WEBVTT 00:00:00.080 --> 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 --> 00:00:17.217 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.217 --> 00:00:17.842 at ocw.mit.edu. 00:00:20.270 --> 00:00:21.770 PROFESSOR: Today, what we want to do 00:00:21.770 --> 00:00:24.760 is finish off our discussion of Lotka-Volterra competition 00:00:24.760 --> 00:00:25.666 models. 00:00:25.666 --> 00:00:27.540 So starting with this idea of the two species 00:00:27.540 --> 00:00:29.950 interacting competitively but then moving on to try to 00:00:29.950 --> 00:00:31.366 think about the general properties 00:00:31.366 --> 00:00:32.450 of Lotka-Volterra systems. 00:00:32.450 --> 00:00:34.199 In particular, when you have more species, 00:00:34.199 --> 00:00:36.100 the kinds of dynamics that you can get. 00:00:36.100 --> 00:00:39.270 Also, we're going to talk about these non-transitive 00:00:39.270 --> 00:00:41.770 interactions, which are the rock-paper-scissors type 00:00:41.770 --> 00:00:45.550 interactions that may facilitate the maintenance of diversity 00:00:45.550 --> 00:00:47.640 in populations or ecosystems, in particular, 00:00:47.640 --> 00:00:49.910 in the presence of some sort of spatial structure. 00:00:49.910 --> 00:00:53.204 And so we'll talk both about this demonstration 00:00:53.204 --> 00:00:54.870 of rock-paper-scissors type interactions 00:00:54.870 --> 00:00:56.580 in the context of male mating strategies 00:00:56.580 --> 00:00:59.850 in lizards, this paper that was by Curt Lively. 00:00:59.850 --> 00:01:02.424 And then we'll talk about another paper 00:01:02.424 --> 00:01:04.590 in the microbial realm, where they showed that there 00:01:04.590 --> 00:01:07.500 are rock-paper-scissors type interactions in the context 00:01:07.500 --> 00:01:10.940 of colicin production, toxin production 00:01:10.940 --> 00:01:12.870 in the context of bacteria. 00:01:12.870 --> 00:01:17.720 Then, at the end, we'll talk about these population waves. 00:01:17.720 --> 00:01:19.761 There was a rather mathematical reading 00:01:19.761 --> 00:01:22.260 that I had originally proposed, but then we kind of switched 00:01:22.260 --> 00:01:24.730 things up a bit, so that you could, instead, 00:01:24.730 --> 00:01:30.739 read that rock-paper-scissors paper-- that is confusing-- 00:01:30.739 --> 00:01:32.030 that I think is maybe more fun. 00:01:32.030 --> 00:01:34.580 But I'll tell you kind of this basic idea of the population 00:01:34.580 --> 00:01:38.242 waves, what happens, if you have a combination of some growth 00:01:38.242 --> 00:01:40.200 process together with some effective diffusion, 00:01:40.200 --> 00:01:42.100 you can get these population waves that 00:01:42.100 --> 00:01:44.890 correspond to this process of range expansion, 00:01:44.890 --> 00:01:46.840 where a population expands into new territory. 00:01:50.870 --> 00:01:52.950 I just started by putting up what 00:01:52.950 --> 00:01:54.780 we had from the last class. 00:01:54.780 --> 00:01:58.070 So this is the two species Lotka-Volterra competition 00:01:58.070 --> 00:01:58.930 model. 00:01:58.930 --> 00:02:02.870 So what we found was that, for this to be competition, 00:02:02.870 --> 00:02:05.960 for the species to be kind of bad for each other, 00:02:05.960 --> 00:02:10.025 that that corresponds to the betas being positive. 00:02:12.740 --> 00:02:19.720 Now, what we found is there are four cases 00:02:19.720 --> 00:02:22.675 in terms of the outcome of this two species interaction. 00:02:22.675 --> 00:02:24.280 And we wanted to, at least, try to get 00:02:24.280 --> 00:02:29.040 some sense of why that was and what the trajectories 00:02:29.040 --> 00:02:30.180 might look like. 00:02:30.180 --> 00:02:33.370 If you look at and N1 and N2, we can draw these nullclines 00:02:33.370 --> 00:02:35.470 and then get a sense of where the trajectories are 00:02:35.470 --> 00:02:36.180 going to go. 00:02:36.180 --> 00:02:38.660 Now, the basic outcome of this two species Lotka-Volterra 00:02:38.660 --> 00:02:42.460 competition model are really exactly the same 00:02:42.460 --> 00:02:45.360 as the possible outcomes when we're 00:02:45.360 --> 00:02:48.160 thinking about frequency dependent selection, where 00:02:48.160 --> 00:02:54.310 we could get, in this case, that species 1 dominates or species 00:02:54.310 --> 00:02:57.329 2 dominates, 1 or 2. 00:02:57.329 --> 00:02:59.620 But then we could also get coexistence or bi-stability. 00:03:03.620 --> 00:03:05.460 And indeed, there is, in general, 00:03:05.460 --> 00:03:09.770 a mapping from the Lotka-Volterra kind of approach 00:03:09.770 --> 00:03:15.560 here and the approach that was kind of in Martin Nonwak's 00:03:15.560 --> 00:03:19.444 book of thinking about frequency dependent selection 00:03:19.444 --> 00:03:20.235 in this population. 00:03:23.080 --> 00:03:24.790 And so then it's not a surprise that you 00:03:24.790 --> 00:03:29.920 get the same four outcomes between these two situations. 00:03:29.920 --> 00:03:32.260 And I think that this is also highlighting 00:03:32.260 --> 00:03:35.850 some very interesting and deep connections between evolution, 00:03:35.850 --> 00:03:38.910 which is changes in, say, allele frequency in a population 00:03:38.910 --> 00:03:42.020 of a single species, over time, and then 00:03:42.020 --> 00:03:45.250 some of these ecological kind of processes, where you're really 00:03:45.250 --> 00:03:47.000 thinking about these as different species. 00:03:51.440 --> 00:03:55.947 Now, of course, in the case of the evolutionary dynamics 00:03:55.947 --> 00:03:57.780 that we analyze Martin's book, though, those 00:03:57.780 --> 00:04:02.220 were the evolution of different clonal populations 00:04:02.220 --> 00:04:06.487 in asexually reproducing populations. 00:04:06.487 --> 00:04:08.320 Do you guys remember what I'm talking about? 00:04:08.320 --> 00:04:10.950 All right. 00:04:10.950 --> 00:04:14.619 So I just want to draw a couple of these sorts of diagrams. 00:04:14.619 --> 00:04:16.410 I'm not going to draw out all four of them, 00:04:16.410 --> 00:04:18.220 because it does take some time. 00:04:18.220 --> 00:04:21.990 But hopefully, we can reconstruct where we were. 00:04:25.710 --> 00:04:33.590 The case that we were analyzing before, the N1 dot equals 0. 00:04:33.590 --> 00:04:35.280 We're going to have it as a dashed line. 00:04:35.280 --> 00:04:37.640 And N2 is going to be-- well, maybe 00:04:37.640 --> 00:04:39.040 we'll try to use thick chalk. 00:04:39.040 --> 00:04:41.380 Oh, wow, I missed. 00:04:41.380 --> 00:04:42.860 These are thick lines. 00:04:42.860 --> 00:04:47.310 So we'll draw these nullclines over here. 00:04:47.310 --> 00:04:49.470 So the N1 dot-- and indeed, one of the comments 00:04:49.470 --> 00:04:51.920 is that it would be nice to draw where the nullclines are 00:04:51.920 --> 00:04:54.330 on the axes as well. 00:04:54.330 --> 00:04:56.410 And indeed, we can do that. 00:04:56.410 --> 00:04:57.800 We aim to please. 00:04:57.800 --> 00:05:02.530 So the dashed lines correspond to N1 dot being to 0. 00:05:02.530 --> 00:05:06.600 So here's N1, N2. 00:05:06.600 --> 00:05:11.520 So N1 equal to 0 corresponds to this thing. 00:05:14.480 --> 00:05:17.260 And then we have this other guy that's a line, here. 00:05:23.660 --> 00:05:26.230 Now what we found is that, if N2 is equal to 0, 00:05:26.230 --> 00:05:33.650 this intersects at K1, whereas the intersection over here 00:05:33.650 --> 00:05:35.050 is at K1 divided by beta 12. 00:05:40.330 --> 00:05:43.390 Now, we have our other nullclines 00:05:43.390 --> 00:05:47.160 that correspond to N2 dot equal to 0. 00:05:47.160 --> 00:05:50.770 Now one of those lines is indeed going to be along here. 00:05:55.410 --> 00:05:59.570 And the other line can fall-- there 00:05:59.570 --> 00:06:01.510 are four different possibilities for how 00:06:01.510 --> 00:06:04.780 we might draw it in relation to this N1 dot equal to 0 00:06:04.780 --> 00:06:06.420 nullcline. 00:06:06.420 --> 00:06:08.640 And depending on the orientation of that, 00:06:08.640 --> 00:06:11.760 we'll end up getting these four different possible outcomes 00:06:11.760 --> 00:06:15.170 of 1 dominating, irrespective of the initial conditions, 00:06:15.170 --> 00:06:17.950 2 dominating, irrespective of initial conditions, 00:06:17.950 --> 00:06:20.440 or coexistence or bi-stability. 00:06:20.440 --> 00:06:22.940 So bi-stability is the only-- so if you 00:06:22.940 --> 00:06:26.220 start with a finite number of each of these two species, 00:06:26.220 --> 00:06:29.090 then bi-stability is the only case where the outcome depends 00:06:29.090 --> 00:06:30.490 on the starting condition. 00:06:30.490 --> 00:06:33.400 So, of course, if you start out without one of the two species, 00:06:33.400 --> 00:06:35.640 then you won't get creation of those species, right? 00:06:35.640 --> 00:06:39.060 Because the only way to get creation of species 1 00:06:39.060 --> 00:06:41.650 is to have some species 1 individual in this model. 00:06:45.790 --> 00:06:49.080 So we're going to get another line, here, corresponding to N2 00:06:49.080 --> 00:06:50.740 dot equal to 0. 00:06:50.740 --> 00:06:53.790 And I think that the one that we were trying to analyze 00:06:53.790 --> 00:06:57.120 was with the solid line underneath. 00:06:57.120 --> 00:07:01.390 Is that consistent with people's notes? 00:07:01.390 --> 00:07:03.195 So we can draw some other line here. 00:07:03.195 --> 00:07:06.580 It doesn't have to the same slope. 00:07:06.580 --> 00:07:08.419 Now it's good to be clear about where 00:07:08.419 --> 00:07:09.710 these things are going to fall. 00:07:09.710 --> 00:07:12.190 So K2 is this point. 00:07:12.190 --> 00:07:16.450 And now K2 divided by beta 21 is over here. 00:07:16.450 --> 00:07:20.730 Now recall that beta 12 is telling us 00:07:20.730 --> 00:07:23.640 about how much species 2 is reducing 00:07:23.640 --> 00:07:25.570 the growth of species 1. 00:07:25.570 --> 00:07:28.217 Whereas beta 21 is how much species 00:07:28.217 --> 00:07:29.800 1 is reducing the growth of species 2. 00:07:33.290 --> 00:07:36.240 Now, everything comes down to the relative ordering 00:07:36.240 --> 00:07:38.724 of these two quantities and these two quantities. 00:07:38.724 --> 00:07:40.640 And since there are two possibilities on each, 00:07:40.640 --> 00:07:42.348 that gives us the four possible outcomes. 00:07:44.820 --> 00:07:47.290 And broadly, the idea here is just that, 00:07:47.290 --> 00:07:51.130 if the species are weakly interfering with each other, 00:07:51.130 --> 00:07:52.130 then what should happen? 00:07:57.449 --> 00:07:58.990 Yes, then you should get coexistence. 00:07:58.990 --> 00:08:04.860 Coexistence is when the betas are small. 00:08:04.860 --> 00:08:06.846 Of course, this is a concrete model, 00:08:06.846 --> 00:08:08.720 so you have to define what you mean by small. 00:08:08.720 --> 00:08:11.450 And indeed, small here ends up being 00:08:11.450 --> 00:08:14.740 relative to the ratios of these carrying capacities. 00:08:14.740 --> 00:08:18.710 If the carrying capacities are just equal to, say, 1, 00:08:18.710 --> 00:08:24.670 then that's saying that the betas-- sorry, 00:08:24.670 --> 00:08:27.595 if the carrying capacities are the same, then the simple way 00:08:27.595 --> 00:08:29.470 to think about this is just whether the betas 00:08:29.470 --> 00:08:34.390 are larger or smaller than 1, whether each species interferes 00:08:34.390 --> 00:08:37.940 with the other species more than a member of that other species. 00:08:40.346 --> 00:08:41.970 So if carrying capacities are the same, 00:08:41.970 --> 00:08:46.600 that's what demarcates the different zones. 00:08:46.600 --> 00:08:49.440 So what we want to do is take this sort of diagram 00:08:49.440 --> 00:08:53.120 and try to figure out where will the trajectories be 00:08:53.120 --> 00:08:54.970 on this diagram? 00:08:54.970 --> 00:08:59.440 Now, it's always good to locate the fixed points. 00:08:59.440 --> 00:09:01.150 The fixed points of the system are? 00:09:01.150 --> 00:09:05.250 And somebody, words? 00:09:05.250 --> 00:09:08.190 How would we define fixed points in this system? 00:09:08.190 --> 00:09:09.662 AUDIENCE: [INAUDIBLE]. 00:09:09.662 --> 00:09:10.620 PROFESSOR: What's that? 00:09:10.620 --> 00:09:11.786 Both lines intersect, right? 00:09:11.786 --> 00:09:14.150 So when the dashed line intersects the solid lines, 00:09:14.150 --> 00:09:14.950 right? 