1 00:00:00,080 --> 00:00:02,500 The following content is provided under a Creative 2 00:00:02,500 --> 00:00:04,019 Commons license. 3 00:00:04,019 --> 00:00:06,360 Your support will help MIT OpenCourseWare 4 00:00:06,360 --> 00:00:10,730 continue to offer high quality, educational resources for free. 5 00:00:10,730 --> 00:00:13,330 To make a donation or view additional materials 6 00:00:13,330 --> 00:00:17,217 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,217 --> 00:00:17,842 at ocw.mit.edu. 8 00:00:20,270 --> 00:00:21,770 PROFESSOR: Today, what we want to do 9 00:00:21,770 --> 00:00:24,760 is finish off our discussion of Lotka-Volterra competition 10 00:00:24,760 --> 00:00:25,666 models. 11 00:00:25,666 --> 00:00:27,540 So starting with this idea of the two species 12 00:00:27,540 --> 00:00:29,950 interacting competitively but then moving on to try to 13 00:00:29,950 --> 00:00:31,366 think about the general properties 14 00:00:31,366 --> 00:00:32,450 of Lotka-Volterra systems. 15 00:00:32,450 --> 00:00:34,199 In particular, when you have more species, 16 00:00:34,199 --> 00:00:36,100 the kinds of dynamics that you can get. 17 00:00:36,100 --> 00:00:39,270 Also, we're going to talk about these non-transitive 18 00:00:39,270 --> 00:00:41,770 interactions, which are the rock-paper-scissors type 19 00:00:41,770 --> 00:00:45,550 interactions that may facilitate the maintenance of diversity 20 00:00:45,550 --> 00:00:47,640 in populations or ecosystems, in particular, 21 00:00:47,640 --> 00:00:49,910 in the presence of some sort of spatial structure. 22 00:00:49,910 --> 00:00:53,204 And so we'll talk both about this demonstration 23 00:00:53,204 --> 00:00:54,870 of rock-paper-scissors type interactions 24 00:00:54,870 --> 00:00:56,580 in the context of male mating strategies 25 00:00:56,580 --> 00:00:59,850 in lizards, this paper that was by Curt Lively. 26 00:00:59,850 --> 00:01:02,424 And then we'll talk about another paper 27 00:01:02,424 --> 00:01:04,590 in the microbial realm, where they showed that there 28 00:01:04,590 --> 00:01:07,500 are rock-paper-scissors type interactions in the context 29 00:01:07,500 --> 00:01:10,940 of colicin production, toxin production 30 00:01:10,940 --> 00:01:12,870 in the context of bacteria. 31 00:01:12,870 --> 00:01:17,720 Then, at the end, we'll talk about these population waves. 32 00:01:17,720 --> 00:01:19,761 There was a rather mathematical reading 33 00:01:19,761 --> 00:01:22,260 that I had originally proposed, but then we kind of switched 34 00:01:22,260 --> 00:01:24,730 things up a bit, so that you could, instead, 35 00:01:24,730 --> 00:01:30,739 read that rock-paper-scissors paper-- that is confusing-- 36 00:01:30,739 --> 00:01:32,030 that I think is maybe more fun. 37 00:01:32,030 --> 00:01:34,580 But I'll tell you kind of this basic idea of the population 38 00:01:34,580 --> 00:01:38,242 waves, what happens, if you have a combination of some growth 39 00:01:38,242 --> 00:01:40,200 process together with some effective diffusion, 40 00:01:40,200 --> 00:01:42,100 you can get these population waves that 41 00:01:42,100 --> 00:01:44,890 correspond to this process of range expansion, 42 00:01:44,890 --> 00:01:46,840 where a population expands into new territory. 43 00:01:50,870 --> 00:01:52,950 I just started by putting up what 44 00:01:52,950 --> 00:01:54,780 we had from the last class. 45 00:01:54,780 --> 00:01:58,070 So this is the two species Lotka-Volterra competition 46 00:01:58,070 --> 00:01:58,930 model. 47 00:01:58,930 --> 00:02:02,870 So what we found was that, for this to be competition, 48 00:02:02,870 --> 00:02:05,960 for the species to be kind of bad for each other, 49 00:02:05,960 --> 00:02:10,025 that that corresponds to the betas being positive. 50 00:02:12,740 --> 00:02:19,720 Now, what we found is there are four cases 51 00:02:19,720 --> 00:02:22,675 in terms of the outcome of this two species interaction. 52 00:02:22,675 --> 00:02:24,280 And we wanted to, at least, try to get 53 00:02:24,280 --> 00:02:29,040 some sense of why that was and what the trajectories 54 00:02:29,040 --> 00:02:30,180 might look like. 55 00:02:30,180 --> 00:02:33,370 If you look at and N1 and N2, we can draw these nullclines 56 00:02:33,370 --> 00:02:35,470 and then get a sense of where the trajectories are 57 00:02:35,470 --> 00:02:36,180 going to go. 58 00:02:36,180 --> 00:02:38,660 Now, the basic outcome of this two species Lotka-Volterra 59 00:02:38,660 --> 00:02:42,460 competition model are really exactly the same 60 00:02:42,460 --> 00:02:45,360 as the possible outcomes when we're 61 00:02:45,360 --> 00:02:48,160 thinking about frequency dependent selection, where 62 00:02:48,160 --> 00:02:54,310 we could get, in this case, that species 1 dominates or species 63 00:02:54,310 --> 00:02:57,329 2 dominates, 1 or 2. 64 00:02:57,329 --> 00:02:59,620 But then we could also get coexistence or bi-stability. 65 00:03:03,620 --> 00:03:05,460 And indeed, there is, in general, 66 00:03:05,460 --> 00:03:09,770 a mapping from the Lotka-Volterra kind of approach 67 00:03:09,770 --> 00:03:15,560 here and the approach that was kind of in Martin Nonwak's 68 00:03:15,560 --> 00:03:19,444 book of thinking about frequency dependent selection 69 00:03:19,444 --> 00:03:20,235 in this population. 70 00:03:23,080 --> 00:03:24,790 And so then it's not a surprise that you 71 00:03:24,790 --> 00:03:29,920 get the same four outcomes between these two situations. 72 00:03:29,920 --> 00:03:32,260 And I think that this is also highlighting 73 00:03:32,260 --> 00:03:35,850 some very interesting and deep connections between evolution, 74 00:03:35,850 --> 00:03:38,910 which is changes in, say, allele frequency in a population 75 00:03:38,910 --> 00:03:42,020 of a single species, over time, and then 76 00:03:42,020 --> 00:03:45,250 some of these ecological kind of processes, where you're really 77 00:03:45,250 --> 00:03:47,000 thinking about these as different species. 78 00:03:51,440 --> 00:03:55,947 Now, of course, in the case of the evolutionary dynamics 79 00:03:55,947 --> 00:03:57,780 that we analyze Martin's book, though, those 80 00:03:57,780 --> 00:04:02,220 were the evolution of different clonal populations 81 00:04:02,220 --> 00:04:06,487 in asexually reproducing populations. 82 00:04:06,487 --> 00:04:08,320 Do you guys remember what I'm talking about? 83 00:04:08,320 --> 00:04:10,950 All right. 84 00:04:10,950 --> 00:04:14,619 So I just want to draw a couple of these sorts of diagrams. 85 00:04:14,619 --> 00:04:16,410 I'm not going to draw out all four of them, 86 00:04:16,410 --> 00:04:18,220 because it does take some time. 87 00:04:18,220 --> 00:04:21,990 But hopefully, we can reconstruct where we were. 88 00:04:25,710 --> 00:04:33,590 The case that we were analyzing before, the N1 dot equals 0. 89 00:04:33,590 --> 00:04:35,280 We're going to have it as a dashed line. 90 00:04:35,280 --> 00:04:37,640 And N2 is going to be-- well, maybe 91 00:04:37,640 --> 00:04:39,040 we'll try to use thick chalk. 92 00:04:39,040 --> 00:04:41,380 Oh, wow, I missed. 93 00:04:41,380 --> 00:04:42,860 These are thick lines. 94 00:04:42,860 --> 00:04:47,310 So we'll draw these nullclines over here. 95 00:04:47,310 --> 00:04:49,470 So the N1 dot-- and indeed, one of the comments 96 00:04:49,470 --> 00:04:51,920 is that it would be nice to draw where the nullclines are 97 00:04:51,920 --> 00:04:54,330 on the axes as well. 98 00:04:54,330 --> 00:04:56,410 And indeed, we can do that. 99 00:04:56,410 --> 00:04:57,800 We aim to please. 100 00:04:57,800 --> 00:05:02,530 So the dashed lines correspond to N1 dot being to 0. 101 00:05:02,530 --> 00:05:06,600 So here's N1, N2. 102 00:05:06,600 --> 00:05:11,520 So N1 equal to 0 corresponds to this thing. 103 00:05:14,480 --> 00:05:17,260 And then we have this other guy that's a line, here. 104 00:05:23,660 --> 00:05:26,230 Now what we found is that, if N2 is equal to 0, 105 00:05:26,230 --> 00:05:33,650 this intersects at K1, whereas the intersection over here 106 00:05:33,650 --> 00:05:35,050 is at K1 divided by beta 12. 107 00:05:40,330 --> 00:05:43,390 Now, we have our other nullclines 108 00:05:43,390 --> 00:05:47,160 that correspond to N2 dot equal to 0. 109 00:05:47,160 --> 00:05:50,770 Now one of those lines is indeed going to be along here. 110 00:05:55,410 --> 00:05:59,570 And the other line can fall-- there 111 00:05:59,570 --> 00:06:01,510 are four different possibilities for how 112 00:06:01,510 --> 00:06:04,780 we might draw it in relation to this N1 dot equal to 0 113 00:06:04,780 --> 00:06:06,420 nullcline. 114 00:06:06,420 --> 00:06:08,640 And depending on the orientation of that, 115 00:06:08,640 --> 00:06:11,760 we'll end up getting these four different possible outcomes 116 00:06:11,760 --> 00:06:15,170 of 1 dominating, irrespective of the initial conditions, 117 00:06:15,170 --> 00:06:17,950 2 dominating, irrespective of initial conditions, 118 00:06:17,950 --> 00:06:20,440 or coexistence or bi-stability. 119 00:06:20,440 --> 00:06:22,940 So bi-stability is the only-- so if you 120 00:06:22,940 --> 00:06:26,220 start with a finite number of each of these two species, 121 00:06:26,220 --> 00:06:29,090 then bi-stability is the only case where the outcome depends 122 00:06:29,090 --> 00:06:30,490 on the starting condition. 123 00:06:30,490 --> 00:06:33,400 So, of course, if you start out without one of the two species, 124 00:06:33,400 --> 00:06:35,640 then you won't get creation of those species, right? 125 00:06:35,640 --> 00:06:39,060 Because the only way to get creation of species 1 126 00:06:39,060 --> 00:06:41,650 is to have some species 1 individual in this model. 127 00:06:45,790 --> 00:06:49,080 So we're going to get another line, here, corresponding to N2 128 00:06:49,080 --> 00:06:50,740 dot equal to 0. 129 00:06:50,740 --> 00:06:53,790 And I think that the one that we were trying to analyze 130 00:06:53,790 --> 00:06:57,120 was with the solid line underneath. 