both cases we have the crossing of the lines. 01:10:44.280 --> 01:10:48.070 So maybe does that mean they're the same outcome, 01:10:48.070 --> 01:10:51.640 the same qualitative outcome? 01:10:51.640 --> 01:10:53.860 No. 01:10:53.860 --> 01:10:57.980 And what will end up happening here is that we're 01:10:57.980 --> 01:11:02.650 going to get cases of-- 01:11:08.680 --> 01:11:10.450 So you basically can get that species 01:11:10.450 --> 01:11:14.310 one dominates independent of starting condition, 01:11:14.310 --> 01:11:16.920 assuming that both are present at the beginning. 01:11:16.920 --> 01:11:18.620 Species two could dominate. 01:11:22.420 --> 01:11:29.010 They could coexist, or you get bistability. 01:11:29.010 --> 01:11:31.945 History dependent, mutual exclusion. 01:11:38.260 --> 01:11:40.210 I just want to draw. 01:11:40.210 --> 01:11:45.820 So this one here was the-- We had a dashed line for n1 dot 01:11:45.820 --> 01:11:48.850 and we had the solid line for this. 01:11:48.850 --> 01:11:49.574 Yes? 01:11:49.574 --> 01:11:51.615 AUDIENCE: Where is [? the breaking ?] [INAUDIBLE] 01:12:03.780 --> 01:12:08.670 PROFESSOR: So I guess if you swap the labels, 01:12:08.670 --> 01:12:09.590 or so, you're saying? 01:12:09.590 --> 01:12:12.100 I guess in the case that if one of them 01:12:12.100 --> 01:12:17.324 is just above the other one, then you can distinguish them. 01:12:17.324 --> 01:12:18.740 So if you have a dashed line here, 01:12:18.740 --> 01:12:22.264 you have a solid line here, that means that they-- 01:12:22.264 --> 01:12:24.724 AUDIENCE: How do I distinguish case 2 and case 3? 01:12:28.190 --> 01:12:28.690 [INAUDIBLE] 01:12:34.290 --> 01:12:38.840 PROFESSOR: Maybe we should try to figure out 01:12:38.840 --> 01:12:44.740 which one's which, and then we can figure it out from there. 01:12:44.740 --> 01:12:47.670 Can somebody remind us, what are these lines again? 01:12:52.070 --> 01:12:53.988 Why are we drawing them, or what do they mean? 01:12:58.260 --> 01:13:03.400 So they're not actually quite a fixed point. 01:13:03.400 --> 01:13:04.170 It's a no incline. 01:13:04.170 --> 01:13:05.961 What's the difference between these things? 01:13:12.260 --> 01:13:14.500 If we have a fixed point here, what we're saying 01:13:14.500 --> 01:13:18.080 is that both n1 dot and n2 dot are equal to 0. 01:13:18.080 --> 01:13:20.080 Fixed point means that if you start right there, 01:13:20.080 --> 01:13:21.080 you'll stay right there. 01:13:21.080 --> 01:13:23.600 We're not yet say anything about stability. 01:13:23.600 --> 01:13:28.130 And this is going to be very relevant for Sam's question. 01:13:28.130 --> 01:13:29.005 AUDIENCE: [INAUDIBLE] 01:13:35.800 --> 01:13:37.780 PROFESSOR: The axes are definitely labeled. 01:13:37.780 --> 01:13:45.320 I think I may agree, but I'm a little worried that I'm-- If 01:13:45.320 --> 01:13:47.130 you're happy I'm happy. 01:13:47.130 --> 01:13:55.530 OK let's-- A fixed point of this pair of equations will be when 01:13:55.530 --> 01:13:58.420 both of the derivatives are equal 0. 01:13:58.420 --> 01:14:00.820 And that's not the line that we've drawn. 01:14:00.820 --> 01:14:03.930 The lines that we've drawn are just when one or the other one 01:14:03.930 --> 01:14:04.610 is equal to 0. 01:14:08.770 --> 01:14:12.080 So what does that mean about in case four, 01:14:12.080 --> 01:14:15.940 is it possible well to have coexistence? 01:14:15.940 --> 01:14:16.440 No. 01:14:16.440 --> 01:14:18.980 Because there's no fixed one in the middle. 01:14:18.980 --> 01:14:21.140 The only case it's possible to get coexistence 01:14:21.140 --> 01:14:23.600 is either line two or line three. 01:14:26.970 --> 01:14:29.520 But that tells us there's a fixed point, 01:14:29.520 --> 01:14:33.794 it doesn't tell us about the stability of the fixed point, 01:14:33.794 --> 01:14:35.170 though. 01:14:35.170 --> 01:14:37.752 What we'll find is that in one case, 01:14:37.752 --> 01:14:39.710 it's a stable fixed point, you get coexistence. 01:14:39.710 --> 01:14:41.811 In the other case, it's an unstable fixed point, 01:14:41.811 --> 01:14:42.810 and you get bistability. 01:14:46.972 --> 01:14:48.430 What we're going to do now is we're 01:14:48.430 --> 01:14:51.180 going to ask and try to figure out in which case, 01:14:51.180 --> 01:14:54.060 how do we label these things? 