00:09:14.950 --> 00:09:19.380 So we have one such fixed point here. 00:09:19.380 --> 00:09:23.666 We have another fixed point here and another fixed point here. 00:09:27.046 --> 00:09:28.420 Have we figured out the stability 00:09:28.420 --> 00:09:30.460 of those fixed points? 00:09:30.460 --> 00:09:30.960 No. 00:09:34.997 --> 00:09:36.705 What's the stability of this fixed point? 00:09:42.175 --> 00:09:43.050 It's unstable, right? 00:09:43.050 --> 00:09:45.760 Because we've already, previously 00:09:45.760 --> 00:09:49.784 assumed that these r's are greater than 0. 00:09:49.784 --> 00:09:51.450 We're assuming that the species would be 00:09:51.450 --> 00:09:53.590 able to survive on their own. 00:09:53.590 --> 00:09:55.520 And that's actually true for both species. 00:09:55.520 --> 00:10:01.000 So this thing is unstable kind of in both directions, right? 00:10:01.000 --> 00:10:02.770 And what are the eigenvectors associated 00:10:02.770 --> 00:10:03.686 with this fixed point? 00:10:06.420 --> 00:10:09.910 On the count of three, draw, use your arms 00:10:09.910 --> 00:10:14.400 like the hands of a clock to indicate the directions 00:10:14.400 --> 00:10:16.210 of the eigenvectors. 00:10:16.210 --> 00:10:21.180 All right, ready, three, two, one. 00:10:21.180 --> 00:10:23.714 All right, all right. 00:10:23.714 --> 00:10:24.880 There's no diagonals, right? 00:10:28.520 --> 00:10:29.780 Of course, you could also. 00:10:29.780 --> 00:10:32.450 I was waiting for somebody to be kind of obnoxious and point 00:10:32.450 --> 00:10:33.900 in the other direction. 00:10:33.900 --> 00:10:37.086 A surprisingly not obnoxious class we have. 00:10:37.086 --> 00:10:38.960 So this is just saying that, if you start out 00:10:38.960 --> 00:10:40.290 just a little bit of one species, 00:10:40.290 --> 00:10:41.748 you'll just stay with that species. 00:10:41.748 --> 00:10:43.490 It makes sense. 00:10:43.490 --> 00:10:46.330 Now, what do these lines tell us about the directions 00:10:46.330 --> 00:10:53.827 of the trajectories or the orientations 00:10:53.827 --> 00:10:54.660 of the trajectories? 00:11:03.990 --> 00:11:04.990 Why did I draw them? 00:11:08.500 --> 00:11:09.000 Yes? 00:11:13.360 --> 00:11:15.175 AUDIENCE: These trajectories are-- 00:11:15.175 --> 00:11:18.055 these are lines are nonlines. 00:11:18.055 --> 00:11:20.440 And one of the derivatives [INAUDIBLE]. 00:11:22.495 --> 00:11:24.370 PROFESSOR: One of the derivatives, all right. 00:11:24.370 --> 00:11:30.810 And in particular, let's look at this line here. 00:11:30.810 --> 00:11:33.310 Are the trajectories? 00:11:33.310 --> 00:11:37.000 Again, we're going to do our arms 00:11:37.000 --> 00:11:40.410 to indicate the orientation of the trajectories. 00:11:40.410 --> 00:11:42.790 In particular, there's a trajectory right here. 00:11:42.790 --> 00:11:45.900 What direction will that trajectory be pointing? 00:11:45.900 --> 00:11:48.460 There's only going to be one arm. 00:11:48.460 --> 00:11:53.670 Ready, three, two, one. 00:11:53.670 --> 00:11:55.990 All right, we got a lot of people not voting. 00:11:55.990 --> 00:11:59.560 All right, that means we need to turn our neighbors and discuss. 00:11:59.560 --> 00:12:01.710 If you didn't vote, it means, I think, 00:12:01.710 --> 00:12:04.002 not following what we're talking about. 00:12:04.002 --> 00:12:04.710 Turn to somebody. 00:12:08.560 --> 00:12:10.849 If your neighbor agrees with you about which direction 00:12:10.849 --> 00:12:12.390 you should be pointing your arm, then 00:12:12.390 --> 00:12:15.680 try to figure out whether we know 00:12:15.680 --> 00:12:19.301 which orientation the arrow should actually be in. 00:12:19.301 --> 00:12:22.460 [SIDE DISCUSSIONS] 00:12:47.515 --> 00:12:48.890 PROFESSOR: We'll figure this out. 00:12:48.890 --> 00:12:52.292 [SIDE DISCUSSIONS] 00:13:02.684 --> 00:13:04.350 PROFESSOR: Let's go ahead and reconvene, 00:13:04.350 --> 00:13:06.580 because it seems like some people are being quiet. 00:13:06.580 --> 00:13:09.540 But I'm not sure if that's because they think 00:13:09.540 --> 00:13:12.600 they know what's going on or they're just very much unhappy 00:13:12.600 --> 00:13:14.440 with the situation. 00:13:14.440 --> 00:13:16.460 Let me see a fresh voting. 00:13:16.460 --> 00:13:18.775 All right, ready, three, two, one. 00:13:21.312 --> 00:13:23.020 Definitely, it's going up or down, right? 00:13:23.020 --> 00:13:26.870 Because the definition of this dashed line is that N1 dot 00:13:26.870 --> 00:13:28.680 is equal to 0. 00:13:28.680 --> 00:13:30.544 We don't know what N2 dot is. 00:13:30.544 --> 00:13:31.960 We'll figure that out in a moment. 00:13:31.960 --> 00:13:34.100 But what we know is that the trajectories 00:13:34.100 --> 00:13:38.180 we should be something, lines here, either up or down. 00:13:38.180 --> 00:13:42.320 And we're going to find that they're down but here. 00:13:42.320 --> 00:13:47.369 Now, on this solid line, quickly, the orientation of 00:13:47.369 --> 00:13:47.910 trajectories. 00:13:47.910 --> 00:13:49.900 Ready, three, two, one. 00:13:49.900 --> 00:13:53.930 All right, perfect. 00:13:53.930 --> 00:13:55.870 So we know that N2 dot is 0 here. 00:13:58.860 --> 00:14:02.150 Now, an actual direction of the trajectory, at this point, 00:14:02.150 --> 00:14:03.230 right here. 00:14:03.230 --> 00:14:06.050 Ready, three, two, one. 00:14:06.050 --> 00:14:07.370 OK, good. 00:14:07.370 --> 00:14:10.776 Because in the absence of N2, we know 00:14:10.776 --> 00:14:13.150 that species N1 should just come to carrying capacity K1. 00:14:15.880 --> 00:14:20.760 Same thing over here, we should get arrows coming down. 00:14:20.760 --> 00:14:23.500 So indeed, what you can see is that the arrows are coming down 00:14:23.500 --> 00:14:23.650 here. 00:14:23.650 --> 00:14:26.024 That means they actually do have to come down immediately 00:14:26.024 --> 00:14:27.825 to the right of them. 00:14:27.825 --> 00:14:29.200 Whereas over here, the trajectory 00:14:29.200 --> 00:14:33.310 is point right here, so we can kind of figure out 00:14:33.310 --> 00:14:37.780 that-- so they're coming here. 00:14:37.780 --> 00:14:39.780 And from far away, they're coming here. 00:14:39.780 --> 00:14:42.630 And you can see that they have to come across here. 00:14:42.630 --> 00:14:44.740 And then they're going to come into this point. 00:14:44.740 --> 00:14:48.010 So this is going to be our stable fixed point. 00:14:48.010 --> 00:14:51.371 I'll color it in to indicate that. 00:14:51.371 --> 00:14:53.370 From here, they're going to curve around though. 00:14:53.370 --> 00:14:56.085 So if you start out right here, you kind of do this business. 00:15:05.140 --> 00:15:06.810 From here, we come in. 00:15:12.580 --> 00:15:14.990 Are these lines allowed to cross each other? 00:15:14.990 --> 00:15:15.490 No. 00:15:23.290 --> 00:15:25.330 Now, indeed, you could actually see 00:15:25.330 --> 00:15:28.960 here, what is the direction of the other eigenvector 00:15:28.960 --> 00:15:31.180 at this point? 00:15:31.180 --> 00:15:37.640 Using a hand, arm, ready, three, two, one. 00:15:37.640 --> 00:15:39.299 Right, it's kind of something in there. 00:15:39.299 --> 00:15:41.590 Because there's all these trajectories are coming here, 00:15:41.590 --> 00:15:44.420 and then they approach this fixed point from this point. 00:15:44.420 --> 00:15:46.360 Because here, the other eigenvector 00:15:46.360 --> 00:15:49.110 is still horizontal, right? 00:15:49.110 --> 00:15:52.180 Because we know, if we don't have N2, then we just have N1. 00:15:52.180 --> 00:15:55.662 But this other eigenvector is not purely straight up. 00:15:55.662 --> 00:15:57.870 The other one is along here, because the trajectories 00:15:57.870 --> 00:16:01.310 are coming in along that. 00:16:01.310 --> 00:16:04.010 Does that make sense? 00:16:04.010 --> 00:16:06.900 So in this case, we should just be clear. 00:16:06.900 --> 00:16:12.050 We are in a situation where K1 over beta 12 00:16:12.050 --> 00:16:13.460 is greater than K2. 00:16:16.740 --> 00:16:20.580 Another way of writing that is that beta 12 is less than K1 00:16:20.580 --> 00:16:22.770 over K2. 00:16:22.770 --> 00:16:29.930 So that means species 2 does not strongly harm species 1. 00:16:29.930 --> 00:16:36.930 Yet, we know that K1 is greater than K2 divided by beta 21. 00:16:36.930 --> 00:16:42.120 So that means that beta 21 is greater than K2 over K1. 00:16:42.120 --> 00:16:46.250 This is telling us that species 1 is strongly 00:16:46.250 --> 00:16:48.625 harming species 2. 00:16:48.625 --> 00:16:49.520 And that makes sense. 00:16:49.520 --> 00:16:51.420 In that case, species 1 wins. 00:16:56.290 --> 00:17:01.430 Does that outcome change if we change the r values, 00:17:01.430 --> 00:17:02.520 the division rates? 00:17:07.900 --> 00:17:08.960 A is yes. 00:17:08.960 --> 00:17:09.950 B is no. 00:17:09.950 --> 00:17:12.390 I'm going to give you 10 seconds. 00:17:12.390 --> 00:17:15.219 What I just said, if I change r's does that 00:17:15.219 --> 00:17:16.010 change the outcome? 00:17:16.010 --> 00:17:19.575 Ready, three, two, one. 00:17:22.165 --> 00:17:25.710 So we've got a majority of B. So the answer is no. 00:17:25.710 --> 00:17:37.506 This statement that, in this situation, species 1 dominates, 00:17:37.506 --> 00:17:38.755 that's independent of the r's. 00:17:44.112 --> 00:17:45.570 And what you see is the conditions, 00:17:45.570 --> 00:17:48.350 here, only depend on the what's in here. 00:17:48.350 --> 00:17:51.390 So the actual shape of those trajectories will depend upon 00:17:51.390 --> 00:17:51.920 the r's. 00:17:54.520 --> 00:17:58.420 So if it's the case that species 2 is just 00:17:58.420 --> 00:18:00.921 a faster grower than species 1, then you 00:18:00.921 --> 00:18:02.420 might end up with a situation where, 00:18:02.420 --> 00:18:04.045 if you start with a little bit of each, 00:18:04.045 --> 00:18:05.940 you might come way up here-- well, no. 00:18:05.940 --> 00:18:10.196 I guess you might come close to this fixed point. 00:18:10.196 --> 00:18:11.820 So you might think that it really looks 00:18:11.820 --> 00:18:13.390 like species 2 is about to win. 00:18:13.390 --> 00:18:17.500 But eventually, they'll curve over and come back. 00:18:17.500 --> 00:18:20.930 And indeed, in the Strogatz book, one of the chapters 00:18:20.930 --> 00:18:24.770 that I recommended, he has an example of sheeps and rabbits, 00:18:24.770 --> 00:18:27.560 the idea is that they are competing species, 00:18:27.560 --> 00:18:28.884 maybe eating similar foods. 00:18:28.884 --> 00:18:30.050 I don't know if that's true. 00:18:30.050 --> 00:18:32.410 But the rabbits can divide more rapidly. 00:18:32.410 --> 00:18:36.750 So here, the idea would be, well, if the sheep can really 00:18:36.750 --> 00:18:38.930 displace the rabbits, because it's just bigger 00:18:38.930 --> 00:18:40.650 and push them aside, then what can happen 00:18:40.650 --> 00:18:42.892 is that the rabbits first divide rapidly. 00:18:42.892 --> 00:18:44.600 It looks like they're going to take over, 00:18:44.600 --> 00:18:48.379 but, over time, eventually, the sheep population kind of 00:18:48.379 --> 00:18:50.170 grow up, and they start displacing rabbits. 00:18:50.170 --> 00:18:52.770 And you end up excluding the rabbits. 00:18:52.770 --> 00:18:55.130 This is this phenomenon of competitive exclusion. 00:19:00.570 --> 00:19:06.570 And depending upon the context-- OK, that's an e-- 00:19:06.570 --> 00:19:08.790 this is either more or less maybe formally phrased. 00:19:08.790 --> 00:19:14.210 But the idea is that, if there are two species that 00:19:14.210 --> 00:19:16.026 are too similar, and in particular, 00:19:16.026 --> 00:19:17.650 if they're somehow perfect competitors, 00:19:17.650 --> 00:19:19.060 they're really just trying to eat the same thing, 00:19:19.060 --> 00:19:20.850 then you should only end up with one 00:19:20.850 --> 00:19:23.370 of the two species surviving. 00:19:23.370 --> 00:19:26.220 And that's the kind of idea here. 00:19:26.220 --> 00:19:29.840 Although, I think, you can argue about the mapping I think. 00:19:29.840 --> 00:19:32.205 So this is one of the four outcomes. 00:19:35.360 --> 00:19:36.979 And of course, it takes 15 minutes 00:19:36.979 --> 00:19:38.520 to go through each of these examples. 