131 00:06:57,120 --> 00:07:01,390 Is that consistent with people's notes? 132 00:07:01,390 --> 00:07:03,195 So we can draw some other line here. 133 00:07:03,195 --> 00:07:06,580 It doesn't have to the same slope. 134 00:07:06,580 --> 00:07:08,419 Now it's good to be clear about where 135 00:07:08,419 --> 00:07:09,710 these things are going to fall. 136 00:07:09,710 --> 00:07:12,190 So K2 is this point. 137 00:07:12,190 --> 00:07:16,450 And now K2 divided by beta 21 is over here. 138 00:07:16,450 --> 00:07:20,730 Now recall that beta 12 is telling us 139 00:07:20,730 --> 00:07:23,640 about how much species 2 is reducing 140 00:07:23,640 --> 00:07:25,570 the growth of species 1. 141 00:07:25,570 --> 00:07:28,217 Whereas beta 21 is how much species 142 00:07:28,217 --> 00:07:29,800 1 is reducing the growth of species 2. 143 00:07:33,290 --> 00:07:36,240 Now, everything comes down to the relative ordering 144 00:07:36,240 --> 00:07:38,724 of these two quantities and these two quantities. 145 00:07:38,724 --> 00:07:40,640 And since there are two possibilities on each, 146 00:07:40,640 --> 00:07:42,348 that gives us the four possible outcomes. 147 00:07:44,820 --> 00:07:47,290 And broadly, the idea here is just that, 148 00:07:47,290 --> 00:07:51,130 if the species are weakly interfering with each other, 149 00:07:51,130 --> 00:07:52,130 then what should happen? 150 00:07:57,449 --> 00:07:58,990 Yes, then you should get coexistence. 151 00:07:58,990 --> 00:08:04,860 Coexistence is when the betas are small. 152 00:08:04,860 --> 00:08:06,846 Of course, this is a concrete model, 153 00:08:06,846 --> 00:08:08,720 so you have to define what you mean by small. 154 00:08:08,720 --> 00:08:11,450 And indeed, small here ends up being 155 00:08:11,450 --> 00:08:14,740 relative to the ratios of these carrying capacities. 156 00:08:14,740 --> 00:08:18,710 If the carrying capacities are just equal to, say, 1, 157 00:08:18,710 --> 00:08:24,670 then that's saying that the betas-- sorry, 158 00:08:24,670 --> 00:08:27,595 if the carrying capacities are the same, then the simple way 159 00:08:27,595 --> 00:08:29,470 to think about this is just whether the betas 160 00:08:29,470 --> 00:08:34,390 are larger or smaller than 1, whether each species interferes 161 00:08:34,390 --> 00:08:37,940 with the other species more than a member of that other species. 162 00:08:40,346 --> 00:08:41,970 So if carrying capacities are the same, 163 00:08:41,970 --> 00:08:46,600 that's what demarcates the different zones. 164 00:08:46,600 --> 00:08:49,440 So what we want to do is take this sort of diagram 165 00:08:49,440 --> 00:08:53,120 and try to figure out where will the trajectories be 166 00:08:53,120 --> 00:08:54,970 on this diagram? 167 00:08:54,970 --> 00:08:59,440 Now, it's always good to locate the fixed points. 168 00:08:59,440 --> 00:09:01,150 The fixed points of the system are? 169 00:09:01,150 --> 00:09:05,250 And somebody, words? 170 00:09:05,250 --> 00:09:08,190 How would we define fixed points in this system? 171 00:09:08,190 --> 00:09:09,662 AUDIENCE: [INAUDIBLE]. 172 00:09:09,662 --> 00:09:10,620 PROFESSOR: What's that? 173 00:09:10,620 --> 00:09:11,786 Both lines intersect, right? 174 00:09:11,786 --> 00:09:14,150 So when the dashed line intersects the solid lines, 175 00:09:14,150 --> 00:09:14,950 right? 176 00:09:14,950 --> 00:09:19,380 So we have one such fixed point here. 177 00:09:19,380 --> 00:09:23,666 We have another fixed point here and another fixed point here. 178 00:09:27,046 --> 00:09:28,420 Have we figured out the stability 179 00:09:28,420 --> 00:09:30,460 of those fixed points? 180 00:09:30,460 --> 00:09:30,960 No. 181 00:09:34,997 --> 00:09:36,705 What's the stability of this fixed point? 182 00:09:42,175 --> 00:09:43,050 It's unstable, right? 183 00:09:43,050 --> 00:09:45,760 Because we've already, previously 184 00:09:45,760 --> 00:09:49,784 assumed that these r's are greater than 0. 185 00:09:49,784 --> 00:09:51,450 We're assuming that the species would be 186 00:09:51,450 --> 00:09:53,590 able to survive on their own. 187 00:09:53,590 --> 00:09:55,520 And that's actually true for both species. 188 00:09:55,520 --> 00:10:01,000 So this thing is unstable kind of in both directions, right? 189 00:10:01,000 --> 00:10:02,770 And what are the eigenvectors associated 190 00:10:02,770 --> 00:10:03,686 with this fixed point? 191 00:10:06,420 --> 00:10:09,910 On the count of three, draw, use your arms 192 00:10:09,910 --> 00:10:14,400 like the hands of a clock to indicate the directions 193 00:10:14,400 --> 00:10:16,210 of the eigenvectors. 194 00:10:16,210 --> 00:10:21,180 All right, ready, three, two, one. 195 00:10:21,180 --> 00:10:23,714 All right, all right. 196 00:10:23,714 --> 00:10:24,880 There's no diagonals, right? 197 00:10:28,520 --> 00:10:29,780 Of course, you could also. 198 00:10:29,780 --> 00:10:32,450 I was waiting for somebody to be kind of obnoxious and point 199 00:10:32,450 --> 00:10:33,900 in the other direction. 200 00:10:33,900 --> 00:10:37,086 A surprisingly not obnoxious class we have. 201 00:10:37,086 --> 00:10:38,960 So this is just saying that, if you start out 202 00:10:38,960 --> 00:10:40,290 just a little bit of one species, 203 00:10:40,290 --> 00:10:41,748 you'll just stay with that species. 204 00:10:41,748 --> 00:10:43,490 It makes sense. 205 00:10:43,490 --> 00:10:46,330 Now, what do these lines tell us about the directions 206 00:10:46,330 --> 00:10:53,827 of the trajectories or the orientations 207 00:10:53,827 --> 00:10:54,660 of the trajectories? 208 00:11:03,990 --> 00:11:04,990 Why did I draw them? 209 00:11:08,500 --> 00:11:09,000 Yes? 210 00:11:13,360 --> 00:11:15,175 AUDIENCE: These trajectories are-- 211 00:11:15,175 --> 00:11:18,055 these are lines are nonlines. 212 00:11:18,055 --> 00:11:20,440 And one of the derivatives [INAUDIBLE]. 213 00:11:22,495 --> 00:11:24,370 PROFESSOR: One of the derivatives, all right. 214 00:11:24,370 --> 00:11:30,810 And in particular, let's look at this line here. 215 00:11:30,810 --> 00:11:33,310 Are the trajectories? 216 00:11:33,310 --> 00:11:37,000 Again, we're going to do our arms 217 00:11:37,000 --> 00:11:40,410 to indicate the orientation of the trajectories. 218 00:11:40,410 --> 00:11:42,790 In particular, there's a trajectory right here. 219 00:11:42,790 --> 00:11:45,900 What direction will that trajectory be pointing? 220 00:11:45,900 --> 00:11:48,460 There's only going to be one arm. 221 00:11:48,460 --> 00:11:53,670 Ready, three, two, one. 222 00:11:53,670 --> 00:11:55,990 All right, we got a lot of people not voting. 223 00:11:55,990 --> 00:11:59,560 All right, that means we need to turn our neighbors and discuss. 224 00:11:59,560 --> 00:12:01,710 If you didn't vote, it means, I think, 225 00:12:01,710 --> 00:12:04,002 not following what we're talking about. 226 00:12:04,002 --> 00:12:04,710 Turn to somebody. 227 00:12:08,560 --> 00:12:10,849 If your neighbor agrees with you about which direction 228 00:12:10,849 --> 00:12:12,390 you should be pointing your arm, then 229 00:12:12,390 --> 00:12:15,680 try to figure out whether we know 230 00:12:15,680 --> 00:12:19,301 which orientation the arrow should actually be in. 231 00:12:19,301 --> 00:12:22,460 [SIDE DISCUSSIONS] 232 00:12:47,515 --> 00:12:48,890 PROFESSOR: We'll figure this out. 233 00:12:48,890 --> 00:12:52,292 [SIDE DISCUSSIONS] 234 00:13:02,684 --> 00:13:04,350 PROFESSOR: Let's go ahead and reconvene, 235 00:13:04,350 --> 00:13:06,580 because it seems like some people are being quiet. 236 00:13:06,580 --> 00:13:09,540 But I'm not sure if that's because they think 237 00:13:09,540 --> 00:13:12,600 they know what's going on or they're just very much unhappy 238 00:13:12,600 --> 00:13:14,440 with the situation. 239 00:13:14,440 --> 00:13:16,460 Let me see a fresh voting. 240 00:13:16,460 --> 00:13:18,775 All right, ready, three, two, one. 241 00:13:21,312 --> 00:13:23,020 Definitely, it's going up or down, right? 242 00:13:23,020 --> 00:13:26,870 Because the definition of this dashed line is that N1 dot 243 00:13:26,870 --> 00:13:28,680 is equal to 0. 244 00:13:28,680 --> 00:13:30,544 We don't know what N2 dot is. 245 00:13:30,544 --> 00:13:31,960 We'll figure that out in a moment. 246 00:13:31,960 --> 00:13:34,100 But what we know is that the trajectories 247 00:13:34,100 --> 00:13:38,180 we should be something, lines here, either up or down. 248 00:13:38,180 --> 00:13:42,320 And we're going to find that they're down but here. 249 00:13:42,320 --> 00:13:47,369 Now, on this solid line, quickly, the orientation of 250 00:13:47,369 --> 00:13:47,910 trajectories. 251 00:13:47,910 --> 00:13:49,900 Ready, three, two, one. 252 00:13:49,900 --> 00:13:53,930 All right, perfect. 253 00:13:53,930 --> 00:13:55,870 So we know that N2 dot is 0 here. 254 00:13:58,860 --> 00:14:02,150 Now, an actual direction of the trajectory, at this point, 255 00:14:02,150 --> 00:14:03,230 right here. 256 00:14:03,230 --> 00:14:06,050 Ready, three, two, one. 257 00:14:06,050 --> 00:14:07,370 OK, good. 258 00:14:07,370 --> 00:14:10,776 Because in the absence of N2, we know 259 00:14:10,776 --> 00:14:13,150 that species N1 should just come to carrying capacity K1. 260 00:14:15,880 --> 00:14:20,760 Same thing over here, we should get arrows coming down. 261 00:14:20,760 --> 00:14:23,500 So indeed, what you can see is that the arrows are coming down 262 00:14:23,500 --> 00:14:23,650 here. 263 00:14:23,650 --> 00:14:26,024 That means they actually do have to come down immediately 264 00:14:26,024 --> 00:14:27,825 to the right of them. 265 00:14:27,825 --> 00:14:29,200 Whereas over here, the trajectory 266 00:14:29,200 --> 00:14:33,310 is point right here, so we can kind of figure out 267 00:14:33,310 --> 00:14:37,780 that-- so they're coming here. 268 00:14:37,780 --> 00:14:39,780 And from far away, they're coming here. 269 00:14:39,780 --> 00:14:42,630 And you can see that they have to come across here. 270 00:14:42,630 --> 00:14:44,740 And then they're going to come into this point. 