01:14:54.060 --> 01:15:04.530 So what we'll do is ask, first of all in line one, 01:15:04.530 --> 01:15:07.130 we want to figure out, is that a situation 01:15:07.130 --> 01:15:09.970 where we know it's not going to be 01:15:09.970 --> 01:15:12.650 coexistence are bistability, but is it going to a situation 01:15:12.650 --> 01:15:16.530 where a species one wins, or when species two wins? 01:15:16.530 --> 01:15:18.830 But, if you want, you can vote for one of the others. 01:15:18.830 --> 01:15:20.750 I don't want to constrain your choices. 01:15:20.750 --> 01:15:29.730 So it's going to be a is one wins, B, two wins, not three 01:15:29.730 --> 01:15:30.230 wins. 01:15:42.402 --> 01:15:43.360 This is something else. 01:15:46.060 --> 01:15:50.380 I'm going to give you a minute to think about what 01:15:50.380 --> 01:15:52.110 might be going on here. 01:15:52.110 --> 01:15:54.480 So the question is put in situation one. 01:16:26.830 --> 01:16:29.230 Oh no, I did that wrong. 01:16:29.230 --> 01:16:31.280 In this. 01:16:31.280 --> 01:16:34.400 For the solid line, this is what we're 01:16:34.400 --> 01:16:35.690 asking for n1 is equal to 0. 01:16:35.690 --> 01:16:38.125 This is just equal to k2. 01:16:38.125 --> 01:16:39.462 AUDIENCE: [INAUDIBLE] 01:16:39.462 --> 01:16:40.420 PROFESSOR: What's that? 01:16:43.228 --> 01:16:45.980 AUDIENCE: There's other [? null ?] lines. 01:16:45.980 --> 01:16:47.730 PROFESSOR: You want more [? null lines? ?] 01:16:47.730 --> 01:16:50.994 I feel like there are lots of lines up there already. 01:16:50.994 --> 01:16:55.467 AUDIENCE: Where n equals 0 [? null ?] lines. 01:16:55.467 --> 01:16:57.952 The axes are actual [? null ?] lines. 01:17:02.162 --> 01:17:03.370 PROFESSOR: Yes, that's right. 01:17:03.370 --> 01:17:05.292 Yeah, that's true. 01:17:05.292 --> 01:17:09.450 AUDIENCE: That helps-- I'm confused between labels. 01:17:09.450 --> 01:17:10.590 PROFESSOR: Oh, I see. 01:17:13.680 --> 01:17:16.370 OK I'm hesitant to draw anything more up on the board. 01:17:16.370 --> 01:17:20.950 But it is true that the axes are also [? null ?] lines. 01:17:20.950 --> 01:17:22.540 Because we have n's up there. 01:17:26.736 --> 01:17:28.360 I'll give you another 20 seconds to try 01:17:28.360 --> 01:17:29.526 to figure out this case one. 01:17:34.650 --> 01:17:36.887 Which of the species is going to win? 01:18:22.587 --> 01:18:24.670 And let's go ahead and vote, and see where we are. 01:18:24.670 --> 01:18:25.170 Ready? 01:18:25.170 --> 01:18:27.960 Three, two, one. 01:18:32.960 --> 01:18:35.231 So there's a slight majority that 01:18:35.231 --> 01:18:36.980 are agreeing that in this case, it's going 01:18:36.980 --> 01:18:40.320 to be that one is going to win. 01:18:40.320 --> 01:18:46.700 And I think that it's a little bit hard to figure it out 01:18:46.700 --> 01:18:50.550 completely, but what I will say is that in the case where we're 01:18:50.550 --> 01:18:54.940 down here, what this means is that k1 divided nu beta 1,2 01:18:54.940 --> 01:18:58.020 is larger than k2. 01:18:58.020 --> 01:18:59.500 Now the question says, what happens 01:18:59.500 --> 01:19:05.520 in the limit of-- If species two does not hurt species one? 01:19:05.520 --> 01:19:10.330 Well that's when beta 1,2 goes to 0. 01:19:10.330 --> 01:19:14.380 And that's the situation where this goes up. 01:19:14.380 --> 01:19:17.360 So this is saying that if the beta 1,2-- 01:19:17.360 --> 01:19:19.290 And then we can figure out where this line is 01:19:19.290 --> 01:19:25.370 as well, because this thing is K2 divided by beta 2,1. 01:19:25.370 --> 01:19:28.150 So if species to doesn't hurt to see one, 01:19:28.150 --> 01:19:31.597 but species one really hurts species two, 01:19:31.597 --> 01:19:33.513 that's going to be the limit where species one 01:19:33.513 --> 01:19:34.731 is going to win. 01:19:42.530 --> 01:19:45.870 We are out of time. 01:19:45.870 --> 01:19:52.000 Maybe what we'll do is start class on Thursday 01:19:52.000 --> 01:19:54.760 by-- I'll start with this on the board, 01:19:54.760 --> 01:20:00.017 and then we'll complete these four possibilities, 01:20:00.017 --> 01:20:01.600 to try to get some intuition about it. 01:20:01.600 --> 01:20:04.050 And also draw these options.