00:19:38.520 --> 00:19:41.430 So we're not going to go through all of them. 00:19:41.430 --> 00:19:45.280 But you should be able to, for a given 00:19:45.280 --> 00:19:49.600 a combination of betas and K's, be able to figure out, 00:19:49.600 --> 00:19:53.410 using some combination of algebra, derivatives, 00:19:53.410 --> 00:19:57.185 fixed point stability analyses, and drawing of things, well, 00:19:57.185 --> 00:19:58.935 you should be able to do all of the above. 00:20:03.720 --> 00:20:05.805 Are there any questions about where we are here? 00:20:13.512 --> 00:20:14.970 AUDIENCE: I guess that none of this 00:20:14.970 --> 00:20:18.571 holds in the stochastic [INAUDIBLE]. 00:20:18.571 --> 00:20:20.570 PROFESSOR: Yeah, that's an interesting question. 00:20:20.570 --> 00:20:23.762 AUDIENCE: For example, in that case, if r2 is much bigger, 00:20:23.762 --> 00:20:28.063 then you're going get almost, very close to like K2, 00:20:28.063 --> 00:20:31.615 and then maybe just the last individual [INAUDIBLE] 00:20:31.615 --> 00:20:34.530 and then you just add that fixed point. 00:20:34.530 --> 00:20:36.520 PROFESSOR: Right. 00:20:36.520 --> 00:20:39.660 So it's certainly the case that, once you 00:20:39.660 --> 00:20:41.690 have stochastic extinction-- the thing is 00:20:41.690 --> 00:20:43.839 that, you would probably be most susceptible 00:20:43.839 --> 00:20:44.880 to stochastic extinction. 00:20:44.880 --> 00:20:46.300 In the case you were talking about, 00:20:46.300 --> 00:20:47.920 you would be most susceptible to stochastic extinction 00:20:47.920 --> 00:20:49.760 when you're around here, actually. 00:20:49.760 --> 00:20:58.800 These trajectories are still always moving up in N1 space. 00:20:58.800 --> 00:21:00.700 I think I know what you're saying. 00:21:00.700 --> 00:21:06.300 We're going to maybe zoom in onto this N1, N2. 00:21:06.300 --> 00:21:08.780 Because we have this unstable fixed point here. 00:21:08.780 --> 00:21:13.190 And the claim was that, if you start out over here, 00:21:13.190 --> 00:21:16.500 then the trajectory might look something like this. 00:21:16.500 --> 00:21:19.500 And you'd say, oh, well, you might go extinct here. 00:21:19.500 --> 00:21:21.850 AUDIENCE: The idea was just the statement 00:21:21.850 --> 00:21:25.620 that [INAUDIBLE] r1, r2 are very dynamical [INAUDIBLE]. 00:21:31.380 --> 00:21:36.150 PROFESSOR: It's true that, I guess, things change. 00:21:36.150 --> 00:21:39.320 There are a number of things you might want to say. 00:21:39.320 --> 00:21:43.540 First of all, this is a purely continuous and deterministic 00:21:43.540 --> 00:21:46.090 description of the setting. 00:21:46.090 --> 00:21:49.670 It allows for fractional individuals. 00:21:49.670 --> 00:21:52.440 There's no shocks or perturbations 00:21:52.440 --> 00:21:54.016 that you have to worry about. 00:21:54.016 --> 00:21:55.640 I guess the only thing I wanted to say, 00:21:55.640 --> 00:21:58.790 in regards to your question, is that I 00:21:58.790 --> 00:22:01.739 think the stochastic extinction will not be dominated. 00:22:01.739 --> 00:22:04.030 We're talking about stochastic extinction of species 1. 00:22:04.030 --> 00:22:06.529 It will not be dominated due to a stochastic extinction here 00:22:06.529 --> 00:22:10.820 but stochastic extinction at the beginning. 00:22:10.820 --> 00:22:13.920 I haven't drawn this very, very well, but, in this case, 00:22:13.920 --> 00:22:15.580 I think these trajectories are always 00:22:15.580 --> 00:22:20.900 are going up in numbers of the 1 species, which 00:22:20.900 --> 00:22:22.910 means that you're most likely to experience 00:22:22.910 --> 00:22:25.048 stochastic extinction at the beginning. 00:22:25.048 --> 00:22:25.548 Yeah? 00:22:25.548 --> 00:22:28.008 AUDIENCE: r1 and r2 should still really 00:22:28.008 --> 00:22:30.468 mess with stochastic extinction a lot, right? 00:22:30.468 --> 00:22:35.830 Because the larger r1 and r2 are the quicker 00:22:35.830 --> 00:22:39.670 we get out of the regime of stochastic extinction. 00:22:39.670 --> 00:22:41.140 PROFESSOR: That's all true. 00:22:45.420 --> 00:22:49.420 We have to be careful about many of these things. 00:22:49.420 --> 00:22:52.350 In particular, if you go and you do a stochastic simulation 00:22:52.350 --> 00:22:55.420 of this, so let's say you plug this thing into a Gillespie 00:22:55.420 --> 00:23:01.430 simulation, can you get stochastic extinction? 00:23:01.430 --> 00:23:06.900 A yes, or B no, ready, three, two, one. 00:23:06.900 --> 00:23:09.020 No. 00:23:09.020 --> 00:23:11.748 So the answer is no but why? 00:23:11.748 --> 00:23:12.666 AUDIENCE: It depends. 00:23:12.666 --> 00:23:16.810 If you take r combination [INAUDIBLE]. 00:23:16.810 --> 00:23:17.790 PROFESSOR: Exactly. 00:23:17.790 --> 00:23:20.555 So right now, as written, there's no death. 00:23:22.577 --> 00:23:24.660 Although, I guess you could say that this a death. 00:23:27.710 --> 00:23:31.420 There's a question of how you partition things. 00:23:31.420 --> 00:23:34.850 So in principle, this is the difference between the growth 00:23:34.850 --> 00:23:36.450 and the death rate. 00:23:36.450 --> 00:23:39.640 But the most straightforward way of doing such a simulation 00:23:39.640 --> 00:23:44.220 is that you put this whole thing in here as a rate for birth. 00:23:47.732 --> 00:23:50.065 Somebody is going to say, ah, that's not how I was going 00:23:50.065 --> 00:23:51.267 to do the simulation, right? 00:23:51.267 --> 00:23:53.350 Well, that's probably how you were going to do it. 00:23:53.350 --> 00:23:54.850 AUDIENCE: That's a terrible way. 00:23:54.850 --> 00:23:56.670 PROFESSOR: But if you did that, if there's only birth, 00:23:56.670 --> 00:23:59.020 then you can't get stochastic extinction, obviously. 00:23:59.020 --> 00:24:01.829 But in general-- and this is one of things 00:24:01.829 --> 00:24:03.370 we spend our time a lot time thinking 00:24:03.370 --> 00:24:07.040 about in this semester-- there are multiple ways 00:24:07.040 --> 00:24:09.410 of doing kind of a stochastic simulation 00:24:09.410 --> 00:24:11.420 from a deterministic equation. 00:24:11.420 --> 00:24:13.966 And this thing, you could be more explicit 00:24:13.966 --> 00:24:15.340 and say, oh, this thing is really 00:24:15.340 --> 00:24:19.625 a B2 minus a D2, so a birth rate minus a death rate for example. 00:24:19.625 --> 00:24:21.750 And form the standpoint of a differential equation, 00:24:21.750 --> 00:24:23.280 it doesn't make any difference. 00:24:23.280 --> 00:24:27.160 But if you do the Fokker-Planck approximation 00:24:27.160 --> 00:24:29.520 or you do a simulation or whatnot, 00:24:29.520 --> 00:24:32.720 then these lead to different things. 00:24:32.720 --> 00:24:34.970 In particular, the rate of, say, stochastic extinction 00:24:34.970 --> 00:24:38.650 here increases as B and D increase, because that leads 00:24:38.650 --> 00:24:40.095 to more of these fluctuations. 00:24:44.560 --> 00:24:48.911 So there are many things that are different once you include 00:24:48.911 --> 00:24:49.910 the stochastic dynamics. 00:24:49.910 --> 00:24:54.450 But it's always good to get a base sense of the dynamics 00:24:54.450 --> 00:24:57.910 from the standpoint of just deterministic differential 00:24:57.910 --> 00:25:01.860 equations before you think too much about the stochastic 00:25:01.860 --> 00:25:04.304 dynamics, because otherwise you get overwhelmed quickly. 00:25:08.180 --> 00:25:10.370 Any other questions about that? 00:25:25.510 --> 00:25:28.880 What I want to do is just spend a little bit of time 00:25:28.880 --> 00:25:30.890 to think about the more generalized case of more 00:25:30.890 --> 00:25:31.390 species. 00:25:40.280 --> 00:25:43.970 And in particular, we could convert this set of equations. 00:25:43.970 --> 00:25:47.600 We can normalize by each of their carrying capacities. 00:25:47.600 --> 00:25:54.394 And we can convert a set of equations 00:25:54.394 --> 00:25:55.560 to look something like this. 00:25:55.560 --> 00:25:59.760 So now we just have Xi dot is equal to-- there's 00:25:59.760 --> 00:26:03.850 some ri Xi, 1 minus. 00:26:03.850 --> 00:26:06.270 And what we can actually do is normalize everything 00:26:06.270 --> 00:26:07.950 so that it's just written like this. 00:26:12.220 --> 00:26:15.020 And normally, what we assume is that we've 00:26:15.020 --> 00:26:21.990 done things such that alpha i i is equal to 1 for all i. 00:26:21.990 --> 00:26:24.760 So this is just saying that this is the normalization such 00:26:24.760 --> 00:26:29.450 that each species inhibits itself in a way 00:26:29.450 --> 00:26:33.060 that it's just give a simple logistic growth. 00:26:33.060 --> 00:26:39.545 And it's going to be logistic growth with a carrying 00:26:39.545 --> 00:26:41.980 capacity equal to 1. 00:26:41.980 --> 00:26:45.130 And then once you've done that, then a species 00:26:45.130 --> 00:26:48.720 inhibits itself with alpha i i 1. 00:26:48.720 --> 00:26:53.210 And then everything comes down to what this alpha matrix is. 00:26:57.720 --> 00:27:02.380 And I would say that, as always, it's really very, very 00:27:02.380 --> 00:27:04.180 important that you can go back and forth 00:27:04.180 --> 00:27:07.920 between the non-dimensionalized versions of equations 00:27:07.920 --> 00:27:09.420 and the base version. 00:27:09.420 --> 00:27:12.280 This was something that, on exam number two, 00:27:12.280 --> 00:27:21.720 there were quite a lot of problems where we asked about, 00:27:21.720 --> 00:27:23.260 how does the parameter change when 00:27:23.260 --> 00:27:25.760 you change the strength of expression or this or that, 00:27:25.760 --> 00:27:26.620 right? 00:27:26.620 --> 00:27:28.880 So this is something that I think is very important. 00:27:28.880 --> 00:27:30.380 Because this comes up a lot. 00:27:33.100 --> 00:27:35.020 So in this case, the alpha matrix 00:27:35.020 --> 00:27:37.330 tells you kind of everything. 00:27:37.330 --> 00:27:39.270 And then there are a number of things 00:27:39.270 --> 00:27:41.450 that, well, the mathematicians have proven 00:27:41.450 --> 00:27:42.742 about these sorts of equations. 00:27:42.742 --> 00:27:45.116 And I just want to point you towards some of those things 00:27:45.116 --> 00:27:45.960 to think about. 00:27:45.960 --> 00:27:52.590 So first, I'm considering a case where, again, alpha i 00:27:52.590 --> 00:28:01.290 j is greater than 0, again, for all i and ji, 00:28:01.290 --> 00:28:03.810 or greater than or equal to 0. 00:28:07.310 --> 00:28:10.240 So some of them can be 0. 00:28:10.240 --> 00:28:13.055 But the interactions, when they exist, are competitive. 00:28:16.360 --> 00:28:23.040 So first, if you start out in the region where 00:28:23.040 --> 00:28:25.850 all of the species start, between 0 and 1, then 00:28:25.850 --> 00:28:28.150 you stay in that region. 00:28:33.340 --> 00:28:36.100 If this is true initially, then it will be true forever. 00:28:40.590 --> 00:28:41.550 That's good. 00:28:41.550 --> 00:28:45.560 Negative abundances, you maybe were not so worried about. 00:28:45.560 --> 00:28:49.050 But it's not as obvious that you can't get above one, 00:28:49.050 --> 00:28:51.440 because there's nothing saying that, in principle, you 00:28:51.440 --> 00:28:53.090 couldn't have started there, right? 00:28:53.090 --> 00:28:56.939 Or in principle, you couldn't have gotten there? 00:28:56.939 --> 00:28:58.480 And certainly, it's physical to think 00:28:58.480 --> 00:29:00.063 about starting outside of that region, 00:29:00.063 --> 00:29:02.170 because we often talk about carrying capacities 00:29:02.170 --> 00:29:03.211 as something that's like. 00:29:06.830 --> 00:29:10.146 If you think about, this is an Xi as a function of time. 00:29:13.096 --> 00:29:15.720 So in these situations, it's not crazy to think about something 00:29:15.720 --> 00:29:18.017 above the carrying capacity. 00:29:18.017 --> 00:29:20.350 But this mathematical statement about the Latka-Volterra 00:29:20.350 --> 00:29:21.650 framework is that, if you start out 00:29:21.650 --> 00:29:23.500 with everything below its carrying capacity, 00:29:23.500 --> 00:29:25.500 then everything will always stay there. 00:29:29.810 --> 00:29:42.060 And all the dynamics occur-- this is for i equal to 1 to N 00:29:42.060 --> 00:29:43.710 Do I want to use a big N or little n? 00:29:43.710 --> 00:29:44.420 Does it matter? 00:29:50.840 --> 00:29:52.750 So this is big N, different species. 00:29:52.750 --> 00:29:57.650 The dynamics occur on an N minus one manifold. 