271 00:14:44,740 --> 00:14:48,010 So this is going to be our stable fixed point. 272 00:14:48,010 --> 00:14:51,371 I'll color it in to indicate that. 273 00:14:51,371 --> 00:14:53,370 From here, they're going to curve around though. 274 00:14:53,370 --> 00:14:56,085 So if you start out right here, you kind of do this business. 275 00:15:05,140 --> 00:15:06,810 From here, we come in. 276 00:15:12,580 --> 00:15:14,990 Are these lines allowed to cross each other? 277 00:15:14,990 --> 00:15:15,490 No. 278 00:15:23,290 --> 00:15:25,330 Now, indeed, you could actually see 279 00:15:25,330 --> 00:15:28,960 here, what is the direction of the other eigenvector 280 00:15:28,960 --> 00:15:31,180 at this point? 281 00:15:31,180 --> 00:15:37,640 Using a hand, arm, ready, three, two, one. 282 00:15:37,640 --> 00:15:39,299 Right, it's kind of something in there. 283 00:15:39,299 --> 00:15:41,590 Because there's all these trajectories are coming here, 284 00:15:41,590 --> 00:15:44,420 and then they approach this fixed point from this point. 285 00:15:44,420 --> 00:15:46,360 Because here, the other eigenvector 286 00:15:46,360 --> 00:15:49,110 is still horizontal, right? 287 00:15:49,110 --> 00:15:52,180 Because we know, if we don't have N2, then we just have N1. 288 00:15:52,180 --> 00:15:55,662 But this other eigenvector is not purely straight up. 289 00:15:55,662 --> 00:15:57,870 The other one is along here, because the trajectories 290 00:15:57,870 --> 00:16:01,310 are coming in along that. 291 00:16:01,310 --> 00:16:04,010 Does that make sense? 292 00:16:04,010 --> 00:16:06,900 So in this case, we should just be clear. 293 00:16:06,900 --> 00:16:12,050 We are in a situation where K1 over beta 12 294 00:16:12,050 --> 00:16:13,460 is greater than K2. 295 00:16:16,740 --> 00:16:20,580 Another way of writing that is that beta 12 is less than K1 296 00:16:20,580 --> 00:16:22,770 over K2. 297 00:16:22,770 --> 00:16:29,930 So that means species 2 does not strongly harm species 1. 298 00:16:29,930 --> 00:16:36,930 Yet, we know that K1 is greater than K2 divided by beta 21. 299 00:16:36,930 --> 00:16:42,120 So that means that beta 21 is greater than K2 over K1. 300 00:16:42,120 --> 00:16:46,250 This is telling us that species 1 is strongly 301 00:16:46,250 --> 00:16:48,625 harming species 2. 302 00:16:48,625 --> 00:16:49,520 And that makes sense. 303 00:16:49,520 --> 00:16:51,420 In that case, species 1 wins. 304 00:16:56,290 --> 00:17:01,430 Does that outcome change if we change the r values, 305 00:17:01,430 --> 00:17:02,520 the division rates? 306 00:17:07,900 --> 00:17:08,960 A is yes. 307 00:17:08,960 --> 00:17:09,950 B is no. 308 00:17:09,950 --> 00:17:12,390 I'm going to give you 10 seconds. 309 00:17:12,390 --> 00:17:15,219 What I just said, if I change r's does that 310 00:17:15,219 --> 00:17:16,010 change the outcome? 311 00:17:16,010 --> 00:17:19,575 Ready, three, two, one. 312 00:17:22,165 --> 00:17:25,710 So we've got a majority of B. So the answer is no. 313 00:17:25,710 --> 00:17:37,506 This statement that, in this situation, species 1 dominates, 314 00:17:37,506 --> 00:17:38,755 that's independent of the r's. 315 00:17:44,112 --> 00:17:45,570 And what you see is the conditions, 316 00:17:45,570 --> 00:17:48,350 here, only depend on the what's in here. 317 00:17:48,350 --> 00:17:51,390 So the actual shape of those trajectories will depend upon 318 00:17:51,390 --> 00:17:51,920 the r's. 319 00:17:54,520 --> 00:17:58,420 So if it's the case that species 2 is just 320 00:17:58,420 --> 00:18:00,921 a faster grower than species 1, then you 321 00:18:00,921 --> 00:18:02,420 might end up with a situation where, 322 00:18:02,420 --> 00:18:04,045 if you start with a little bit of each, 323 00:18:04,045 --> 00:18:05,940 you might come way up here-- well, no. 324 00:18:05,940 --> 00:18:10,196 I guess you might come close to this fixed point. 325 00:18:10,196 --> 00:18:11,820 So you might think that it really looks 326 00:18:11,820 --> 00:18:13,390 like species 2 is about to win. 327 00:18:13,390 --> 00:18:17,500 But eventually, they'll curve over and come back. 328 00:18:17,500 --> 00:18:20,930 And indeed, in the Strogatz book, one of the chapters 329 00:18:20,930 --> 00:18:24,770 that I recommended, he has an example of sheeps and rabbits, 330 00:18:24,770 --> 00:18:27,560 the idea is that they are competing species, 331 00:18:27,560 --> 00:18:28,884 maybe eating similar foods. 332 00:18:28,884 --> 00:18:30,050 I don't know if that's true. 333 00:18:30,050 --> 00:18:32,410 But the rabbits can divide more rapidly. 334 00:18:32,410 --> 00:18:36,750 So here, the idea would be, well, if the sheep can really 335 00:18:36,750 --> 00:18:38,930 displace the rabbits, because it's just bigger 336 00:18:38,930 --> 00:18:40,650 and push them aside, then what can happen 337 00:18:40,650 --> 00:18:42,892 is that the rabbits first divide rapidly. 338 00:18:42,892 --> 00:18:44,600 It looks like they're going to take over, 339 00:18:44,600 --> 00:18:48,379 but, over time, eventually, the sheep population kind of 340 00:18:48,379 --> 00:18:50,170 grow up, and they start displacing rabbits. 341 00:18:50,170 --> 00:18:52,770 And you end up excluding the rabbits. 342 00:18:52,770 --> 00:18:55,130 This is this phenomenon of competitive exclusion. 343 00:19:00,570 --> 00:19:06,570 And depending upon the context-- OK, that's an e-- 344 00:19:06,570 --> 00:19:08,790 this is either more or less maybe formally phrased. 345 00:19:08,790 --> 00:19:14,210 But the idea is that, if there are two species that 346 00:19:14,210 --> 00:19:16,026 are too similar, and in particular, 347 00:19:16,026 --> 00:19:17,650 if they're somehow perfect competitors, 348 00:19:17,650 --> 00:19:19,060 they're really just trying to eat the same thing, 349 00:19:19,060 --> 00:19:20,850 then you should only end up with one 350 00:19:20,850 --> 00:19:23,370 of the two species surviving. 351 00:19:23,370 --> 00:19:26,220 And that's the kind of idea here. 352 00:19:26,220 --> 00:19:29,840 Although, I think, you can argue about the mapping I think. 353 00:19:29,840 --> 00:19:32,205 So this is one of the four outcomes. 354 00:19:35,360 --> 00:19:36,979 And of course, it takes 15 minutes 355 00:19:36,979 --> 00:19:38,520 to go through each of these examples. 356 00:19:38,520 --> 00:19:41,430 So we're not going to go through all of them. 357 00:19:41,430 --> 00:19:45,280 But you should be able to, for a given 358 00:19:45,280 --> 00:19:49,600 a combination of betas and K's, be able to figure out, 359 00:19:49,600 --> 00:19:53,410 using some combination of algebra, derivatives, 360 00:19:53,410 --> 00:19:57,185 fixed point stability analyses, and drawing of things, well, 361 00:19:57,185 --> 00:19:58,935 you should be able to do all of the above. 362 00:20:03,720 --> 00:20:05,805 Are there any questions about where we are here? 363 00:20:13,512 --> 00:20:14,970 AUDIENCE: I guess that none of this 364 00:20:14,970 --> 00:20:18,571 holds in the stochastic [INAUDIBLE]. 365 00:20:18,571 --> 00:20:20,570 PROFESSOR: Yeah, that's an interesting question. 366 00:20:20,570 --> 00:20:23,762 AUDIENCE: For example, in that case, if r2 is much bigger, 367 00:20:23,762 --> 00:20:28,063 then you're going get almost, very close to like K2, 368 00:20:28,063 --> 00:20:31,615 and then maybe just the last individual [INAUDIBLE] 369 00:20:31,615 --> 00:20:34,530 and then you just add that fixed point. 370 00:20:34,530 --> 00:20:36,520 PROFESSOR: Right. 371 00:20:36,520 --> 00:20:39,660 So it's certainly the case that, once you 372 00:20:39,660 --> 00:20:41,690 have stochastic extinction-- the thing is 373 00:20:41,690 --> 00:20:43,839 that, you would probably be most susceptible 374 00:20:43,839 --> 00:20:44,880 to stochastic extinction. 375 00:20:44,880 --> 00:20:46,300 In the case you were talking about, 376 00:20:46,300 --> 00:20:47,920 you would be most susceptible to stochastic extinction 377 00:20:47,920 --> 00:20:49,760 when you're around here, actually. 378 00:20:49,760 --> 00:20:58,800 These trajectories are still always moving up in N1 space. 379 00:20:58,800 --> 00:21:00,700 I think I know what you're saying. 380 00:21:00,700 --> 00:21:06,300 We're going to maybe zoom in onto this N1, N2. 381 00:21:06,300 --> 00:21:08,780 Because we have this unstable fixed point here. 382 00:21:08,780 --> 00:21:13,190 And the claim was that, if you start out over here, 383 00:21:13,190 --> 00:21:16,500 then the trajectory might look something like this. 384 00:21:16,500 --> 00:21:19,500 And you'd say, oh, well, you might go extinct here. 385 00:21:19,500 --> 00:21:21,850 AUDIENCE: The idea was just the statement 386 00:21:21,850 --> 00:21:25,620 that [INAUDIBLE] r1, r2 are very dynamical [INAUDIBLE]. 387 00:21:31,380 --> 00:21:36,150 PROFESSOR: It's true that, I guess, things change. 388 00:21:36,150 --> 00:21:39,320 There are a number of things you might want to say. 389 00:21:39,320 --> 00:21:43,540 First of all, this is a purely continuous and deterministic 390 00:21:43,540 --> 00:21:46,090 description of the setting. 391 00:21:46,090 --> 00:21:49,670 It allows for fractional individuals. 392 00:21:49,670 --> 00:21:52,440 There's no shocks or perturbations 393 00:21:52,440 --> 00:21:54,016 that you have to worry about. 394 00:21:54,016 --> 00:21:55,640 I guess the only thing I wanted to say, 395 00:21:55,640 --> 00:21:58,790 in regards to your question, is that I 396 00:21:58,790 --> 00:22:01,739 think the stochastic extinction will not be dominated. 397 00:22:01,739 --> 00:22:04,030 We're talking about stochastic extinction of species 1. 398 00:22:04,030 --> 00:22:06,529 It will not be dominated due to a stochastic extinction here 399 00:22:06,529 --> 00:22:10,820 but stochastic extinction at the beginning. 