00:29:57.650 --> 00:29:59.150 If we have many mathematicians, they 00:29:59.150 --> 00:30:01.270 can explain what this technically means. 00:30:01.270 --> 00:30:03.760 But basically, what it's saying is 00:30:03.760 --> 00:30:06.945 that there's going to be some N minus 1 dimensional surface 00:30:06.945 --> 00:30:09.660 or volume or whatnot where all the dynamics are 00:30:09.660 --> 00:30:10.980 going end up being on. 00:30:10.980 --> 00:30:15.920 And what that means is that, in particular, a limit cycle 00:30:15.920 --> 00:30:28.440 requires two dimensions, requires 00:30:28.440 --> 00:30:31.510 2D, which means that to get a limit cycle 00:30:31.510 --> 00:30:36.990 requires-- so then N has to be greater than or equal to 3 00:30:36.990 --> 00:30:39.320 to get a limit cycle. 00:30:39.320 --> 00:30:41.500 Can you see what I'm saying? 00:30:41.500 --> 00:30:47.897 Whereas chaos requires 3D, and that 00:30:47.897 --> 00:30:49.980 means that N has to be greater than or equal to 4. 00:30:53.980 --> 00:30:56.941 And indeed, you can get limit cycles with N equal to 3. 00:30:56.941 --> 00:30:58.440 And you get chaos with N equal to 4. 00:31:01.796 --> 00:31:03.170 There's another theorem that says 00:31:03.170 --> 00:31:10.360 that any dynamics are possible for N equal to 5 or larger. 00:31:10.360 --> 00:31:13.550 And chaos seems as much as I would want to ask for. 00:31:13.550 --> 00:31:20.620 But there's, apparently, a 4D torus, 00:31:20.620 --> 00:31:22.764 something that is different. 00:31:22.764 --> 00:31:24.930 This is something to think about in your spare time. 00:31:33.010 --> 00:31:35.900 Well, you know, like a lot of things. 00:31:35.900 --> 00:31:41.440 And this is worth spending a moment talking about. 00:31:41.440 --> 00:31:44.550 First of all, so here's a two dimensional thing. 00:31:44.550 --> 00:31:46.550 The special thing, as you might remember, 00:31:46.550 --> 00:31:50.950 about continuous dynamics is that these trajectories are not 00:31:50.950 --> 00:31:53.670 allowed to cross each other, right? 00:31:53.670 --> 00:31:58.990 Which means that you just can't draw a chaotic trajectory, 00:31:58.990 --> 00:32:02.210 because you're going to have to cross yourself again. 00:32:02.210 --> 00:32:06.550 And that's also why a limit cycle requires two, 00:32:06.550 --> 00:32:10.070 because we originally tried to draw a limit cycle in one 00:32:10.070 --> 00:32:11.400 dimension, and it didn't work. 00:32:11.400 --> 00:32:13.080 Do you remember this? 00:32:13.080 --> 00:32:15.360 And the same thing with-- you can imagine, here's 00:32:15.360 --> 00:32:17.959 a nice limit cycle. 00:32:17.959 --> 00:32:19.000 And that could be stable. 00:32:21.730 --> 00:32:24.770 Oh, and incidentally, you were right about. 00:32:24.770 --> 00:32:28.440 I was getting confused about the Poincare-Bendixson theorem. 00:32:28.440 --> 00:32:30.390 Because I think there are these funny things. 00:32:30.390 --> 00:32:32.280 It wasn't. 00:32:32.280 --> 00:32:34.065 OK, well, you know, all these people 00:32:34.065 --> 00:32:36.890 that complain about what I say. 00:32:36.890 --> 00:32:40.499 Because if you have the trajectories coming in, 00:32:40.499 --> 00:32:43.040 then what I said was that, if you had an unstable fixed point 00:32:43.040 --> 00:32:44.950 coming out, then you could draw this region, 00:32:44.950 --> 00:32:46.575 then you could be guaranteed that there 00:32:46.575 --> 00:32:47.930 was a limit cycle in there. 00:32:47.930 --> 00:32:52.887 But you cannot, just based on what I had said, say that, 00:32:52.887 --> 00:32:55.220 if it's a stable fixed point, then you won't get a limit 00:32:55.220 --> 00:32:56.140 cycle. 00:32:56.140 --> 00:32:59.820 I mean in some other cases you can prove that it doesn't work. 00:32:59.820 --> 00:33:04.570 Particularly, like in the Latka-Volterra, 00:33:04.570 --> 00:33:08.439 there in one of the crossings, it's 00:33:08.439 --> 00:33:09.980 now going to be a stable fixed point. 00:33:09.980 --> 00:33:11.800 In that situation, you can prove, mathematically, 00:33:11.800 --> 00:33:13.420 that you can never get a limit cycle oscillation 00:33:13.420 --> 00:33:15.670 because of some divergence condition of some function 00:33:15.670 --> 00:33:18.311 and so forth. 00:33:18.311 --> 00:33:20.310 But in the example that I was telling you about, 00:33:20.310 --> 00:33:23.870 in the predator-prey, it was true in that particular case 00:33:23.870 --> 00:33:26.569 that, when that thing is stable, you don't get oscillations. 00:33:26.569 --> 00:33:27.860 And when it's unstable, you do. 00:33:27.860 --> 00:33:29.734 But both directions do not follow 00:33:29.734 --> 00:33:30.900 from the Poincare-Bendixson. 00:33:35.580 --> 00:33:37.540 This is a limit cycle. 00:33:37.540 --> 00:33:41.020 Now, in a chaotic situation, you have to be able to do something 00:33:41.020 --> 00:33:43.440 where you kind of come around and then, 00:33:43.440 --> 00:33:45.190 every now and then, you kind go over here. 00:33:45.190 --> 00:33:48.410 And then you go around and around. 00:33:48.410 --> 00:33:49.970 Something crazy happens. 00:33:49.970 --> 00:33:53.720 But you can see that this situation doesn't 00:33:53.720 --> 00:33:57.450 work if it's only 2D, right? 00:33:57.450 --> 00:33:59.430 Do you see why I'm saying that? 00:33:59.430 --> 00:34:00.410 AUDIENCE: Yeah. 00:34:00.410 --> 00:34:01.380 PROFESSOR: You don't? 00:34:01.380 --> 00:34:03.040 You don't agree. 00:34:03.040 --> 00:34:05.436 It's just, if it's 2D, then these lines cannot cross. 00:34:05.436 --> 00:34:06.810 So you need to a third dimension, 00:34:06.810 --> 00:34:09.730 so that they can just shoot above and below each other. 00:34:09.730 --> 00:34:11.505 AUDIENCE: And what's with the definition? 00:34:11.505 --> 00:34:13.790 Because you can come up with trajectories, 00:34:13.790 --> 00:34:16.639 like you can have trajectories that asymptotically approach 00:34:16.639 --> 00:34:21.112 the limit cycle but that never cross themselves, right? 00:34:21.112 --> 00:34:21.778 PROFESSOR: Yeah. 00:34:21.778 --> 00:34:23.920 AUDIENCE: Those are obviously not chaotic, but the y-- 00:34:23.920 --> 00:34:24.628 PROFESSOR: Right. 00:34:26.989 --> 00:34:29.310 So this is not actually class in non-linear dynamics, 00:34:29.310 --> 00:34:30.139 so we're not. 00:34:30.139 --> 00:34:33.510 So normally, you characterize this Lyapunov exponent, 00:34:33.510 --> 00:34:35.642 which tells you about how the phase space is 00:34:35.642 --> 00:34:36.850 kind of growing or shrinking. 00:34:36.850 --> 00:34:38.725 In the case that you were just talking about, 00:34:38.725 --> 00:34:41.150 that's a case where all the trajectories come together. 00:34:41.150 --> 00:34:43.774 Because in this case where this trajectory comes into the limit 00:34:43.774 --> 00:34:46.920 cycle, if you draw like a blob of phase space, 00:34:46.920 --> 00:34:49.020 it's going to come together over time. 00:34:49.020 --> 00:34:52.480 Whereas in a chaotic system, if you have a blob of phase space, 00:34:52.480 --> 00:34:55.409 it's going to diverge and fold and do all the craziness. 00:35:00.289 --> 00:35:01.992 Yes? 00:35:01.992 --> 00:35:03.867 AUDIENCE: So you said [INAUDIBLE] essentially 00:35:03.867 --> 00:35:06.145 you need n greater than or equal to 3. 00:35:06.145 --> 00:35:08.100 Is that just for this-- 00:35:08.100 --> 00:35:10.021 PROFESSOR: Yeah. 00:35:10.021 --> 00:35:10.520 Yes. 00:35:10.520 --> 00:35:12.920 So this is in this Latka-Volterra. 00:35:12.920 --> 00:35:16.700 Because, in general, you can get a limit cycle 00:35:16.700 --> 00:35:19.040 with two equations. 00:35:19.040 --> 00:35:20.150 And that's with the 2D. 00:35:20.150 --> 00:35:23.270 And in general, you get a chaos with three equations 00:35:23.270 --> 00:35:24.890 or three variables. 00:35:24.890 --> 00:35:29.870 But in the Latka-Volterra model, it requires three and four, 00:35:29.870 --> 00:35:30.990 respectively. 00:35:30.990 --> 00:35:34.139 That doesn't mean that every four species 00:35:34.139 --> 00:35:35.430 Latka-Volterra will have chaos. 00:35:35.430 --> 00:35:37.221 But it means that it's possible to get one. 00:35:39.059 --> 00:35:41.100 And I think one thing that's just rather striking 00:35:41.100 --> 00:35:45.830 is that this really is kind of the simplest, possible model 00:35:45.830 --> 00:35:49.870 you can ever write down describing 00:35:49.870 --> 00:35:52.587 how species might interact or variables 00:35:52.587 --> 00:35:53.670 might interact or whatnot. 00:35:53.670 --> 00:35:55.630 And so it's really kind of incredible to me 00:35:55.630 --> 00:35:57.800 that you get all these crazy dynamics. 00:36:00.205 --> 00:36:00.705 Yes? 00:36:00.705 --> 00:36:03.200 AUDIENCE: I was going to ask about the n minus 1 thing. 00:36:03.200 --> 00:36:06.415 This dynamic [INAUDIBLE] n minus 1 [INAUDIBLE]. 00:36:06.415 --> 00:36:07.081 PROFESSOR: Yeah. 00:36:07.081 --> 00:36:08.705 AUDIENCE: Over here it just looks to me 00:36:08.705 --> 00:36:11.464 like-- we have n equals 2 and the dynamics 00:36:11.464 --> 00:36:17.004 are occuring on a plane, on a two-dimensional plane. 00:36:17.004 --> 00:36:17.670 PROFESSOR: Yeah. 00:36:22.280 --> 00:36:24.860 The transience or whatever can occur. 00:36:24.860 --> 00:36:28.050 It requires the full N dimensions to describe. 00:36:28.050 --> 00:36:30.890 Because you can start-- to describe 00:36:30.890 --> 00:36:35.120 all of the trajectories, clearly requires all dimensions, right? 00:36:35.120 --> 00:36:36.670 Because anywhere you start, you have 00:36:36.670 --> 00:36:39.550 to have specify it by five dimensions. 00:36:39.550 --> 00:36:43.870 But the dynamics, as far as like-- and here, 00:36:43.870 --> 00:36:46.300 there aren't even any dynamics. 00:36:46.300 --> 00:36:48.290 I think that you go to a point. 00:36:51.571 --> 00:36:53.565 AUDIENCE: What do you mean by dynamics? 00:36:53.565 --> 00:36:55.680 [INAUDIBLE] 00:36:55.680 --> 00:36:57.480 PROFESSOR: Yeah, I agree that there 00:36:57.480 --> 00:37:02.850 is a-- so in the case of the limit cycles 00:37:02.850 --> 00:37:07.410 and so forth, this is really like, at steady state, 00:37:07.410 --> 00:37:08.830 it's doing something. 00:37:08.830 --> 00:37:11.640 So here, steady state, it only goes to the fixed points. 00:37:13.639 --> 00:37:15.430 But if you have a three dimensional system, 00:37:15.430 --> 00:37:19.890 then you can have it steady state, the trajectories 00:37:19.890 --> 00:37:21.070 on like a plane. 00:37:30.620 --> 00:37:31.120 Yeah? 00:37:31.120 --> 00:37:33.347 AUDIENCE: Is it because there's like some concept 00:37:33.347 --> 00:37:39.550 of [INAUDIBLE] that can be [INAUDIBLE] for the system? 00:37:39.550 --> 00:37:43.100 PROFESSOR: I don't think that that-- I'm not 00:37:43.100 --> 00:37:44.800 aware of that being the case. 00:37:44.800 --> 00:37:48.605 But I'm hesitant to say that it's not true. 00:37:55.750 --> 00:37:58.160 I want to switch gears a little bit 00:37:58.160 --> 00:38:01.389 and think about three species interactions 00:38:01.389 --> 00:38:03.430 and, in particular the three species interactions 00:38:03.430 --> 00:38:05.460 when they're non-transitive. 00:38:05.460 --> 00:38:08.120 Because this is thought to be potentially 00:38:08.120 --> 00:38:11.355 a significant stabilizer for diversity in populations. 00:38:28.388 --> 00:38:36.240 So this is non-transitive interactions. 00:38:36.240 --> 00:38:37.990 And we often just say rock-paper-scissors. 00:38:44.630 --> 00:38:47.130 Is everybody from a cultural that plays rock-paper-scissors? 00:38:52.410 --> 00:38:52.956 Yes? 00:38:52.956 --> 00:38:53.580 AUDIENCE: Yeah. 00:38:53.580 --> 00:38:56.580 PROFESSOR: So this is a true, human universal. 00:38:56.580 --> 00:39:00.015 Although I think that what we call it does vary and so forth. 00:39:00.015 --> 00:39:02.180 You know how sometimes the linguists 00:39:02.180 --> 00:39:06.310 try to find this ur-language that our ancestors spoke 00:39:06.310 --> 00:39:07.590 50,000 years ago or whatever? 00:39:07.590 --> 00:39:09.250 I think that you probably do something 00:39:09.250 --> 00:39:10.625 similar with rock-paper-scissors, 00:39:10.625 --> 00:39:14.300 because it seems to be a pretty common theme. 00:39:14.300 --> 00:39:16.710 But the idea here is that you have-- wait, 00:39:16.710 --> 00:39:19.810 which direction does it go? 00:39:19.810 --> 00:39:23.610 So paper beats rock, scissors beats paper, 00:39:23.610 --> 00:39:25.574 but rock beats the scissors. 00:39:30.460 --> 00:39:33.570 So you can imagine that this kind of dynamic 00:39:33.570 --> 00:39:38.000 can be captured in the Latka-Volterra type framework. 00:39:38.000 --> 00:39:40.480 So you just have to set up these betas or alphas 00:39:40.480 --> 00:39:43.900 so that this is true. 00:39:43.900 --> 00:39:46.500 And again, the way to think about this 00:39:46.500 --> 00:39:49.280 would be-- the simplest thing is to think about it as dominance. 00:39:49.280 --> 00:39:52.250 So if you have the rock species and the scissor species 00:39:52.250 --> 00:39:56.050 together, then the rock species will drive the scissor species 00:39:56.050 --> 00:39:58.900 extinct and so forth. 00:39:58.900 --> 00:40:01.080 Now, this is the kind of situation 00:40:01.080 --> 00:40:04.600 that, in principle, can lead to very complicated dynamics 00:40:04.600 --> 00:40:08.240 in multi-species ecosystems that, at least in many models, 00:40:08.240 --> 00:40:13.720 can stabilize the coexistence of multiple species via some 00:40:13.720 --> 00:40:16.930 of these complex, crazy dynamics that we were just 00:40:16.930 --> 00:40:17.690 talking about. 00:40:17.690 --> 00:40:20.210 I would say that, as a mechanism for the stabilization 00:40:20.210 --> 00:40:23.630 of diversity, I don't know how convincing 00:40:23.630 --> 00:40:27.240 that is in terms of being what explains why it is there's 00:40:27.240 --> 00:40:30.970 so much diversity outside, when you look outside the window. 00:40:30.970 --> 00:40:32.970 But, at least, it's in principle true. 00:40:32.970 --> 00:40:37.966 And one of the topics of these papers-- 00:40:37.966 --> 00:40:39.340 certainly the one you just read-- 00:40:39.340 --> 00:40:41.880 is that rock-paper-scissors, i.e. 00:40:41.880 --> 00:40:44.220 non-transitive interactions on their own, 00:40:44.220 --> 00:40:46.082 may not be sufficient, but, in the presence 00:40:46.082 --> 00:40:47.540 of spatial structure, maybe it does 00:40:47.540 --> 00:40:52.750 allow for long-term coexistence of these species or strategies 00:40:52.750 --> 00:40:53.430 and so forth. 00:40:53.430 --> 00:40:56.680 And again, it's not always clear in these situations 00:40:56.680 --> 00:40:58.560 whether you're thinking about ecology, 00:40:58.560 --> 00:40:59.976 where these are different species, 00:40:59.976 --> 00:41:01.720 or you're thinking about evolution, where 00:41:01.720 --> 00:41:03.570 these are different genotypes. 00:41:03.570 --> 00:41:06.090 And indeed, in the Kerr paper that you read, 00:41:06.090 --> 00:41:07.407 these are all E. coli. 00:41:07.407 --> 00:41:09.240 So it's all one species, it's just that they 00:41:09.240 --> 00:41:11.270 have different mutations. 00:41:11.270 --> 00:41:13.360 So that's really rock-paper-scissors 00:41:13.360 --> 00:41:16.630 in the context of evolution. 00:41:16.630 --> 00:41:19.560 Whereas in the male mating strategies 00:41:19.560 --> 00:41:24.120 paper by Curt Lively, that's more of an ecological context, 00:41:24.120 --> 00:41:26.540 but it still is evolution within the species. 00:41:26.540 --> 00:41:30.910 Because these mating strategies are heritable. 00:41:36.750 --> 00:41:41.850 I did tell you the base idea of this lizard mating strategy 00:41:41.850 --> 00:41:42.386 business? 00:41:42.386 --> 00:41:44.010 Or did I never say anything about that? 00:41:44.010 --> 00:41:44.850 Not really? 00:41:44.850 --> 00:41:45.690 OK. 00:41:45.690 --> 00:41:47.900 For some reason, I thought that I'd alluded to it. 00:41:47.900 --> 00:41:50.960 Well, let me explain it to you. 00:41:50.960 --> 00:41:54.800 I think it's kind of an incredible paper. 00:41:54.800 --> 00:41:56.990 So this is a paper by my Curt Lively. 00:41:56.990 --> 00:41:59.010 So it's Sinervo and Lively. 00:42:03.970 --> 00:42:05.220 It was Nature in '96. 00:42:11.970 --> 00:42:14.010 And it's called "The Rock-Paper-Scissors 00:42:14.010 --> 00:42:16.450 Game and the Evolution of Alternative Male Strategies." 00:42:16.450 --> 00:42:19.550 So what was known is that there are 00:42:19.550 --> 00:42:24.660 many examples of alternative mating strategies in males. 00:42:24.660 --> 00:42:27.810 In particular, it's rather common that there are, 00:42:27.810 --> 00:42:34.310 what you might call, territorial males 00:42:34.310 --> 00:42:36.620 and what they often call sneaker males. 00:42:41.930 --> 00:42:45.180 This is observed in fish and various land animals. 00:42:45.180 --> 00:42:47.960 And in many of the cases, these sneaker males 00:42:47.960 --> 00:42:51.655 really do look phenotypically like females. 00:43:04.490 --> 00:43:06.750 And this has been measured using various kinds 00:43:06.750 --> 00:43:08.980 of observational, experimental approaches. 00:43:08.980 --> 00:43:10.900 There's often what you call negative frequency 00:43:10.900 --> 00:43:14.150 dependent selection between these strategies. 00:43:17.180 --> 00:43:21.340 The sneakers can often, when rare, 00:43:21.340 --> 00:43:25.050 spread in population of the territorial and vice versa. 00:43:25.050 --> 00:43:27.350 But this was, at the least the first case 00:43:27.350 --> 00:43:31.230 I'm aware of where these ideas had been demonstrated, 00:43:31.230 --> 00:43:32.920 that there were really three strategies. 00:43:32.920 --> 00:43:34.880 And the three strategies implemented, 00:43:34.880 --> 00:43:38.620 one of these rock-paper-scissors interactions. 00:43:38.620 --> 00:43:40.987 I want to maybe make a little more space. 00:43:40.987 --> 00:43:42.570 If you guys are available after class, 00:43:42.570 --> 00:43:45.267 I encourage you to come up and look at the paper, 00:43:45.267 --> 00:43:47.100 because they actually have pictures of them, 00:43:47.100 --> 00:43:51.250 so you can identify the sneaker males. 00:43:51.250 --> 00:43:55.420 Because they actually look different 00:43:55.420 --> 00:43:58.470 based on the coloring of their throat. 00:43:58.470 --> 00:44:03.910 So these are lizards that live in the mountains up outside 00:44:03.910 --> 00:44:06.310 of the Bay Area in California, in Merced County. 00:44:06.310 --> 00:44:08.790 They're side-blotched lizards. 00:44:08.790 --> 00:44:11.470 I don't know anything about that. 00:44:11.470 --> 00:44:15.190 And what they showed was that there's 00:44:15.190 --> 00:44:23.920 these guys with orange throats that are kind of aggressive. 00:44:23.920 --> 00:44:28.820 And they defend a very large territory 00:44:28.820 --> 00:44:31.190 with a large number of females. 00:44:31.190 --> 00:44:34.250 And they fight off any males that come. 00:44:34.250 --> 00:44:40.250 Then there are the dark blue. 00:44:43.130 --> 00:44:44.960 Sorry, I should have put that over here. 00:44:44.960 --> 00:44:51.450 So there are other lizards here that are dark blue throats. 00:44:51.450 --> 00:44:54.700 And these guys are less aggressive with smaller 00:44:54.700 --> 00:44:55.200 territories. 00:44:59.080 --> 00:45:04.260 So you can guess, if it were just the orange-throated guys 00:45:04.260 --> 00:45:07.280 and the dark-- and these are genetically encoded strategies, 00:45:07.280 --> 00:45:09.870 in the sense that they do seem to be passed on, 00:45:09.870 --> 00:45:14.930 and it's determined by the genes that the male inherits-- less 00:45:14.930 --> 00:45:18.740 aggressive and small territory. 00:45:18.740 --> 00:45:20.979 Can you guess which one wins between these two, 00:45:20.979 --> 00:45:21.770 against each other? 00:45:25.410 --> 00:45:25.910 What's that? 00:45:25.910 --> 00:45:27.540 AUDIENCE: The two aggressive ones 00:45:27.540 --> 00:45:30.123 PROFESSOR: Yeah, but if you just have the two aggressive ones? 00:45:30.123 --> 00:45:30.940 AUDIENCE: Yeah. 00:45:30.940 --> 00:45:34.320 PROFESSOR: This guy's going to beat this one, right? 00:45:34.320 --> 00:45:36.760 That's just because this aggressive male has a larger 00:45:36.760 --> 00:45:40.400 territory, and they pass on more genes 00:45:40.400 --> 00:45:42.324 than the less aggressive ones. 00:45:42.324 --> 00:45:44.990 However, what they found is that there's a third mating strategy 00:45:44.990 --> 00:45:47.920 here, which are these sneaker males. 00:45:47.920 --> 00:45:52.640 And they indeed look like the females. 00:45:52.640 --> 00:45:59.210 They have these yellow stripes on their throats. 00:45:59.210 --> 00:46:01.740 They look like the female, and they have no territory. 00:46:01.740 --> 00:46:05.330 Sneaker males, so they don't defend any territory. 00:46:05.330 --> 00:46:08.850 Instead, they just sneak into the territory 00:46:08.850 --> 00:46:11.040 of the other males and try to mate 00:46:11.040 --> 00:46:12.540 with the females in that territory. 00:46:12.540 --> 00:46:17.270 And what they show is that, over the course of seven years, 00:46:17.270 --> 00:46:20.304 in the mountains of Merced, they see 00:46:20.304 --> 00:46:21.720 that the frequency of these things 00:46:21.720 --> 00:46:25.950 goes through an entire oscillation. 00:46:25.950 --> 00:46:31.280 So they see just over one period of this oscillation. 00:46:31.280 --> 00:46:33.560 And their argument is that, although it's true 00:46:33.560 --> 00:46:36.090 that the aggressive males can outcompete the less aggressive 00:46:36.090 --> 00:46:39.990 ones, the sneakers actually outcompete 00:46:39.990 --> 00:46:41.700 these aggressive ones, basically, 00:46:41.700 --> 00:46:45.527 because these aggressive males, it's 00:46:45.527 --> 00:46:47.110 just too large of a territory for them 00:46:47.110 --> 00:46:49.670 to effectively defend it. 00:46:49.670 --> 00:46:53.370 So the sneakers can actually outcompete 00:46:53.370 --> 00:46:54.530 the aggressive males. 00:46:54.530 --> 00:46:58.120 Yet the less aggressive ones can outcompete the sneakers, 00:46:58.120 --> 00:47:01.980 because they are trying to defend a smaller territory. 00:47:01.980 --> 00:47:05.910 So they basically measured the frequency 00:47:05.910 --> 00:47:08.980 of these strategies and also the number of females 00:47:08.980 --> 00:47:16.670 in the different territories, from 1990 to '96 or so, 00:47:16.670 --> 00:47:20.250 and saw that this thing kind of went around in some circle 00:47:20.250 --> 00:47:22.780 in this frequency space of alternative male strategies. 00:47:31.280 --> 00:47:35.520 Kind of an incredible paper. 00:47:35.520 --> 00:47:38.750 So the idea here is this is a situation where 00:47:38.750 --> 00:47:40.284 it is non-transitive. 00:47:40.284 --> 00:47:41.950 So there's this rock-paper-scissors type 00:47:41.950 --> 00:47:42.600 dynamic. 00:47:42.600 --> 00:47:47.877 And it's also spatial, because these lizards are 00:47:47.877 --> 00:47:49.960 in some particular place, they have some territory 00:47:49.960 --> 00:47:50.680 and so forth. 00:47:50.680 --> 00:47:53.002 And in this other paper, by Benjamin Kerr, what 00:47:53.002 --> 00:47:54.960 he wanted to do was try to understand something 00:47:54.960 --> 00:47:57.400 about what the role of that spatial structure 00:47:57.400 --> 00:48:00.010 might be in maintaining the diversity. 00:48:00.010 --> 00:48:03.742 So in this case, the argument was that, if you go out 00:48:03.742 --> 00:48:05.450 and you look at these lizards, the reason 00:48:05.450 --> 00:48:07.199 that you see all three of these strategies 00:48:07.199 --> 00:48:10.640 is because of this rock-paper-scissors dynamic. 00:48:10.640 --> 00:48:12.510 If one of the strategies becomes more rare, 00:48:12.510 --> 00:48:13.940 it's going to have an advantage relative to the others. 00:48:13.940 --> 00:48:15.299 It's going to spread. 00:48:15.299 --> 00:48:16.965 And what Benjamin Kerr wanted to explore 00:48:16.965 --> 00:48:22.300 is whether that statement could be made-- somehow 00:48:22.300 --> 00:48:25.430 you can distinguish between whether the spatial component 00:48:25.430 --> 00:48:26.760 is important or not. 00:48:26.760 --> 00:48:29.416 Of course, it's hard to do that in the case of the lizards, 00:48:29.416 --> 00:48:30.790 but he was able to implement this 00:48:30.790 --> 00:48:36.780 in the case some chemical warfare behavior in bacteria. 00:48:36.780 --> 00:48:42.080 So in this paper by Kerr, and it's also 00:48:42.080 --> 00:48:53.790 a Nature paper from 2002. 00:48:53.790 --> 00:48:56.150 So this paper is called "Local Dispersal 00:48:56.150 --> 00:48:57.890 of Promotes Biodiversity in a Real Life 00:48:57.890 --> 00:49:01.437 Game of Rock-Paper-Scissors." 