400 00:22:10,820 --> 00:22:13,920 I haven't drawn this very, very well, but, in this case, 401 00:22:13,920 --> 00:22:15,580 I think these trajectories are always 402 00:22:15,580 --> 00:22:20,900 are going up in numbers of the 1 species, which 403 00:22:20,900 --> 00:22:22,910 means that you're most likely to experience 404 00:22:22,910 --> 00:22:25,048 stochastic extinction at the beginning. 405 00:22:25,048 --> 00:22:25,548 Yeah? 406 00:22:25,548 --> 00:22:28,008 AUDIENCE: r1 and r2 should still really 407 00:22:28,008 --> 00:22:30,468 mess with stochastic extinction a lot, right? 408 00:22:30,468 --> 00:22:35,830 Because the larger r1 and r2 are the quicker 409 00:22:35,830 --> 00:22:39,670 we get out of the regime of stochastic extinction. 410 00:22:39,670 --> 00:22:41,140 PROFESSOR: That's all true. 411 00:22:45,420 --> 00:22:49,420 We have to be careful about many of these things. 412 00:22:49,420 --> 00:22:52,350 In particular, if you go and you do a stochastic simulation 413 00:22:52,350 --> 00:22:55,420 of this, so let's say you plug this thing into a Gillespie 414 00:22:55,420 --> 00:23:01,430 simulation, can you get stochastic extinction? 415 00:23:01,430 --> 00:23:06,900 A yes, or B no, ready, three, two, one. 416 00:23:06,900 --> 00:23:09,020 No. 417 00:23:09,020 --> 00:23:11,748 So the answer is no but why? 418 00:23:11,748 --> 00:23:12,666 AUDIENCE: It depends. 419 00:23:12,666 --> 00:23:16,810 If you take r combination [INAUDIBLE]. 420 00:23:16,810 --> 00:23:17,790 PROFESSOR: Exactly. 421 00:23:17,790 --> 00:23:20,555 So right now, as written, there's no death. 422 00:23:22,577 --> 00:23:24,660 Although, I guess you could say that this a death. 423 00:23:27,710 --> 00:23:31,420 There's a question of how you partition things. 424 00:23:31,420 --> 00:23:34,850 So in principle, this is the difference between the growth 425 00:23:34,850 --> 00:23:36,450 and the death rate. 426 00:23:36,450 --> 00:23:39,640 But the most straightforward way of doing such a simulation 427 00:23:39,640 --> 00:23:44,220 is that you put this whole thing in here as a rate for birth. 428 00:23:47,732 --> 00:23:50,065 Somebody is going to say, ah, that's not how I was going 429 00:23:50,065 --> 00:23:51,267 to do the simulation, right? 430 00:23:51,267 --> 00:23:53,350 Well, that's probably how you were going to do it. 431 00:23:53,350 --> 00:23:54,850 AUDIENCE: That's a terrible way. 432 00:23:54,850 --> 00:23:56,670 PROFESSOR: But if you did that, if there's only birth, 433 00:23:56,670 --> 00:23:59,020 then you can't get stochastic extinction, obviously. 434 00:23:59,020 --> 00:24:01,829 But in general-- and this is one of things 435 00:24:01,829 --> 00:24:03,370 we spend our time a lot time thinking 436 00:24:03,370 --> 00:24:07,040 about in this semester-- there are multiple ways 437 00:24:07,040 --> 00:24:09,410 of doing kind of a stochastic simulation 438 00:24:09,410 --> 00:24:11,420 from a deterministic equation. 439 00:24:11,420 --> 00:24:13,966 And this thing, you could be more explicit 440 00:24:13,966 --> 00:24:15,340 and say, oh, this thing is really 441 00:24:15,340 --> 00:24:19,625 a B2 minus a D2, so a birth rate minus a death rate for example. 442 00:24:19,625 --> 00:24:21,750 And form the standpoint of a differential equation, 443 00:24:21,750 --> 00:24:23,280 it doesn't make any difference. 444 00:24:23,280 --> 00:24:27,160 But if you do the Fokker-Planck approximation 445 00:24:27,160 --> 00:24:29,520 or you do a simulation or whatnot, 446 00:24:29,520 --> 00:24:32,720 then these lead to different things. 447 00:24:32,720 --> 00:24:34,970 In particular, the rate of, say, stochastic extinction 448 00:24:34,970 --> 00:24:38,650 here increases as B and D increase, because that leads 449 00:24:38,650 --> 00:24:40,095 to more of these fluctuations. 450 00:24:44,560 --> 00:24:48,911 So there are many things that are different once you include 451 00:24:48,911 --> 00:24:49,910 the stochastic dynamics. 452 00:24:49,910 --> 00:24:54,450 But it's always good to get a base sense of the dynamics 453 00:24:54,450 --> 00:24:57,910 from the standpoint of just deterministic differential 454 00:24:57,910 --> 00:25:01,860 equations before you think too much about the stochastic 455 00:25:01,860 --> 00:25:04,304 dynamics, because otherwise you get overwhelmed quickly. 456 00:25:08,180 --> 00:25:10,370 Any other questions about that? 457 00:25:25,510 --> 00:25:28,880 What I want to do is just spend a little bit of time 458 00:25:28,880 --> 00:25:30,890 to think about the more generalized case of more 459 00:25:30,890 --> 00:25:31,390 species. 460 00:25:40,280 --> 00:25:43,970 And in particular, we could convert this set of equations. 461 00:25:43,970 --> 00:25:47,600 We can normalize by each of their carrying capacities. 462 00:25:47,600 --> 00:25:54,394 And we can convert a set of equations 463 00:25:54,394 --> 00:25:55,560 to look something like this. 464 00:25:55,560 --> 00:25:59,760 So now we just have Xi dot is equal to-- there's 465 00:25:59,760 --> 00:26:03,850 some ri Xi, 1 minus. 466 00:26:03,850 --> 00:26:06,270 And what we can actually do is normalize everything 467 00:26:06,270 --> 00:26:07,950 so that it's just written like this. 468 00:26:12,220 --> 00:26:15,020 And normally, what we assume is that we've 469 00:26:15,020 --> 00:26:21,990 done things such that alpha i i is equal to 1 for all i. 470 00:26:21,990 --> 00:26:24,760 So this is just saying that this is the normalization such 471 00:26:24,760 --> 00:26:29,450 that each species inhibits itself in a way 472 00:26:29,450 --> 00:26:33,060 that it's just give a simple logistic growth. 473 00:26:33,060 --> 00:26:39,545 And it's going to be logistic growth with a carrying 474 00:26:39,545 --> 00:26:41,980 capacity equal to 1. 475 00:26:41,980 --> 00:26:45,130 And then once you've done that, then a species 476 00:26:45,130 --> 00:26:48,720 inhibits itself with alpha i i 1. 477 00:26:48,720 --> 00:26:53,210 And then everything comes down to what this alpha matrix is. 478 00:26:57,720 --> 00:27:02,380 And I would say that, as always, it's really very, very 479 00:27:02,380 --> 00:27:04,180 important that you can go back and forth 480 00:27:04,180 --> 00:27:07,920 between the non-dimensionalized versions of equations 481 00:27:07,920 --> 00:27:09,420 and the base version. 482 00:27:09,420 --> 00:27:12,280 This was something that, on exam number two, 483 00:27:12,280 --> 00:27:21,720 there were quite a lot of problems where we asked about, 484 00:27:21,720 --> 00:27:23,260 how does the parameter change when 485 00:27:23,260 --> 00:27:25,760 you change the strength of expression or this or that, 486 00:27:25,760 --> 00:27:26,620 right? 487 00:27:26,620 --> 00:27:28,880 So this is something that I think is very important. 488 00:27:28,880 --> 00:27:30,380 Because this comes up a lot. 489 00:27:33,100 --> 00:27:35,020 So in this case, the alpha matrix 490 00:27:35,020 --> 00:27:37,330 tells you kind of everything. 491 00:27:37,330 --> 00:27:39,270 And then there are a number of things 492 00:27:39,270 --> 00:27:41,450 that, well, the mathematicians have proven 493 00:27:41,450 --> 00:27:42,742 about these sorts of equations. 494 00:27:42,742 --> 00:27:45,116 And I just want to point you towards some of those things 495 00:27:45,116 --> 00:27:45,960 to think about. 496 00:27:45,960 --> 00:27:52,590 So first, I'm considering a case where, again, alpha i 497 00:27:52,590 --> 00:28:01,290 j is greater than 0, again, for all i and ji, 498 00:28:01,290 --> 00:28:03,810 or greater than or equal to 0. 499 00:28:07,310 --> 00:28:10,240 So some of them can be 0. 500 00:28:10,240 --> 00:28:13,055 But the interactions, when they exist, are competitive. 501 00:28:16,360 --> 00:28:23,040 So first, if you start out in the region where 502 00:28:23,040 --> 00:28:25,850 all of the species start, between 0 and 1, then 503 00:28:25,850 --> 00:28:28,150 you stay in that region. 504 00:28:33,340 --> 00:28:36,100 If this is true initially, then it will be true forever. 505 00:28:40,590 --> 00:28:41,550 That's good. 506 00:28:41,550 --> 00:28:45,560 Negative abundances, you maybe were not so worried about. 507 00:28:45,560 --> 00:28:49,050 But it's not as obvious that you can't get above one, 508 00:28:49,050 --> 00:28:51,440 because there's nothing saying that, in principle, you 509 00:28:51,440 --> 00:28:53,090 couldn't have started there, right? 510 00:28:53,090 --> 00:28:56,939 Or in principle, you couldn't have gotten there? 511 00:28:56,939 --> 00:28:58,480 And certainly, it's physical to think 512 00:28:58,480 --> 00:29:00,063 about starting outside of that region, 513 00:29:00,063 --> 00:29:02,170 because we often talk about carrying capacities 514 00:29:02,170 --> 00:29:03,211 as something that's like. 515 00:29:06,830 --> 00:29:10,146 If you think about, this is an Xi as a function of time. 516 00:29:13,096 --> 00:29:15,720 So in these situations, it's not crazy to think about something 517 00:29:15,720 --> 00:29:18,017 above the carrying capacity. 518 00:29:18,017 --> 00:29:20,350 But this mathematical statement about the Latka-Volterra 519 00:29:20,350 --> 00:29:21,650 framework is that, if you start out 520 00:29:21,650 --> 00:29:23,500 with everything below its carrying capacity, 521 00:29:23,500 --> 00:29:25,500 then everything will always stay there. 522 00:29:29,810 --> 00:29:42,060 And all the dynamics occur-- this is for i equal to 1 to N 523 00:29:42,060 --> 00:29:43,710 Do I want to use a big N or little n? 524 00:29:43,710 --> 00:29:44,420 Does it matter? 525 00:29:50,840 --> 00:29:52,750 So this is big N, different species. 526 00:29:52,750 --> 00:29:57,650 The dynamics occur on an N minus one manifold. 527 00:29:57,650 --> 00:29:59,150 If we have many mathematicians, they 528 00:29:59,150 --> 00:30:01,270 can explain what this technically means. 