00:49:01.437 --> 00:49:02.270 So you guys read it. 00:49:02.270 --> 00:49:04.025 So what were the three strategies? 00:49:12.252 --> 00:49:14.370 AUDIENCE: [INAUDIBLE]. 00:49:14.370 --> 00:49:18.400 PROFESSOR: So C is the colicin producers. 00:49:23.040 --> 00:49:25.430 Incidentally, just this colicin production 00:49:25.430 --> 00:49:29.510 is kind of an incredible phenomenon already. 00:49:29.510 --> 00:49:34.280 So these are proteins that are produced that bind 00:49:34.280 --> 00:49:37.615 to other bacteria and can often make pores in the membrane 00:49:37.615 --> 00:49:38.840 and kill them. 00:49:38.840 --> 00:49:41.620 But the amazing thing about the colicin production 00:49:41.620 --> 00:49:45.730 is that in E. coli and other gram 00:49:45.730 --> 00:49:48.790 negative bacteria, the only way that these colicins are 00:49:48.790 --> 00:49:52.370 released is by cell lysis. 00:49:52.370 --> 00:49:54.270 So it's not just that the cell is engaging 00:49:54.270 --> 00:49:58.210 in some costly behavior in order to make this protein that 00:49:58.210 --> 00:49:59.200 will kill other cells. 00:49:59.200 --> 00:50:01.830 But the only way that the toxin is released 00:50:01.830 --> 00:50:05.840 is by the cell actually bursting open. 00:50:05.840 --> 00:50:09.420 So it's clear then that this has to be supported 00:50:09.420 --> 00:50:14.950 by some kind of group level or kin-selection kind of argument. 00:50:14.950 --> 00:50:17.930 This can never be good for the individual, 00:50:17.930 --> 00:50:22.330 because the individual has had to spill its guts 00:50:22.330 --> 00:50:23.890 in order to harm other cells. 00:50:23.890 --> 00:50:26.420 So the only way that this can be supported 00:50:26.420 --> 00:50:30.200 is by inhibiting the growth of competitors 00:50:30.200 --> 00:50:32.480 and allowing your kind of kin mates 00:50:32.480 --> 00:50:36.060 or other cells that also have this plasmid, and, therefore, 00:50:36.060 --> 00:50:40.200 also the immunity protein, allowing them to grow better. 00:50:40.200 --> 00:50:45.350 So this is a very neat example of an altruistic, 00:50:45.350 --> 00:50:46.505 kind of warlike behavior. 00:50:49.810 --> 00:50:51.250 So this is one of the strategies. 00:50:51.250 --> 00:50:53.150 What was the other two. 00:50:56.394 --> 00:50:58.350 AUDIENCE: [INAUDIBLE]. 00:50:58.350 --> 00:50:59.236 PROFESSOR: Resistant. 00:50:59.236 --> 00:51:00.610 So there's R, which is resistant. 00:51:06.150 --> 00:51:09.200 And between the C and R, who wins? 00:51:09.200 --> 00:51:10.080 AUDIENCE: R. 00:51:10.080 --> 00:51:12.060 PROFESSOR: R, that's right. 00:51:12.060 --> 00:51:13.560 There might be some costs associated 00:51:13.560 --> 00:51:15.860 with being resistant, but the cost 00:51:15.860 --> 00:51:17.670 is not as large as actually bursting open. 00:51:21.280 --> 00:51:22.190 What's the last one? 00:51:31.307 --> 00:51:32.640 AUDIENCE: [INAUDIBLE] sensitive. 00:51:32.640 --> 00:51:34.680 PROFESSOR: Sensitive, perfect. 00:51:34.680 --> 00:51:36.360 So this is just the normal bacteria. 00:51:40.720 --> 00:51:43.332 And the argument is that there's often 00:51:43.332 --> 00:51:45.790 a cost to be resistant, which means that sensitive bacteria 00:51:45.790 --> 00:51:47.630 will outcompete resistant. 00:51:47.630 --> 00:51:51.380 Yet, if it's just the sensitive and the colicinogenic strains, 00:51:51.380 --> 00:51:55.305 then this strain can beat this strain. 00:51:55.305 --> 00:51:57.430 So this is the idea of the rock-paper-scissors game 00:51:57.430 --> 00:51:58.730 in this system. 00:51:58.730 --> 00:52:01.420 They do say that, in some situations, 00:52:01.420 --> 00:52:04.710 the fitnesses are such that it's like this. 00:52:04.710 --> 00:52:07.750 And it's good to take these sentences seriously, 00:52:07.750 --> 00:52:13.150 because, if they thought that every time that you isolate 00:52:13.150 --> 00:52:16.890 a resistant bacterium, that it would satisfy this, 00:52:16.890 --> 00:52:19.830 they would have said, when you do this, this is what you see. 00:52:19.830 --> 00:52:22.700 Their phrasing tells you that actually, 00:52:22.700 --> 00:52:26.170 depending upon which strain you get here or there, 00:52:26.170 --> 00:52:28.607 you may or may not see this. 00:52:28.607 --> 00:52:29.690 So you have to be careful. 00:52:29.690 --> 00:52:32.148 Just because there's a nice paper that's written about this 00:52:32.148 --> 00:52:34.840 doesn't mean that, if you go out and you find particular strains 00:52:34.840 --> 00:52:37.360 that have these properties, that it will always 00:52:37.360 --> 00:52:39.440 yield this particular outcome. 00:52:42.940 --> 00:52:49.840 So they argue that these strains, just because they 00:52:49.840 --> 00:52:51.450 have a non-transitive interaction, 00:52:51.450 --> 00:52:53.010 does not necessarily mean that they 00:52:53.010 --> 00:52:59.660 will be able to coexist in a well mixed environment 00:52:59.660 --> 00:53:02.130 in particular. 00:53:02.130 --> 00:53:06.880 And in their simulations and the experiments, 00:53:06.880 --> 00:53:10.130 where they did experiments in a test tube, 00:53:10.130 --> 00:53:11.480 which strain died first? 00:53:16.918 --> 00:53:18.240 The sensitive strain. 00:53:18.240 --> 00:53:20.660 Does that make sense? 00:53:20.660 --> 00:53:22.640 Yeah, right? 00:53:22.640 --> 00:53:25.580 And the other thing to remember is that these strains, there's 00:53:25.580 --> 00:53:27.860 no reason that they should be accurately described 00:53:27.860 --> 00:53:31.647 by a Latka-Volterra type formulation. 00:53:31.647 --> 00:53:33.480 And in particular, it could just be the case 00:53:33.480 --> 00:53:35.980 that, if you have enough producers 00:53:35.980 --> 00:53:37.480 and they make enough of the colicin, 00:53:37.480 --> 00:53:39.271 then the sensitive cells are just all dead. 00:53:41.620 --> 00:53:46.437 And that will, in general, be hard to capture 00:53:46.437 --> 00:53:47.520 in this sort of framework. 00:53:51.540 --> 00:53:55.680 So the idea is that if you start with a bunch of N's, N 00:53:55.680 --> 00:53:58.220 for each of these three, then first you 00:53:58.220 --> 00:54:01.680 see that the sensitive cells die. 00:54:01.680 --> 00:54:03.730 And once the sensitive cells have died, 00:54:03.730 --> 00:54:06.100 then you're really just playing an interaction 00:54:06.100 --> 00:54:08.240 between these two. 00:54:08.240 --> 00:54:13.040 In that case, you get the colcinogenic strain dying, 00:54:13.040 --> 00:54:16.890 and you're left with just the resistant strain. 00:54:24.537 --> 00:54:26.120 One thing I want to caution you about, 00:54:26.120 --> 00:54:29.980 though, is that, just because in this experiment 00:54:29.980 --> 00:54:34.230 they saw that coexistence of three rock-paper-scissors type 00:54:34.230 --> 00:54:36.720 strains was not possible in a well mixed environment, 00:54:36.720 --> 00:54:39.445 it does not mean that it will always be the case. 00:54:42.305 --> 00:54:44.180 This is a very well known paper in the field. 00:54:44.180 --> 00:54:48.500 And the thing is that it's easy to forget what a paper shows 00:54:48.500 --> 00:54:49.810 and what it doesn't show. 00:54:49.810 --> 00:54:53.515 So what this shows is that there is maybe 00:54:53.515 --> 00:54:54.890 a set of these three strains that 00:54:54.890 --> 00:54:57.170 have a rock-paper-scissors type interaction. 00:54:57.170 --> 00:54:59.687 And in those particular three strains, 00:54:59.687 --> 00:55:01.520 that are interacting in this particular way, 00:55:01.520 --> 00:55:03.500 via colicin killing da-duh, then they 00:55:03.500 --> 00:55:06.655 don't coexist in this particular well mixed and maybe other well 00:55:06.655 --> 00:55:07.780 mixed environments as well. 00:55:07.780 --> 00:55:09.524 But this does not necessarily show 00:55:09.524 --> 00:55:11.190 that any rock-paper-scissors interaction 00:55:11.190 --> 00:55:16.317 in a well mixed environment will not support coexistence. 00:55:16.317 --> 00:55:18.650 And in particular, you might remember, in Martin Nowak's 00:55:18.650 --> 00:55:21.879 book, there are very reasonable equations 00:55:21.879 --> 00:55:24.170 that display rock-paper-scissors type interactions that 00:55:24.170 --> 00:55:26.395 can lead to coexistence. 00:55:26.395 --> 00:55:28.020 So you can kind of spiral in that space 00:55:28.020 --> 00:55:29.360 to a state of coexistence. 00:55:29.360 --> 00:55:30.762 So it's possible. 00:55:30.762 --> 00:55:32.470 But in this situation, it doesn't happen. 00:55:35.210 --> 00:55:40.592 Can somebody remind us how they implemented the spatially 00:55:40.592 --> 00:55:41.550 structured environment? 00:55:46.194 --> 00:55:47.110 AUDIENCE: [INAUDIBLE]. 00:55:47.110 --> 00:55:47.776 PROFESSOR: Yeah. 00:55:47.776 --> 00:55:49.400 So they used agar plates. 00:55:49.400 --> 00:55:53.190 They did this thing where they took this plate, 00:55:53.190 --> 00:55:56.969 and they kind of used a hexagonal grid of some sort. 00:55:56.969 --> 00:55:58.010 Is it actually hexagonal? 00:55:58.010 --> 00:55:59.194 Yeah. 00:55:59.194 --> 00:56:01.360 I don't know how they decided on the original order, 00:56:01.360 --> 00:56:06.020 but they said, OK, here is a sensitive, sensitive, 00:56:06.020 --> 00:56:10.222 resistant, colicinogenic, resistant. 00:56:10.222 --> 00:56:11.180 I don't know, whatever. 00:56:11.180 --> 00:56:14.630 So they filled it up. 00:56:14.630 --> 00:56:18.680 They put maybe 20 different kind of patches there. 00:56:18.680 --> 00:56:21.290 And then they basically, each day, 00:56:21.290 --> 00:56:23.200 they just used one of these velvets, 00:56:23.200 --> 00:56:26.790 that we often use to do replica plating. 00:56:26.790 --> 00:56:29.730 And then they just kind of took off some of the cells 00:56:29.730 --> 00:56:31.230 and put it on a fresh plate. 00:56:31.230 --> 00:56:36.440 And then they did this for a week or so. 00:56:36.440 --> 00:56:39.240 And then they saw that there was some sense 00:56:39.240 --> 00:56:42.990 that there was a spatial dynamic taking 00:56:42.990 --> 00:56:45.680 place, where the spatial structure was 00:56:45.680 --> 00:56:48.520 kind of reminiscent of this. 00:56:48.520 --> 00:56:51.556 The sensitive kind of moved into the resistant. 00:56:51.556 --> 00:56:54.000 The resistant kind of moved into the colicinogenic. 00:56:54.000 --> 00:56:55.990 Colicinogenic kind of moved into the sensitive. 00:56:55.990 --> 00:56:59.630 But they couldn't actually see all of those, 00:56:59.630 --> 00:57:03.767 because two of the strains, they said, grew to similar density. 00:57:03.767 --> 00:57:06.100 That was between the sensitive and the resistant, right? 00:57:06.100 --> 00:57:08.420 So visually, they can only distinguish 00:57:08.420 --> 00:57:11.710 the colicinogenic strain relative to the others. 00:57:11.710 --> 00:57:15.540 But what they found, though, is that, over a similar amount 00:57:15.540 --> 00:57:19.330 of time, they got coexistence of all three. 00:57:22.440 --> 00:57:25.740 So R, C, S, kind of all-- the number 00:57:25.740 --> 00:57:28.840 is just a function of time-- stuck around in this spatially 00:57:28.840 --> 00:57:31.670 structured environment. 00:57:31.670 --> 00:57:34.230 So their argument is that, in some cases maybe, 00:57:34.230 --> 00:57:36.490 this non-transitive interaction is not 00:57:36.490 --> 00:57:39.850 sufficient to maintain diversity in a population, 00:57:39.850 --> 00:57:43.140 whether it's genetic diversity or species diversity. 00:57:43.140 --> 00:57:45.490 But it may be that it's very important 00:57:45.490 --> 00:57:48.260 to have this spatial component. 00:57:51.850 --> 00:57:53.520 If you're curious about these things, 00:57:53.520 --> 00:57:56.510 there's also a nice computational study 00:57:56.510 --> 00:58:05.260 done by Erwin Frey, who studied these rock-paper-scissors 00:58:05.260 --> 00:58:07.190 dynamics as a function of the mobility 00:58:07.190 --> 00:58:09.290 of the individual agents. 00:58:09.290 --> 00:58:12.964 And in that study, he found that there 00:58:12.964 --> 00:58:14.630 was sort of a critical level of mobility 00:58:14.630 --> 00:58:24.440 in which you kind of switch from a spatially structured 00:58:24.440 --> 00:58:27.660 environment with coexistence to, above some mobility, 00:58:27.660 --> 00:58:31.410 you end up losing the diversity, because it kind of goes 00:58:31.410 --> 00:58:34.470 into this well mixed regime. 