529 00:30:01,270 --> 00:30:03,760 But basically, what it's saying is 530 00:30:03,760 --> 00:30:06,945 that there's going to be some N minus 1 dimensional surface 531 00:30:06,945 --> 00:30:09,660 or volume or whatnot where all the dynamics are 532 00:30:09,660 --> 00:30:10,980 going end up being on. 533 00:30:10,980 --> 00:30:15,920 And what that means is that, in particular, a limit cycle 534 00:30:15,920 --> 00:30:28,440 requires two dimensions, requires 535 00:30:28,440 --> 00:30:31,510 2D, which means that to get a limit cycle 536 00:30:31,510 --> 00:30:36,990 requires-- so then N has to be greater than or equal to 3 537 00:30:36,990 --> 00:30:39,320 to get a limit cycle. 538 00:30:39,320 --> 00:30:41,500 Can you see what I'm saying? 539 00:30:41,500 --> 00:30:47,897 Whereas chaos requires 3D, and that 540 00:30:47,897 --> 00:30:49,980 means that N has to be greater than or equal to 4. 541 00:30:53,980 --> 00:30:56,941 And indeed, you can get limit cycles with N equal to 3. 542 00:30:56,941 --> 00:30:58,440 And you get chaos with N equal to 4. 543 00:31:01,796 --> 00:31:03,170 There's another theorem that says 544 00:31:03,170 --> 00:31:10,360 that any dynamics are possible for N equal to 5 or larger. 545 00:31:10,360 --> 00:31:13,550 And chaos seems as much as I would want to ask for. 546 00:31:13,550 --> 00:31:20,620 But there's, apparently, a 4D torus, 547 00:31:20,620 --> 00:31:22,764 something that is different. 548 00:31:22,764 --> 00:31:24,930 This is something to think about in your spare time. 549 00:31:33,010 --> 00:31:35,900 Well, you know, like a lot of things. 550 00:31:35,900 --> 00:31:41,440 And this is worth spending a moment talking about. 551 00:31:41,440 --> 00:31:44,550 First of all, so here's a two dimensional thing. 552 00:31:44,550 --> 00:31:46,550 The special thing, as you might remember, 553 00:31:46,550 --> 00:31:50,950 about continuous dynamics is that these trajectories are not 554 00:31:50,950 --> 00:31:53,670 allowed to cross each other, right? 555 00:31:53,670 --> 00:31:58,990 Which means that you just can't draw a chaotic trajectory, 556 00:31:58,990 --> 00:32:02,210 because you're going to have to cross yourself again. 557 00:32:02,210 --> 00:32:06,550 And that's also why a limit cycle requires two, 558 00:32:06,550 --> 00:32:10,070 because we originally tried to draw a limit cycle in one 559 00:32:10,070 --> 00:32:11,400 dimension, and it didn't work. 560 00:32:11,400 --> 00:32:13,080 Do you remember this? 561 00:32:13,080 --> 00:32:15,360 And the same thing with-- you can imagine, here's 562 00:32:15,360 --> 00:32:17,959 a nice limit cycle. 563 00:32:17,959 --> 00:32:19,000 And that could be stable. 564 00:32:21,730 --> 00:32:24,770 Oh, and incidentally, you were right about. 565 00:32:24,770 --> 00:32:28,440 I was getting confused about the Poincare-Bendixson theorem. 566 00:32:28,440 --> 00:32:30,390 Because I think there are these funny things. 567 00:32:30,390 --> 00:32:32,280 It wasn't. 568 00:32:32,280 --> 00:32:34,065 OK, well, you know, all these people 569 00:32:34,065 --> 00:32:36,890 that complain about what I say. 570 00:32:36,890 --> 00:32:40,499 Because if you have the trajectories coming in, 571 00:32:40,499 --> 00:32:43,040 then what I said was that, if you had an unstable fixed point 572 00:32:43,040 --> 00:32:44,950 coming out, then you could draw this region, 573 00:32:44,950 --> 00:32:46,575 then you could be guaranteed that there 574 00:32:46,575 --> 00:32:47,930 was a limit cycle in there. 575 00:32:47,930 --> 00:32:52,887 But you cannot, just based on what I had said, say that, 576 00:32:52,887 --> 00:32:55,220 if it's a stable fixed point, then you won't get a limit 577 00:32:55,220 --> 00:32:56,140 cycle. 578 00:32:56,140 --> 00:32:59,820 I mean in some other cases you can prove that it doesn't work. 579 00:32:59,820 --> 00:33:04,570 Particularly, like in the Latka-Volterra, 580 00:33:04,570 --> 00:33:08,439 there in one of the crossings, it's 581 00:33:08,439 --> 00:33:09,980 now going to be a stable fixed point. 582 00:33:09,980 --> 00:33:11,800 In that situation, you can prove, mathematically, 583 00:33:11,800 --> 00:33:13,420 that you can never get a limit cycle oscillation 584 00:33:13,420 --> 00:33:15,670 because of some divergence condition of some function 585 00:33:15,670 --> 00:33:18,311 and so forth. 586 00:33:18,311 --> 00:33:20,310 But in the example that I was telling you about, 587 00:33:20,310 --> 00:33:23,870 in the predator-prey, it was true in that particular case 588 00:33:23,870 --> 00:33:26,569 that, when that thing is stable, you don't get oscillations. 589 00:33:26,569 --> 00:33:27,860 And when it's unstable, you do. 590 00:33:27,860 --> 00:33:29,734 But both directions do not follow 591 00:33:29,734 --> 00:33:30,900 from the Poincare-Bendixson. 592 00:33:35,580 --> 00:33:37,540 This is a limit cycle. 593 00:33:37,540 --> 00:33:41,020 Now, in a chaotic situation, you have to be able to do something 594 00:33:41,020 --> 00:33:43,440 where you kind of come around and then, 595 00:33:43,440 --> 00:33:45,190 every now and then, you kind go over here. 596 00:33:45,190 --> 00:33:48,410 And then you go around and around. 597 00:33:48,410 --> 00:33:49,970 Something crazy happens. 598 00:33:49,970 --> 00:33:53,720 But you can see that this situation doesn't 599 00:33:53,720 --> 00:33:57,450 work if it's only 2D, right? 600 00:33:57,450 --> 00:33:59,430 Do you see why I'm saying that? 601 00:33:59,430 --> 00:34:00,410 AUDIENCE: Yeah. 602 00:34:00,410 --> 00:34:01,380 PROFESSOR: You don't? 603 00:34:01,380 --> 00:34:03,040 You don't agree. 604 00:34:03,040 --> 00:34:05,436 It's just, if it's 2D, then these lines cannot cross. 605 00:34:05,436 --> 00:34:06,810 So you need to a third dimension, 606 00:34:06,810 --> 00:34:09,730 so that they can just shoot above and below each other. 607 00:34:09,730 --> 00:34:11,505 AUDIENCE: And what's with the definition? 608 00:34:11,505 --> 00:34:13,790 Because you can come up with trajectories, 609 00:34:13,790 --> 00:34:16,639 like you can have trajectories that asymptotically approach 610 00:34:16,639 --> 00:34:21,112 the limit cycle but that never cross themselves, right? 611 00:34:21,112 --> 00:34:21,778 PROFESSOR: Yeah. 612 00:34:21,778 --> 00:34:23,920 AUDIENCE: Those are obviously not chaotic, but the y-- 613 00:34:23,920 --> 00:34:24,628 PROFESSOR: Right. 614 00:34:26,989 --> 00:34:29,310 So this is not actually class in non-linear dynamics, 615 00:34:29,310 --> 00:34:30,139 so we're not. 616 00:34:30,139 --> 00:34:33,510 So normally, you characterize this Lyapunov exponent, 617 00:34:33,510 --> 00:34:35,642 which tells you about how the phase space is 618 00:34:35,642 --> 00:34:36,850 kind of growing or shrinking. 619 00:34:36,850 --> 00:34:38,725 In the case that you were just talking about, 620 00:34:38,725 --> 00:34:41,150 that's a case where all the trajectories come together. 621 00:34:41,150 --> 00:34:43,774 Because in this case where this trajectory comes into the limit 622 00:34:43,774 --> 00:34:46,920 cycle, if you draw like a blob of phase space, 623 00:34:46,920 --> 00:34:49,020 it's going to come together over time. 624 00:34:49,020 --> 00:34:52,480 Whereas in a chaotic system, if you have a blob of phase space, 625 00:34:52,480 --> 00:34:55,409 it's going to diverge and fold and do all the craziness. 626 00:35:00,289 --> 00:35:01,992 Yes? 627 00:35:01,992 --> 00:35:03,867 AUDIENCE: So you said [INAUDIBLE] essentially 628 00:35:03,867 --> 00:35:06,145 you need n greater than or equal to 3. 629 00:35:06,145 --> 00:35:08,100 Is that just for this-- 630 00:35:08,100 --> 00:35:10,021 PROFESSOR: Yeah. 631 00:35:10,021 --> 00:35:10,520 Yes. 632 00:35:10,520 --> 00:35:12,920 So this is in this Latka-Volterra. 633 00:35:12,920 --> 00:35:16,700 Because, in general, you can get a limit cycle 634 00:35:16,700 --> 00:35:19,040 with two equations. 635 00:35:19,040 --> 00:35:20,150 And that's with the 2D. 636 00:35:20,150 --> 00:35:23,270 And in general, you get a chaos with three equations 637 00:35:23,270 --> 00:35:24,890 or three variables. 638 00:35:24,890 --> 00:35:29,870 But in the Latka-Volterra model, it requires three and four, 639 00:35:29,870 --> 00:35:30,990 respectively. 640 00:35:30,990 --> 00:35:34,139 That doesn't mean that every four species 641 00:35:34,139 --> 00:35:35,430 Latka-Volterra will have chaos. 642 00:35:35,430 --> 00:35:37,221 But it means that it's possible to get one. 643 00:35:39,059 --> 00:35:41,100 And I think one thing that's just rather striking 644 00:35:41,100 --> 00:35:45,830 is that this really is kind of the simplest, possible model 645 00:35:45,830 --> 00:35:49,870 you can ever write down describing 646 00:35:49,870 --> 00:35:52,587 how species might interact or variables 647 00:35:52,587 --> 00:35:53,670 might interact or whatnot. 648 00:35:53,670 --> 00:35:55,630 And so it's really kind of incredible to me 649 00:35:55,630 --> 00:35:57,800 that you get all these crazy dynamics. 650 00:36:00,205 --> 00:36:00,705 Yes? 651 00:36:00,705 --> 00:36:03,200 AUDIENCE: I was going to ask about the n minus 1 thing. 652 00:36:03,200 --> 00:36:06,415 This dynamic [INAUDIBLE] n minus 1 [INAUDIBLE]. 653 00:36:06,415 --> 00:36:07,081 PROFESSOR: Yeah. 654 00:36:07,081 --> 00:36:08,705 AUDIENCE: Over here it just looks to me 655 00:36:08,705 --> 00:36:11,464 like-- we have n equals 2 and the dynamics 656 00:36:11,464 --> 00:36:17,004 are occuring on a plane, on a two-dimensional plane. 657 00:36:17,004 --> 00:36:17,670 PROFESSOR: Yeah. 658 00:36:22,280 --> 00:36:24,860 The transience or whatever can occur. 659 00:36:24,860 --> 00:36:28,050 It requires the full N dimensions to describe. 660 00:36:28,050 --> 00:36:30,890 Because you can start-- to describe 661 00:36:30,890 --> 00:36:35,120 all of the trajectories, clearly requires all dimensions, right? 662 00:36:35,120 --> 00:36:36,670 Because anywhere you start, you have 663 00:36:36,670 --> 00:36:39,550 to have specify it by five dimensions. 664 00:36:39,550 --> 00:36:43,870 But the dynamics, as far as like-- and here, 665 00:36:43,870 --> 00:36:46,300 there aren't even any dynamics. 666 00:36:46,300 --> 00:36:48,290 I think that you go to a point. 667 00:36:51,571 --> 00:36:53,565 AUDIENCE: What do you mean by dynamics? 