00:58:34.470 --> 00:58:36.760 So if you're curious about these things, 00:58:36.760 --> 00:58:38.904 you can look up Erwin's paper. 00:58:41.766 --> 00:58:47.890 Are there any questions about what they did here, 00:58:47.890 --> 00:58:50.126 what you think it means? 00:58:50.126 --> 00:58:50.626 Yeah? 00:58:50.626 --> 00:58:54.070 AUDIENCE: You said you couldn't use [INAUDIBLE]. 00:58:54.070 --> 00:58:56.038 How did they know if it was [INAUDIBLE]? 00:58:57.740 --> 00:59:00.115 PROFESSOR: They can just scrape off everything and plate. 00:59:02.820 --> 00:59:04.422 AUDIENCE: And then you can tell? 00:59:04.422 --> 00:59:05.130 PROFESSOR: Right. 00:59:05.130 --> 00:59:07.060 Because, well, then you can ask, oh, we're 00:59:07.060 --> 00:59:10.370 those guys sensitive to colicin or not? 00:59:10.370 --> 00:59:12.770 AUDIENCE: Why not actually just inject them with colicin? 00:59:12.770 --> 00:59:13.436 PROFESSOR: Yeah. 00:59:13.436 --> 00:59:16.930 Or plate them on something with colicin. 00:59:16.930 --> 00:59:20.780 This is a system that has been very influential, 00:59:20.780 --> 00:59:24.100 but it's really not yet or has not 00:59:24.100 --> 00:59:27.400 been a domesticated kind of model system in the sense 00:59:27.400 --> 00:59:31.460 that they are not nice, fluorescent proteins. 00:59:31.460 --> 00:59:34.480 This is not on a nice cloning plasmid. 00:59:34.480 --> 00:59:37.230 It was a natural plasmid. 00:59:37.230 --> 00:59:41.390 And this resistant strain, the way that they get it is, 00:59:41.390 --> 00:59:46.060 basically, they take some sensitive cells, 00:59:46.060 --> 00:59:47.370 and they add colicin. 00:59:47.370 --> 00:59:49.060 Let's say they supernatant from here, and they ask, 00:59:49.060 --> 00:59:51.075 which cells grow> and then that's a resistant cell. 00:59:51.075 --> 00:59:52.430 And it's genetically resistant. 00:59:52.430 --> 00:59:55.170 But then each one's going to different. 00:59:55.170 --> 00:59:57.730 They have done sequencing. 00:59:57.730 --> 01:00:00.660 It's a surface receptor that the colicin maybe uses to go in 01:00:00.660 --> 01:00:01.560 or other things. 01:00:01.560 --> 01:00:05.290 But indeed, we actually did some experiments 01:00:05.290 --> 01:00:07.570 with some of these strains. 01:00:07.570 --> 01:00:09.675 And it's a little bit messy. 01:00:09.675 --> 01:00:12.710 And I think that there's a sense that the field maybe 01:00:12.710 --> 01:00:17.722 needs to just make some nice plasmids, with nice colors, 01:00:17.722 --> 01:00:19.430 so that we can really distinguish things. 01:00:19.430 --> 01:00:24.020 Because I think that this system, despite 500 citations 01:00:24.020 --> 01:00:26.410 or so, almost none of those citations 01:00:26.410 --> 01:00:30.250 are experimental papers really exploring this thing. 01:00:30.250 --> 01:00:34.640 Because I think that it still is just a little bit messy. 01:00:34.640 --> 01:00:37.820 And I think that it's also a very pretty system 01:00:37.820 --> 01:00:38.908 to explore these ideas. 01:00:46.376 --> 01:00:46.876 Yeah? 01:00:46.876 --> 01:00:47.792 AUDIENCE: [INAUDIBLE]? 01:00:53.367 --> 01:00:55.200 PROFESSOR: So the question is, why can't you 01:00:55.200 --> 01:00:56.190 distinguish these? 01:00:56.190 --> 01:00:59.450 And they say it's because they're similar densities. 01:00:59.450 --> 01:01:02.790 I don't even know if that's even. 01:01:02.790 --> 01:01:09.650 Of course, well, if you added sensitive cells here 01:01:09.650 --> 01:01:12.200 and sensitive cells here and they grew up, 01:01:12.200 --> 01:01:15.790 it's likely you're not going to see a boundary. 01:01:15.790 --> 01:01:18.780 And just more generally, if you take 01:01:18.780 --> 01:01:21.090 similar strains-- and these are rather similar strains, 01:01:21.090 --> 01:01:23.340 this just has some mutation relative to this-- then 01:01:23.340 --> 01:01:26.810 you won't, often, see a boundary when the colonies kind of grow 01:01:26.810 --> 01:01:28.732 up. 01:01:28.732 --> 01:01:31.212 AUDIENCE: So they just didn't see boundaries. 01:01:31.212 --> 01:01:34.024 It's not like they didn't see-- 01:01:34.024 --> 01:01:35.940 PROFESSOR: So they saw there were cells there, 01:01:35.940 --> 01:01:39.894 but they couldn't necessarily say that it was true 01:01:39.894 --> 01:01:41.310 that the sensitive cells were kind 01:01:41.310 --> 01:01:43.615 of spreading into the resistant cell region, because they 01:01:43.615 --> 01:01:44.698 couldn't see the boundary. 01:01:52.070 --> 01:01:54.960 So in the last 20 minutes here, 15, 20, 01:01:54.960 --> 01:01:58.240 I just want to say a few things about these population waves. 01:02:08.270 --> 01:02:11.380 For many, many purposes, there are strong arguments 01:02:11.380 --> 01:02:14.620 to be made that, in the context of evolution or ecology, 01:02:14.620 --> 01:02:16.960 spatial dynamics matter. 01:02:16.960 --> 01:02:20.030 And the way that we often think about the spatial dynamics 01:02:20.030 --> 01:02:22.857 is via some effective diffusive process. 01:02:22.857 --> 01:02:24.440 And that's convenient, because we know 01:02:24.440 --> 01:02:25.700 a lot about how to model it. 01:02:25.700 --> 01:02:28.280 But it's also maybe a reasonable description 01:02:28.280 --> 01:02:32.340 of the motion of animals and other living things over length 01:02:32.340 --> 01:02:34.590 scales that are large compared to the movements 01:02:34.590 --> 01:02:36.700 of the animals. 01:02:36.700 --> 01:02:40.600 So what we do is we take some equation, such as, 01:02:40.600 --> 01:02:44.260 well, we often have, say, dN/dt. 01:02:44.260 --> 01:02:46.220 So this is going to be about population waves. 01:02:49.700 --> 01:02:52.570 So we take our standard thing where we say, oh, here's 01:02:52.570 --> 01:02:58.180 r N 1 minus N/K. 01:02:58.180 --> 01:03:00.830 And then we just want to add some spatial dynamic. 01:03:00.830 --> 01:03:01.840 So what we're going to do is we're going to say, 01:03:01.840 --> 01:03:03.506 now, the derivative, now it's a density. 01:03:03.506 --> 01:03:05.540 So we're going to use a little n, just for fun. 01:03:09.280 --> 01:03:14.120 And we might still use a K just for simple. 01:03:14.120 --> 01:03:16.550 But then we have to add some diffusive term. 01:03:22.097 --> 01:03:24.430 So there still is going to be a local carrying capacity. 01:03:24.430 --> 01:03:27.100 Now this is in terms of density of the organism. 01:03:27.100 --> 01:03:29.850 And we're going to assume that there is some diffusive type 01:03:29.850 --> 01:03:31.970 motion of the organism. 01:03:31.970 --> 01:03:34.600 Now this is this is, of course, going over the life-scale 01:03:34.600 --> 01:03:38.750 over which the animals are actually doing things 01:03:38.750 --> 01:03:39.880 over shorter time scales. 01:03:39.880 --> 01:03:42.390 So it could be that different organisms 01:03:42.390 --> 01:03:45.040 have very different modes of motility. 01:03:45.040 --> 01:03:47.350 In some cases they walk or swim, in some cases 01:03:47.350 --> 01:03:49.860 they just get picked up by a passing deer. 01:03:49.860 --> 01:03:51.940 So there are a wide range of ways 01:03:51.940 --> 01:03:54.100 in which organisms move around. 01:03:54.100 --> 01:03:57.630 But if you look at it over kind of longer length time scales, 01:03:57.630 --> 01:04:01.890 then maybe it doesn't matter. 01:04:01.890 --> 01:04:05.100 And certainly, if it's an unbiased kind of motion, 01:04:05.100 --> 01:04:09.950 with motility being well-behaved, 01:04:09.950 --> 01:04:14.400 i.e., if the probability distribution of these steps, 01:04:14.400 --> 01:04:17.512 as long as it's not long-tailed. 01:04:17.512 --> 01:04:19.470 Similarly, Oskar Hallatschek, over at Berkeley, 01:04:19.470 --> 01:04:21.136 has been doing a lot of fun work looking 01:04:21.136 --> 01:04:23.070 at epidemic spread in cases where 01:04:23.070 --> 01:04:26.110 this kernel, the kind of step size distribution, 01:04:26.110 --> 01:04:27.710 has long tails. 01:04:27.710 --> 01:04:31.510 It leads to qualitatively different behaviors. 01:04:31.510 --> 01:04:34.200 So if you're curious about such things, check out Oskar's work. 01:04:34.200 --> 01:04:38.387 But for normal kind of the step size distributions, 01:04:38.387 --> 01:04:40.470 then the central limit theorem type considerations 01:04:40.470 --> 01:04:42.920 just tell you that you can maybe just look 01:04:42.920 --> 01:04:47.737 at it like this over time spatial scales that are bigger. 01:04:47.737 --> 01:04:48.237 Yeah? 01:04:48.237 --> 01:04:51.576 AUDIENCE: But is it a unique correlation in your step? 01:04:51.576 --> 01:04:54.676 Because the central limit theorem, as long 01:04:54.676 --> 01:04:56.475 as the variance-- I guess, you're saying? 01:04:56.475 --> 01:04:58.350 PROFESSOR: That's why I'm saying long-tailed. 01:04:58.350 --> 01:05:00.302 AUDIENCE: If the variance is infinite? 01:05:00.302 --> 01:05:02.510 PROFESSOR: Exactly Yeah, so that's what I was saying. 01:05:02.510 --> 01:05:06.130 And indeed, people argue, in the case of disease spread, 01:05:06.130 --> 01:05:08.580 with modern air travel and so forth, 01:05:08.580 --> 01:05:10.290 that the probability distribution 01:05:10.290 --> 01:05:13.440 of kind of step sizes for infected individuals, 01:05:13.440 --> 01:05:16.950 over the next week, is long-tailed, right? 01:05:16.950 --> 01:05:19.130 Because there's a fair chance that you're just 01:05:19.130 --> 01:05:20.730 going to stay around your local neighborhood, 01:05:20.730 --> 01:05:21.890 but there's a smaller chance you're 01:05:21.890 --> 01:05:23.473 going to go to the other side of town. 01:05:23.473 --> 01:05:27.540 But you might go to a business trip over in DC. 01:05:27.540 --> 01:05:30.780 At some small right, though, you also fly to South Africa 01:05:30.780 --> 01:05:33.010 and go to a conference. 01:05:33.010 --> 01:05:36.090 So all of these things have reasonable probabilities, 01:05:36.090 --> 01:05:39.240 and so there are arguments that this is kind of some power law 01:05:39.240 --> 01:05:40.200 type distribution. 01:05:40.200 --> 01:05:42.540 And in those cases, you don't necessarily 01:05:42.540 --> 01:05:46.612 have finite variance, and so then you can't just 01:05:46.612 --> 01:05:47.820 put everything under the rug. 01:05:51.237 --> 01:05:53.320 But we first want to understand what happens here, 01:05:53.320 --> 01:05:54.810 and then we can-- well, we're not 01:05:54.810 --> 01:05:56.310 going to-- but then other people can 01:05:56.310 --> 01:05:58.768 think more deeply about what happens in fancier situations. 01:06:02.830 --> 01:06:04.960 It is worth saying, though, that this approach, 01:06:04.960 --> 01:06:08.420 I mean it looks very physics-y, in the sense that 01:06:08.420 --> 01:06:13.020 physicists like simple equations where we add diffusion 01:06:13.020 --> 01:06:13.610 and so forth. 01:06:13.610 --> 01:06:17.872 But this is not what a physicist came up with. 01:06:17.872 --> 01:06:21.980 These are classic ideas in evolution and ecology. 01:06:21.980 --> 01:06:26.780 The solution to this was originally done by Fisher. 01:06:26.780 --> 01:06:28.920 So this was in the 1920s or so. 01:06:28.920 --> 01:06:30.560 So a long time ago, originally to try 01:06:30.560 --> 01:06:32.440 to understand not the spread of a population 01:06:32.440 --> 01:06:36.260 but the spread of a beneficial allele in a spatial population. 01:06:36.260 --> 01:06:39.290 And once again, this highlights the deep connections 01:06:39.290 --> 01:06:42.090 between evolution and ecology. 01:06:42.090 --> 01:06:44.840 You can have a genetic wave in space 01:06:44.840 --> 01:06:47.460 of a of a beneficial mutant spreading, 01:06:47.460 --> 01:06:50.446 or you can have a population wave of an invasive species 01:06:50.446 --> 01:06:52.821 or whatnot, and you end up getting very similar dynamics. 01:07:00.596 --> 01:07:01.970 So the basic idea, here, is that, 01:07:01.970 --> 01:07:10.730 if you look at the density as a function of position. 01:07:10.730 --> 01:07:14.180 If start with, there's one individual. 01:07:14.180 --> 01:07:15.671 What's going to happen? 01:07:19.150 --> 01:07:22.450 It's going to start dividing, right? 