668 00:36:53,565 --> 00:36:55,680 [INAUDIBLE] 669 00:36:55,680 --> 00:36:57,480 PROFESSOR: Yeah, I agree that there 670 00:36:57,480 --> 00:37:02,850 is a-- so in the case of the limit cycles 671 00:37:02,850 --> 00:37:07,410 and so forth, this is really like, at steady state, 672 00:37:07,410 --> 00:37:08,830 it's doing something. 673 00:37:08,830 --> 00:37:11,640 So here, steady state, it only goes to the fixed points. 674 00:37:13,639 --> 00:37:15,430 But if you have a three dimensional system, 675 00:37:15,430 --> 00:37:19,890 then you can have it steady state, the trajectories 676 00:37:19,890 --> 00:37:21,070 on like a plane. 677 00:37:30,620 --> 00:37:31,120 Yeah? 678 00:37:31,120 --> 00:37:33,347 AUDIENCE: Is it because there's like some concept 679 00:37:33,347 --> 00:37:39,550 of [INAUDIBLE] that can be [INAUDIBLE] for the system? 680 00:37:39,550 --> 00:37:43,100 PROFESSOR: I don't think that that-- I'm not 681 00:37:43,100 --> 00:37:44,800 aware of that being the case. 682 00:37:44,800 --> 00:37:48,605 But I'm hesitant to say that it's not true. 683 00:37:55,750 --> 00:37:58,160 I want to switch gears a little bit 684 00:37:58,160 --> 00:38:01,389 and think about three species interactions 685 00:38:01,389 --> 00:38:03,430 and, in particular the three species interactions 686 00:38:03,430 --> 00:38:05,460 when they're non-transitive. 687 00:38:05,460 --> 00:38:08,120 Because this is thought to be potentially 688 00:38:08,120 --> 00:38:11,355 a significant stabilizer for diversity in populations. 689 00:38:28,388 --> 00:38:36,240 So this is non-transitive interactions. 690 00:38:36,240 --> 00:38:37,990 And we often just say rock-paper-scissors. 691 00:38:44,630 --> 00:38:47,130 Is everybody from a cultural that plays rock-paper-scissors? 692 00:38:52,410 --> 00:38:52,956 Yes? 693 00:38:52,956 --> 00:38:53,580 AUDIENCE: Yeah. 694 00:38:53,580 --> 00:38:56,580 PROFESSOR: So this is a true, human universal. 695 00:38:56,580 --> 00:39:00,015 Although I think that what we call it does vary and so forth. 696 00:39:00,015 --> 00:39:02,180 You know how sometimes the linguists 697 00:39:02,180 --> 00:39:06,310 try to find this ur-language that our ancestors spoke 698 00:39:06,310 --> 00:39:07,590 50,000 years ago or whatever? 699 00:39:07,590 --> 00:39:09,250 I think that you probably do something 700 00:39:09,250 --> 00:39:10,625 similar with rock-paper-scissors, 701 00:39:10,625 --> 00:39:14,300 because it seems to be a pretty common theme. 702 00:39:14,300 --> 00:39:16,710 But the idea here is that you have-- wait, 703 00:39:16,710 --> 00:39:19,810 which direction does it go? 704 00:39:19,810 --> 00:39:23,610 So paper beats rock, scissors beats paper, 705 00:39:23,610 --> 00:39:25,574 but rock beats the scissors. 706 00:39:30,460 --> 00:39:33,570 So you can imagine that this kind of dynamic 707 00:39:33,570 --> 00:39:38,000 can be captured in the Latka-Volterra type framework. 708 00:39:38,000 --> 00:39:40,480 So you just have to set up these betas or alphas 709 00:39:40,480 --> 00:39:43,900 so that this is true. 710 00:39:43,900 --> 00:39:46,500 And again, the way to think about this 711 00:39:46,500 --> 00:39:49,280 would be-- the simplest thing is to think about it as dominance. 712 00:39:49,280 --> 00:39:52,250 So if you have the rock species and the scissor species 713 00:39:52,250 --> 00:39:56,050 together, then the rock species will drive the scissor species 714 00:39:56,050 --> 00:39:58,900 extinct and so forth. 715 00:39:58,900 --> 00:40:01,080 Now, this is the kind of situation 716 00:40:01,080 --> 00:40:04,600 that, in principle, can lead to very complicated dynamics 717 00:40:04,600 --> 00:40:08,240 in multi-species ecosystems that, at least in many models, 718 00:40:08,240 --> 00:40:13,720 can stabilize the coexistence of multiple species via some 719 00:40:13,720 --> 00:40:16,930 of these complex, crazy dynamics that we were just 720 00:40:16,930 --> 00:40:17,690 talking about. 721 00:40:17,690 --> 00:40:20,210 I would say that, as a mechanism for the stabilization 722 00:40:20,210 --> 00:40:23,630 of diversity, I don't know how convincing 723 00:40:23,630 --> 00:40:27,240 that is in terms of being what explains why it is there's 724 00:40:27,240 --> 00:40:30,970 so much diversity outside, when you look outside the window. 725 00:40:30,970 --> 00:40:32,970 But, at least, it's in principle true. 726 00:40:32,970 --> 00:40:37,966 And one of the topics of these papers-- 727 00:40:37,966 --> 00:40:39,340 certainly the one you just read-- 728 00:40:39,340 --> 00:40:41,880 is that rock-paper-scissors, i.e. 729 00:40:41,880 --> 00:40:44,220 non-transitive interactions on their own, 730 00:40:44,220 --> 00:40:46,082 may not be sufficient, but, in the presence 731 00:40:46,082 --> 00:40:47,540 of spatial structure, maybe it does 732 00:40:47,540 --> 00:40:52,750 allow for long-term coexistence of these species or strategies 733 00:40:52,750 --> 00:40:53,430 and so forth. 734 00:40:53,430 --> 00:40:56,680 And again, it's not always clear in these situations 735 00:40:56,680 --> 00:40:58,560 whether you're thinking about ecology, 736 00:40:58,560 --> 00:40:59,976 where these are different species, 737 00:40:59,976 --> 00:41:01,720 or you're thinking about evolution, where 738 00:41:01,720 --> 00:41:03,570 these are different genotypes. 739 00:41:03,570 --> 00:41:06,090 And indeed, in the Kerr paper that you read, 740 00:41:06,090 --> 00:41:07,407 these are all E. coli. 741 00:41:07,407 --> 00:41:09,240 So it's all one species, it's just that they 742 00:41:09,240 --> 00:41:11,270 have different mutations. 743 00:41:11,270 --> 00:41:13,360 So that's really rock-paper-scissors 744 00:41:13,360 --> 00:41:16,630 in the context of evolution. 745 00:41:16,630 --> 00:41:19,560 Whereas in the male mating strategies 746 00:41:19,560 --> 00:41:24,120 paper by Curt Lively, that's more of an ecological context, 747 00:41:24,120 --> 00:41:26,540 but it still is evolution within the species. 748 00:41:26,540 --> 00:41:30,910 Because these mating strategies are heritable. 749 00:41:36,750 --> 00:41:41,850 I did tell you the base idea of this lizard mating strategy 750 00:41:41,850 --> 00:41:42,386 business? 751 00:41:42,386 --> 00:41:44,010 Or did I never say anything about that? 752 00:41:44,010 --> 00:41:44,850 Not really? 753 00:41:44,850 --> 00:41:45,690 OK. 754 00:41:45,690 --> 00:41:47,900 For some reason, I thought that I'd alluded to it. 755 00:41:47,900 --> 00:41:50,960 Well, let me explain it to you. 756 00:41:50,960 --> 00:41:54,800 I think it's kind of an incredible paper. 757 00:41:54,800 --> 00:41:56,990 So this is a paper by my Curt Lively. 758 00:41:56,990 --> 00:41:59,010 So it's Sinervo and Lively. 759 00:42:03,970 --> 00:42:05,220 It was Nature in '96. 760 00:42:11,970 --> 00:42:14,010 And it's called "The Rock-Paper-Scissors 761 00:42:14,010 --> 00:42:16,450 Game and the Evolution of Alternative Male Strategies." 762 00:42:16,450 --> 00:42:19,550 So what was known is that there are 763 00:42:19,550 --> 00:42:24,660 many examples of alternative mating strategies in males. 764 00:42:24,660 --> 00:42:27,810 In particular, it's rather common that there are, 765 00:42:27,810 --> 00:42:34,310 what you might call, territorial males 766 00:42:34,310 --> 00:42:36,620 and what they often call sneaker males. 767 00:42:41,930 --> 00:42:45,180 This is observed in fish and various land animals. 768 00:42:45,180 --> 00:42:47,960 And in many of the cases, these sneaker males 769 00:42:47,960 --> 00:42:51,655 really do look phenotypically like females. 770 00:43:04,490 --> 00:43:06,750 And this has been measured using various kinds 771 00:43:06,750 --> 00:43:08,980 of observational, experimental approaches. 772 00:43:08,980 --> 00:43:10,900 There's often what you call negative frequency 773 00:43:10,900 --> 00:43:14,150 dependent selection between these strategies. 774 00:43:17,180 --> 00:43:21,340 The sneakers can often, when rare, 775 00:43:21,340 --> 00:43:25,050 spread in population of the territorial and vice versa. 776 00:43:25,050 --> 00:43:27,350 But this was, at the least the first case 777 00:43:27,350 --> 00:43:31,230 I'm aware of where these ideas had been demonstrated, 778 00:43:31,230 --> 00:43:32,920 that there were really three strategies. 779 00:43:32,920 --> 00:43:34,880 And the three strategies implemented, 780 00:43:34,880 --> 00:43:38,620 one of these rock-paper-scissors interactions. 781 00:43:38,620 --> 00:43:40,987 I want to maybe make a little more space. 782 00:43:40,987 --> 00:43:42,570 If you guys are available after class, 783 00:43:42,570 --> 00:43:45,267 I encourage you to come up and look at the paper, 784 00:43:45,267 --> 00:43:47,100 because they actually have pictures of them, 785 00:43:47,100 --> 00:43:51,250 so you can identify the sneaker males. 786 00:43:51,250 --> 00:43:55,420 Because they actually look different 787 00:43:55,420 --> 00:43:58,470 based on the coloring of their throat. 788 00:43:58,470 --> 00:44:03,910 So these are lizards that live in the mountains up outside 789 00:44:03,910 --> 00:44:06,310 of the Bay Area in California, in Merced County. 790 00:44:06,310 --> 00:44:08,790 They're side-blotched lizards. 791 00:44:08,790 --> 00:44:11,470 I don't know anything about that. 792 00:44:11,470 --> 00:44:15,190 And what they showed was that there's 793 00:44:15,190 --> 00:44:23,920 these guys with orange throats that are kind of aggressive. 794 00:44:23,920 --> 00:44:28,820 And they defend a very large territory 795 00:44:28,820 --> 00:44:31,190 with a large number of females. 796 00:44:31,190 --> 00:44:34,250 And they fight off any males that come. 797 00:44:34,250 --> 00:44:40,250 Then there are the dark blue. 798 00:44:43,130 --> 00:44:44,960 Sorry, I should have put that over here. 799 00:44:44,960 --> 00:44:51,450 So there are other lizards here that are dark blue throats. 800 00:44:51,450 --> 00:44:54,700 And these guys are less aggressive with smaller 801 00:44:54,700 --> 00:44:55,200 territories. 802 00:44:59,080 --> 00:45:04,260 So you can guess, if it were just the orange-throated guys 803 00:45:04,260 --> 00:45:07,280 and the dark-- and these are genetically encoded strategies, 804 00:45:07,280 --> 00:45:09,870 in the sense that they do seem to be passed on, 805 00:45:09,870 --> 00:45:14,930 and it's determined by the genes that the male inherits-- less 806 00:45:14,930 --> 00:45:18,740 aggressive and small territory. 807 00:45:18,740 --> 00:45:20,979 Can you guess which one wins between these two, 808 00:45:20,979 --> 00:45:21,770 against each other? 809 00:45:25,410 --> 00:45:25,910 What's that? 810 00:45:25,910 --> 00:45:27,540 AUDIENCE: The two aggressive ones 811 00:45:27,540 --> 00:45:30,123 PROFESSOR: Yeah, but if you just have the two aggressive ones? 812 00:45:30,123 --> 00:45:30,940 AUDIENCE: Yeah. 813 00:45:30,940 --> 00:45:34,320 PROFESSOR: This guy's going to beat this one, right? 814 00:45:34,320 --> 00:45:36,760 That's just because this aggressive male has a larger 815 00:45:36,760 --> 00:45:40,400 territory, and they pass on more genes 816 00:45:40,400 --> 00:45:42,324 than the less aggressive ones. 817 00:45:42,324 --> 00:45:44,990 However, what they found is that there's a third mating strategy 818 00:45:44,990 --> 00:45:47,920 here, which are these sneaker males. 819 00:45:47,920 --> 00:45:52,640 And they indeed look like the females. 820 00:45:52,640 --> 00:45:59,210 They have these yellow stripes on their throats. 821 00:45:59,210 --> 00:46:01,740 They look like the female, and they have no territory. 822 00:46:01,740 --> 00:46:05,330 Sneaker males, so they don't defend any territory. 823 00:46:05,330 --> 00:46:08,850 Instead, they just sneak into the territory 824 00:46:08,850 --> 00:46:11,040 of the other males and try to mate 825 00:46:11,040 --> 00:46:12,540 with the females in that territory. 826 00:46:12,540 --> 00:46:17,270 And what they show is that, over the course of seven years, 827 00:46:17,270 --> 00:46:20,304 in the mountains of Merced, they see 828 00:46:20,304 --> 00:46:21,720 that the frequency of these things 829 00:46:21,720 --> 00:46:25,950 goes through an entire oscillation. 830 00:46:25,950 --> 00:46:31,280 So they see just over one period of this oscillation. 831 00:46:31,280 --> 00:46:33,560 And their argument is that, although it's true 832 00:46:33,560 --> 00:46:36,090 that the aggressive males can outcompete the less aggressive 833 00:46:36,090 --> 00:46:39,990 ones, the sneakers actually outcompete 834 00:46:39,990 --> 00:46:41,700 these aggressive ones, basically, 835 00:46:41,700 --> 00:46:45,527 because these aggressive males, it's 836 00:46:45,527 --> 00:46:47,110 just too large of a territory for them 837 00:46:47,110 --> 00:46:49,670 to effectively defend it. 838 00:46:49,670 --> 00:46:53,370 So the sneakers can actually outcompete 839 00:46:53,370 --> 00:46:54,530 the aggressive males. 840 00:46:54,530 --> 00:46:58,120 Yet the less aggressive ones can outcompete the sneakers, 841 00:46:58,120 --> 00:47:01,980 because they are trying to defend a smaller territory. 842 00:47:01,980 --> 00:47:05,910 So they basically measured the frequency 843 00:47:05,910 --> 00:47:08,980 of these strategies and also the number of females 844 00:47:08,980 --> 00:47:16,670 in the different territories, from 1990 to '96 or so, 845 00:47:16,670 --> 00:47:20,250 and saw that this thing kind of went around in some circle 846 00:47:20,250 --> 00:47:22,780 in this frequency space of alternative male strategies. 847 00:47:31,280 --> 00:47:35,520 Kind of an incredible paper. 848 00:47:35,520 --> 00:47:38,750 So the idea here is this is a situation where 849 00:47:38,750 --> 00:47:40,284 it is non-transitive. 850 00:47:40,284 --> 00:47:41,950 So there's this rock-paper-scissors type 851 00:47:41,950 --> 00:47:42,600 dynamic. 852 00:47:42,600 --> 00:47:47,877 And it's also spatial, because these lizards are 853 00:47:47,877 --> 00:47:49,960 in some particular place, they have some territory 854 00:47:49,960 --> 00:47:50,680 and so forth. 855 00:47:50,680 --> 00:47:53,002 And in this other paper, by Benjamin Kerr, what 856 00:47:53,002 --> 00:47:54,960 he wanted to do was try to understand something 857 00:47:54,960 --> 00:47:57,400 about what the role of that spatial structure 858 00:47:57,400 --> 00:48:00,010 might be in maintaining the diversity. 859 00:48:00,010 --> 00:48:03,742 So in this case, the argument was that, if you go out 860 00:48:03,742 --> 00:48:05,450 and you look at these lizards, the reason 861 00:48:05,450 --> 00:48:07,199 that you see all three of these strategies 862 00:48:07,199 --> 00:48:10,640 is because of this rock-paper-scissors dynamic. 863 00:48:10,640 --> 00:48:12,510 If one of the strategies becomes more rare, 864 00:48:12,510 --> 00:48:13,940 it's going to have an advantage relative to the others. 865 00:48:13,940 --> 00:48:15,299 It's going to spread. 866 00:48:15,299 --> 00:48:16,965 And what Benjamin Kerr wanted to explore 867 00:48:16,965 --> 00:48:22,300 is whether that statement could be made-- somehow 868 00:48:22,300 --> 00:48:25,430 you can distinguish between whether the spatial component 869 00:48:25,430 --> 00:48:26,760 is important or not. 870 00:48:26,760 --> 00:48:29,416 Of course, it's hard to do that in the case of the lizards, 871 00:48:29,416 --> 00:48:30,790 but he was able to implement this 872 00:48:30,790 --> 00:48:36,780 in the case some chemical warfare behavior in bacteria. 873 00:48:36,780 --> 00:48:42,080 So in this paper by Kerr, and it's also 874 00:48:42,080 --> 00:48:53,790 a Nature paper from 2002. 875 00:48:53,790 --> 00:48:56,150 So this paper is called "Local Dispersal 876 00:48:56,150 --> 00:48:57,890 of Promotes Biodiversity in a Real Life 877 00:48:57,890 --> 00:49:01,437 Game of Rock-Paper-Scissors." 878 00:49:01,437 --> 00:49:02,270 So you guys read it. 879 00:49:02,270 --> 00:49:04,025 So what were the three strategies? 880 00:49:12,252 --> 00:49:14,370 AUDIENCE: [INAUDIBLE]. 881 00:49:14,370 --> 00:49:18,400 PROFESSOR: So C is the colicin producers. 882 00:49:23,040 --> 00:49:25,430 Incidentally, just this colicin production 883 00:49:25,430 --> 00:49:29,510 is kind of an incredible phenomenon already. 884 00:49:29,510 --> 00:49:34,280 So these are proteins that are produced that bind 885 00:49:34,280 --> 00:49:37,615 to other bacteria and can often make pores in the membrane 886 00:49:37,615 --> 00:49:38,840 and kill them. 887 00:49:38,840 --> 00:49:41,620 But the amazing thing about the colicin production 888 00:49:41,620 --> 00:49:45,730 is that in E. coli and other gram 889 00:49:45,730 --> 00:49:48,790 negative bacteria, the only way that these colicins are 890 00:49:48,790 --> 00:49:52,370 released is by cell lysis. 891 00:49:52,370 --> 00:49:54,270 So it's not just that the cell is engaging 892 00:49:54,270 --> 00:49:58,210 in some costly behavior in order to make this protein that 893 00:49:58,210 --> 00:49:59,200 will kill other cells. 894 00:49:59,200 --> 00:50:01,830 But the only way that the toxin is released 895 00:50:01,830 --> 00:50:05,840 is by the cell actually bursting open. 896 00:50:05,840 --> 00:50:09,420 So it's clear then that this has to be supported 897 00:50:09,420 --> 00:50:14,950 by some kind of group level or kin-selection kind of argument. 898 00:50:14,950 --> 00:50:17,930 This can never be good for the individual, 899 00:50:17,930 --> 00:50:22,330 because the individual has had to spill its guts 900 00:50:22,330 --> 00:50:23,890 in order to harm other cells. 901 00:50:23,890 --> 00:50:26,420 So the only way that this can be supported 902 00:50:26,420 --> 00:50:30,200 is by inhibiting the growth of competitors 903 00:50:30,200 --> 00:50:32,480 and allowing your kind of kin mates 904 00:50:32,480 --> 00:50:36,060 or other cells that also have this plasmid, and, therefore, 905 00:50:36,060 --> 00:50:40,200 also the immunity protein, allowing them to grow better. 906 00:50:40,200 --> 00:50:45,350 So this is a very neat example of an altruistic, 907 00:50:45,350 --> 00:50:46,505 kind of warlike behavior. 908 00:50:49,810 --> 00:50:51,250 So this is one of the strategies. 909 00:50:51,250 --> 00:50:53,150 What was the other two. 910 00:50:56,394 --> 00:50:58,350 AUDIENCE: [INAUDIBLE]. 911 00:50:58,350 --> 00:50:59,236 PROFESSOR: Resistant. 912 00:50:59,236 --> 00:51:00,610 So there's R, which is resistant. 913 00:51:06,150 --> 00:51:09,200 And between the C and R, who wins? 914 00:51:09,200 --> 00:51:10,080 AUDIENCE: R. 915 00:51:10,080 --> 00:51:12,060 PROFESSOR: R, that's right. 916 00:51:12,060 --> 00:51:13,560 There might be some costs associated 917 00:51:13,560 --> 00:51:15,860 with being resistant, but the cost 918 00:51:15,860 --> 00:51:17,670 is not as large as actually bursting open. 919 00:51:21,280 --> 00:51:22,190 What's the last one? 920 00:51:31,307 --> 00:51:32,640 AUDIENCE: [INAUDIBLE] sensitive. 921 00:51:32,640 --> 00:51:34,680 PROFESSOR: Sensitive, perfect. 922 00:51:34,680 --> 00:51:36,360 So this is just the normal bacteria. 923 00:51:40,720 --> 00:51:43,332 And the argument is that there's often 924 00:51:43,332 --> 00:51:45,790 a cost to be resistant, which means that sensitive bacteria 925 00:51:45,790 --> 00:51:47,630 will outcompete resistant. 926 00:51:47,630 --> 00:51:51,380 Yet, if it's just the sensitive and the colicinogenic strains, 927 00:51:51,380 --> 00:51:55,305 then this strain can beat this strain. 928 00:51:55,305 --> 00:51:57,430 So this is the idea of the rock-paper-scissors game 929 00:51:57,430 --> 00:51:58,730 in this system. 930 00:51:58,730 --> 00:52:01,420 They do say that, in some situations, 931 00:52:01,420 --> 00:52:04,710 the fitnesses are such that it's like this. 932 00:52:04,710 --> 00:52:07,750 And it's good to take these sentences seriously, 933 00:52:07,750 --> 00:52:13,150 because, if they thought that every time that you isolate 934 00:52:13,150 --> 00:52:16,890 a resistant bacterium, that it would satisfy this, 935 00:52:16,890 --> 00:52:19,830 they would have said, when you do this, this is what you see. 936 00:52:19,830 --> 00:52:22,700 Their phrasing tells you that actually, 937 00:52:22,700 --> 00:52:26,170 depending upon which strain you get here or there, 938 00:52:26,170 --> 00:52:28,607 you may or may not see this. 939 00:52:28,607 --> 00:52:29,690 So you have to be careful. 