01:07:22.450 --> 01:07:24.337 So we kind of get up. 01:07:24.337 --> 01:07:25.420 And it's going to come up. 01:07:25.420 --> 01:07:27.590 And eventually it's going to saturate 01:07:27.590 --> 01:07:32.900 at this carrying capacity, K. 01:07:32.900 --> 01:07:35.360 And then you end up getting these spreading population 01:07:35.360 --> 01:07:38.467 waves that look like this. 01:07:38.467 --> 01:07:40.050 And the reason we're calling it a wave 01:07:40.050 --> 01:07:44.000 is because the shape of this front is the same over time. 01:07:47.330 --> 01:07:59.450 So it can really to be described as some function x minus vt. 01:07:59.450 --> 01:08:01.840 And we are going to, maybe, typically assume 01:08:01.840 --> 01:08:03.910 that we're in a situation where we don't have 01:08:03.910 --> 01:08:04.810 to think about the left and the right, 01:08:04.810 --> 01:08:06.510 because it's just too complicated. 01:08:06.510 --> 01:08:09.070 So we'll just imagine it being that it's at saturation here, 01:08:09.070 --> 01:08:10.861 and then we're looking at some front that's 01:08:10.861 --> 01:08:12.860 moving to the right. 01:08:12.860 --> 01:08:17.988 Now, by dimensional analysis, we should 01:08:17.988 --> 01:08:20.279 be able to figure out what the velocity is going to be. 01:08:22.707 --> 01:08:24.665 Remember how much we liked dimensional analysis 01:08:24.665 --> 01:08:27.600 in this class? 01:08:27.600 --> 01:08:29.060 Yes. 01:08:29.060 --> 01:08:31.910 So what we're going to do is I'll 01:08:31.910 --> 01:08:33.950 give you some characters that you're 01:08:33.950 --> 01:08:35.765 going to be able to use in your quest. 01:08:40.109 --> 01:08:48.069 So we can use r, K, D. I'll give you a square root in case you 01:08:48.069 --> 01:08:50.950 find it useful. 01:08:50.950 --> 01:08:59.600 And you can raise something to the second power as well. 01:08:59.600 --> 01:09:02.080 So what you can do is you're going to set up your cards 01:09:02.080 --> 01:09:05.330 so that when I look at it, from the left to the right, 01:09:05.330 --> 01:09:08.380 it will describe the velocity of this wave. 01:09:12.988 --> 01:09:14.029 I'll give you 30 seconds. 01:09:17.069 --> 01:09:24.049 So this is dimensional analysis for this wave velocity. 01:10:06.890 --> 01:10:08.310 All right, do you need more time? 01:10:08.310 --> 01:10:08.810 Yes? 01:10:08.810 --> 01:10:10.000 OK, that's fine. 01:11:06.650 --> 01:11:08.800 All right, let's go ahead and vote. 01:11:13.310 --> 01:11:16.730 Construct your answer, remember, from me, from left to right. 01:11:19.530 --> 01:11:22.950 Ready, three, two, one. 01:11:27.160 --> 01:11:28.170 All right. 01:11:28.170 --> 01:11:30.700 We have trouble. 01:11:30.700 --> 01:11:32.850 A key skill is being able to imagine yourself 01:11:32.850 --> 01:11:34.680 in someone else's shoes. 01:11:34.680 --> 01:11:36.800 So if I'm viewing-- but that's OK. 01:11:41.630 --> 01:11:47.250 So we have, here, this is kind of units of 1 over time. 01:11:47.250 --> 01:11:51.850 This is what length squared over time. 01:11:51.850 --> 01:11:54.515 Whereas this is a density. 01:11:57.090 --> 01:12:01.190 If we want something that is a length over time, 01:12:01.190 --> 01:12:04.190 then we're going to end up having to take r times D 01:12:04.190 --> 01:12:06.230 and take a square root. 01:12:06.230 --> 01:12:14.230 So this should be DAC, which is a square root 01:12:14.230 --> 01:12:16.689 of-- of course, it depends on how you're entering it 01:12:16.689 --> 01:12:18.980 into your calculator, if you have scientific calculator 01:12:18.980 --> 01:12:21.920 or something else, maybe. 01:12:21.920 --> 01:12:24.290 And indeed, it ends up, there's a 2 here. 01:12:24.290 --> 01:12:26.100 So this is the velocity. 01:12:31.080 --> 01:12:31.658 Yes? 01:12:31.658 --> 01:12:32.574 AUDIENCE: [INAUDIBLE]. 01:12:47.970 --> 01:12:49.470 PROFESSOR: I see what you're saying. 01:12:49.470 --> 01:12:51.840 But it ends up not being true. 01:12:51.840 --> 01:12:57.030 These derivative signs don't have any units. 01:12:57.030 --> 01:13:00.210 So this still has units of a density 01:13:00.210 --> 01:13:02.150 divided by a length squared. 01:13:02.150 --> 01:13:06.096 So for unit purposes, you just look at this thing and at this. 01:13:06.096 --> 01:13:09.155 This squared does mean there's a length squared 01:13:09.155 --> 01:13:11.530 in the denominator, but this squared doesn't do anything. 01:13:15.140 --> 01:13:17.210 So D is still a length squared over time. 01:13:20.140 --> 01:13:22.380 So this is the famous Fisher velocity. 01:13:25.590 --> 01:13:27.185 There are some mathematical subtleties 01:13:27.185 --> 01:13:29.705 to all of this that we're not really going to get into. 01:13:29.705 --> 01:13:30.630 I don't know. 01:13:30.630 --> 01:13:31.880 I'm having trouble-- velocity. 01:13:34.540 --> 01:13:38.350 There are a few features to highlight. 01:13:38.350 --> 01:13:42.497 If the organism grows faster, the wave 01:13:42.497 --> 01:13:43.580 is going to spread faster. 01:13:43.580 --> 01:13:44.860 That make sense? 01:13:44.860 --> 01:13:48.660 If it has a larger mobility, it also moves faster. 01:13:48.660 --> 01:13:50.000 That makes sense. 01:13:50.000 --> 01:13:54.260 Of course, the velocity is given by both of those things. 01:13:54.260 --> 01:13:59.440 So this wave coming out really is a population level property. 01:13:59.440 --> 01:14:01.130 Because it's not just growth. 01:14:01.130 --> 01:14:03.430 It's not just motion. 01:14:03.430 --> 01:14:08.406 It's a result of the coupled division and diffusion 01:14:08.406 --> 01:14:10.280 that leads to this population wave spreading. 01:14:15.330 --> 01:14:17.440 Importantly, to first order, it doesn't 01:14:17.440 --> 01:14:22.030 depend on the carrying capacity, at least 01:14:22.030 --> 01:14:23.470 within the deterministic regime. 01:14:27.350 --> 01:14:31.957 AUDIENCE: It's interesting that, if the reproductive rate goes 01:14:31.957 --> 01:14:35.110 to 0, suddenly the population stops spreading. 01:14:39.226 --> 01:14:40.350 PROFESSOR: So first of all. 01:14:40.350 --> 01:14:42.890 You'd say, oh, it would be sort of surprising if it really 01:14:42.890 --> 01:14:44.910 kept on spreading in the absence of growth. 01:14:44.910 --> 01:14:46.470 But what you're pointing out is that, 01:14:46.470 --> 01:14:49.232 if you just, at one moment, turn off division, 01:14:49.232 --> 01:14:50.690 then there will still be diffusion. 01:14:50.690 --> 01:14:52.330 It'll still keep on going. 01:14:52.330 --> 01:14:55.820 But this is the velocity of a wave when it's a wave. 01:14:55.820 --> 01:15:00.630 When it's described by a function like this. 01:15:00.630 --> 01:15:03.400 So it's true that you could turn off division, 01:15:03.400 --> 01:15:04.456 and it'll still diffuse. 01:15:04.456 --> 01:15:06.080 But then the shape is changing as well. 01:15:12.840 --> 01:15:18.450 I'm going to draw a few lines describing 01:15:18.450 --> 01:15:22.310 possible populations. 01:15:22.310 --> 01:15:27.850 Now, let's assume that they have the same motion, diffusion. 01:15:27.850 --> 01:15:30.650 I want to know, which one has the largest velocity? 01:15:44.530 --> 01:15:51.110 Is it A, B, C, D? 01:16:09.030 --> 01:16:10.030 It's the same diffusion. 01:16:17.040 --> 01:16:19.700 Per capita growth rate as a function of the density 01:16:19.700 --> 01:16:21.810 for three different organisms. 01:16:44.470 --> 01:16:47.865 Ready, three, two, one. 01:16:50.820 --> 01:16:53.870 All right, so I'd say we have a fair number of B's, D's. 01:16:57.080 --> 01:17:03.670 It seems like it's B versus D. Now, this is tricky because D, 01:17:03.670 --> 01:17:07.950 we have not explicitly considered here. 01:17:07.950 --> 01:17:12.232 But it turns out that the answer is B. Certainly, 01:17:12.232 --> 01:17:13.940 between these three, these are all really 01:17:13.940 --> 01:17:16.500 logistic growth functions. 01:17:16.500 --> 01:17:20.140 And so from the standpoint of here, it's just this r. 01:17:20.140 --> 01:17:25.017 And r is the division rate at 0 cell density. 01:17:25.017 --> 01:17:26.850 The per capita growth rate at 0 cell density 01:17:26.850 --> 01:17:30.050 is what determines the velocity in a Fisher wave. 01:17:30.050 --> 01:17:32.400 And indeed, that's true even if there's 01:17:32.400 --> 01:17:35.583 no decrease in the growth up right until you 01:17:35.583 --> 01:17:37.480 get to some carrying capacity. 01:17:37.480 --> 01:17:41.944 And indeed, all of these cases, the division rate 01:17:41.944 --> 01:17:44.110 and the growth rate that's relevant for the velocity 01:17:44.110 --> 01:17:49.445 is when it hits this axis. 01:17:49.445 --> 01:17:53.100 And indeed, all of these waves are described as Fisher 01:17:53.100 --> 01:17:54.000 or pulled waves. 01:17:56.630 --> 01:17:59.810 Because there's a sense that the entire wave 01:17:59.810 --> 01:18:02.480 is determined by the front of the wave. 01:18:02.480 --> 01:18:05.240 So we drew this profile. 01:18:05.240 --> 01:18:06.522 I didn't do that very well. 01:18:06.522 --> 01:18:07.730 Here, this is an exponential. 01:18:11.540 --> 01:18:16.029 And the exponential actually is what's pulling this wave. 01:18:16.029 --> 01:18:17.820 There ends up being a characteristic length 01:18:17.820 --> 01:18:23.160 scale here that is the square root of D/r. 01:18:23.160 --> 01:18:26.230 So this is the length scale of the exponential. 01:18:26.230 --> 01:18:30.000 And the velocity and the length scale 01:18:30.000 --> 01:18:34.630 are only functions of the division 01:18:34.630 --> 01:18:36.770 rate in the limit below cell density 01:18:36.770 --> 01:18:38.040 or low density of organism. 01:18:42.860 --> 01:18:46.187 The shape of what goes on here changes, indeed, 01:18:46.187 --> 01:18:48.020 the bulk properties of the wave, but doesn't 01:18:48.020 --> 01:18:52.910 change the velocity. 01:18:52.910 --> 01:18:56.370 And I just want to make one comparison of all this 01:18:56.370 --> 01:18:59.020 to-- because there's another qualitatively different kind 01:18:59.020 --> 01:19:02.660 of wave, which is a so-called pushed wave. 01:19:07.680 --> 01:19:10.760 And that's what happens if you have an Allee 01:19:10.760 --> 01:19:13.060 effect, particularly like a strong Allee effect. 01:19:13.060 --> 01:19:18.610 If this thing looks-- like this is certainly possible. 01:19:21.840 --> 01:19:25.230 This is an Allee effect. 01:19:25.230 --> 01:19:27.939 Now, if you just said, oh, the only thing that matters is 01:19:27.939 --> 01:19:30.230 the growth rate at low cell density, you would say, oh, 01:19:30.230 --> 01:19:32.300 this thing cannot possibly expand. 01:19:32.300 --> 01:19:35.130 Although it turns out that it still is possible. 01:19:35.130 --> 01:19:37.770 And in this situation, it would be 01:19:37.770 --> 01:19:41.200 called a push wave, where your profile somehow 01:19:41.200 --> 01:19:42.500 maybe looks kind of similar. 01:19:42.500 --> 01:19:43.910 But instead of it being the front 01:19:43.910 --> 01:19:45.920 of the wave that's pulling the wave, 01:19:45.920 --> 01:19:47.890 it's diffusion around the bulk. 01:19:47.890 --> 01:19:49.430 Because the bulk is the part that 01:19:49.430 --> 01:19:51.840 is actually happily growing. 01:19:51.840 --> 01:19:55.350 Because the front, here, in this case, is dying. 01:19:55.350 --> 01:19:59.610 Yet it still is possible to have a positive velocity. 01:19:59.610 --> 01:20:03.330 And so this is, then, a qualitatively different kind 01:20:03.330 --> 01:20:05.681 of population expansion. 01:20:05.681 --> 01:20:07.180 So cooperatively growing populations 01:20:07.180 --> 01:20:10.280 expand very differently from logistically growing 01:20:10.280 --> 01:20:11.200 populations. 01:20:11.200 --> 01:20:14.200 And one of things that the reading in Physics Today 01:20:14.200 --> 01:20:16.000 talked about is these different rates 01:20:16.000 --> 01:20:20.340 of loss of heterozygocity and so forth in different populations. 01:20:20.340 --> 01:20:23.990 And as you might expect, the pulled waves 01:20:23.990 --> 01:20:26.570 have a smaller effective population size 01:20:26.570 --> 01:20:31.137 than the pushed waves, because, here, the relevant population 01:20:31.137 --> 01:20:32.720 is at the front if it's a low density. 01:20:32.720 --> 01:20:34.261 Whereas here, the relevant population 01:20:34.261 --> 01:20:37.144 is the bulk that's at high density. 01:20:37.144 --> 01:20:38.560 With that, I think we should quit. 01:20:38.560 --> 01:20:39.955 But I will see you on Tuesday. 01:20:39.955 --> 01:20:42.500 And we'll talk about this neutral theory in ecology. 01:20:42.500 --> 01:20:44.050 Thanks.