940 00:52:29,690 --> 00:52:32,148 Just because there's a nice paper that's written about this 941 00:52:32,148 --> 00:52:34,840 doesn't mean that, if you go out and you find particular strains 942 00:52:34,840 --> 00:52:37,360 that have these properties, that it will always 943 00:52:37,360 --> 00:52:39,440 yield this particular outcome. 944 00:52:42,940 --> 00:52:49,840 So they argue that these strains, just because they 945 00:52:49,840 --> 00:52:51,450 have a non-transitive interaction, 946 00:52:51,450 --> 00:52:53,010 does not necessarily mean that they 947 00:52:53,010 --> 00:52:59,660 will be able to coexist in a well mixed environment 948 00:52:59,660 --> 00:53:02,130 in particular. 949 00:53:02,130 --> 00:53:06,880 And in their simulations and the experiments, 950 00:53:06,880 --> 00:53:10,130 where they did experiments in a test tube, 951 00:53:10,130 --> 00:53:11,480 which strain died first? 952 00:53:16,918 --> 00:53:18,240 The sensitive strain. 953 00:53:18,240 --> 00:53:20,660 Does that make sense? 954 00:53:20,660 --> 00:53:22,640 Yeah, right? 955 00:53:22,640 --> 00:53:25,580 And the other thing to remember is that these strains, there's 956 00:53:25,580 --> 00:53:27,860 no reason that they should be accurately described 957 00:53:27,860 --> 00:53:31,647 by a Latka-Volterra type formulation. 958 00:53:31,647 --> 00:53:33,480 And in particular, it could just be the case 959 00:53:33,480 --> 00:53:35,980 that, if you have enough producers 960 00:53:35,980 --> 00:53:37,480 and they make enough of the colicin, 961 00:53:37,480 --> 00:53:39,271 then the sensitive cells are just all dead. 962 00:53:41,620 --> 00:53:46,437 And that will, in general, be hard to capture 963 00:53:46,437 --> 00:53:47,520 in this sort of framework. 964 00:53:51,540 --> 00:53:55,680 So the idea is that if you start with a bunch of N's, N 965 00:53:55,680 --> 00:53:58,220 for each of these three, then first you 966 00:53:58,220 --> 00:54:01,680 see that the sensitive cells die. 967 00:54:01,680 --> 00:54:03,730 And once the sensitive cells have died, 968 00:54:03,730 --> 00:54:06,100 then you're really just playing an interaction 969 00:54:06,100 --> 00:54:08,240 between these two. 970 00:54:08,240 --> 00:54:13,040 In that case, you get the colcinogenic strain dying, 971 00:54:13,040 --> 00:54:16,890 and you're left with just the resistant strain. 972 00:54:24,537 --> 00:54:26,120 One thing I want to caution you about, 973 00:54:26,120 --> 00:54:29,980 though, is that, just because in this experiment 974 00:54:29,980 --> 00:54:34,230 they saw that coexistence of three rock-paper-scissors type 975 00:54:34,230 --> 00:54:36,720 strains was not possible in a well mixed environment, 976 00:54:36,720 --> 00:54:39,445 it does not mean that it will always be the case. 977 00:54:42,305 --> 00:54:44,180 This is a very well known paper in the field. 978 00:54:44,180 --> 00:54:48,500 And the thing is that it's easy to forget what a paper shows 979 00:54:48,500 --> 00:54:49,810 and what it doesn't show. 980 00:54:49,810 --> 00:54:53,515 So what this shows is that there is maybe 981 00:54:53,515 --> 00:54:54,890 a set of these three strains that 982 00:54:54,890 --> 00:54:57,170 have a rock-paper-scissors type interaction. 983 00:54:57,170 --> 00:54:59,687 And in those particular three strains, 984 00:54:59,687 --> 00:55:01,520 that are interacting in this particular way, 985 00:55:01,520 --> 00:55:03,500 via colicin killing da-duh, then they 986 00:55:03,500 --> 00:55:06,655 don't coexist in this particular well mixed and maybe other well 987 00:55:06,655 --> 00:55:07,780 mixed environments as well. 988 00:55:07,780 --> 00:55:09,524 But this does not necessarily show 989 00:55:09,524 --> 00:55:11,190 that any rock-paper-scissors interaction 990 00:55:11,190 --> 00:55:16,317 in a well mixed environment will not support coexistence. 991 00:55:16,317 --> 00:55:18,650 And in particular, you might remember, in Martin Nowak's 992 00:55:18,650 --> 00:55:21,879 book, there are very reasonable equations 993 00:55:21,879 --> 00:55:24,170 that display rock-paper-scissors type interactions that 994 00:55:24,170 --> 00:55:26,395 can lead to coexistence. 995 00:55:26,395 --> 00:55:28,020 So you can kind of spiral in that space 996 00:55:28,020 --> 00:55:29,360 to a state of coexistence. 997 00:55:29,360 --> 00:55:30,762 So it's possible. 998 00:55:30,762 --> 00:55:32,470 But in this situation, it doesn't happen. 999 00:55:35,210 --> 00:55:40,592 Can somebody remind us how they implemented the spatially 1000 00:55:40,592 --> 00:55:41,550 structured environment? 1001 00:55:46,194 --> 00:55:47,110 AUDIENCE: [INAUDIBLE]. 1002 00:55:47,110 --> 00:55:47,776 PROFESSOR: Yeah. 1003 00:55:47,776 --> 00:55:49,400 So they used agar plates. 1004 00:55:49,400 --> 00:55:53,190 They did this thing where they took this plate, 1005 00:55:53,190 --> 00:55:56,969 and they kind of used a hexagonal grid of some sort. 1006 00:55:56,969 --> 00:55:58,010 Is it actually hexagonal? 1007 00:55:58,010 --> 00:55:59,194 Yeah. 1008 00:55:59,194 --> 00:56:01,360 I don't know how they decided on the original order, 1009 00:56:01,360 --> 00:56:06,020 but they said, OK, here is a sensitive, sensitive, 1010 00:56:06,020 --> 00:56:10,222 resistant, colicinogenic, resistant. 1011 00:56:10,222 --> 00:56:11,180 I don't know, whatever. 1012 00:56:11,180 --> 00:56:14,630 So they filled it up. 1013 00:56:14,630 --> 00:56:18,680 They put maybe 20 different kind of patches there. 1014 00:56:18,680 --> 00:56:21,290 And then they basically, each day, 1015 00:56:21,290 --> 00:56:23,200 they just used one of these velvets, 1016 00:56:23,200 --> 00:56:26,790 that we often use to do replica plating. 1017 00:56:26,790 --> 00:56:29,730 And then they just kind of took off some of the cells 1018 00:56:29,730 --> 00:56:31,230 and put it on a fresh plate. 1019 00:56:31,230 --> 00:56:36,440 And then they did this for a week or so. 1020 00:56:36,440 --> 00:56:39,240 And then they saw that there was some sense 1021 00:56:39,240 --> 00:56:42,990 that there was a spatial dynamic taking 1022 00:56:42,990 --> 00:56:45,680 place, where the spatial structure was 1023 00:56:45,680 --> 00:56:48,520 kind of reminiscent of this. 1024 00:56:48,520 --> 00:56:51,556 The sensitive kind of moved into the resistant. 1025 00:56:51,556 --> 00:56:54,000 The resistant kind of moved into the colicinogenic. 1026 00:56:54,000 --> 00:56:55,990 Colicinogenic kind of moved into the sensitive. 1027 00:56:55,990 --> 00:56:59,630 But they couldn't actually see all of those, 1028 00:56:59,630 --> 00:57:03,767 because two of the strains, they said, grew to similar density. 1029 00:57:03,767 --> 00:57:06,100 That was between the sensitive and the resistant, right? 1030 00:57:06,100 --> 00:57:08,420 So visually, they can only distinguish 1031 00:57:08,420 --> 00:57:11,710 the colicinogenic strain relative to the others. 1032 00:57:11,710 --> 00:57:15,540 But what they found, though, is that, over a similar amount 1033 00:57:15,540 --> 00:57:19,330 of time, they got coexistence of all three. 1034 00:57:22,440 --> 00:57:25,740 So R, C, S, kind of all-- the number 1035 00:57:25,740 --> 00:57:28,840 is just a function of time-- stuck around in this spatially 1036 00:57:28,840 --> 00:57:31,670 structured environment. 1037 00:57:31,670 --> 00:57:34,230 So their argument is that, in some cases maybe, 1038 00:57:34,230 --> 00:57:36,490 this non-transitive interaction is not 1039 00:57:36,490 --> 00:57:39,850 sufficient to maintain diversity in a population, 1040 00:57:39,850 --> 00:57:43,140 whether it's genetic diversity or species diversity. 1041 00:57:43,140 --> 00:57:45,490 But it may be that it's very important 1042 00:57:45,490 --> 00:57:48,260 to have this spatial component. 1043 00:57:51,850 --> 00:57:53,520 If you're curious about these things, 1044 00:57:53,520 --> 00:57:56,510 there's also a nice computational study 1045 00:57:56,510 --> 00:58:05,260 done by Erwin Frey, who studied these rock-paper-scissors 1046 00:58:05,260 --> 00:58:07,190 dynamics as a function of the mobility 1047 00:58:07,190 --> 00:58:09,290 of the individual agents. 1048 00:58:09,290 --> 00:58:12,964 And in that study, he found that there 1049 00:58:12,964 --> 00:58:14,630 was sort of a critical level of mobility 1050 00:58:14,630 --> 00:58:24,440 in which you kind of switch from a spatially structured 1051 00:58:24,440 --> 00:58:27,660 environment with coexistence to, above some mobility, 1052 00:58:27,660 --> 00:58:31,410 you end up losing the diversity, because it kind of goes 1053 00:58:31,410 --> 00:58:34,470 into this well mixed regime. 1054 00:58:34,470 --> 00:58:36,760 So if you're curious about these things, 1055 00:58:36,760 --> 00:58:38,904 you can look up Erwin's paper. 1056 00:58:41,766 --> 00:58:47,890 Are there any questions about what they did here, 1057 00:58:47,890 --> 00:58:50,126 what you think it means? 1058 00:58:50,126 --> 00:58:50,626 Yeah? 1059 00:58:50,626 --> 00:58:54,070 AUDIENCE: You said you couldn't use [INAUDIBLE]. 1060 00:58:54,070 --> 00:58:56,038 How did they know if it was [INAUDIBLE]? 1061 00:58:57,740 --> 00:59:00,115 PROFESSOR: They can just scrape off everything and plate. 1062 00:59:02,820 --> 00:59:04,422 AUDIENCE: And then you can tell? 1063 00:59:04,422 --> 00:59:05,130 PROFESSOR: Right. 1064 00:59:05,130 --> 00:59:07,060 Because, well, then you can ask, oh, we're 1065 00:59:07,060 --> 00:59:10,370 those guys sensitive to colicin or not? 1066 00:59:10,370 --> 00:59:12,770 AUDIENCE: Why not actually just inject them with colicin? 1067 00:59:12,770 --> 00:59:13,436 PROFESSOR: Yeah. 1068 00:59:13,436 --> 00:59:16,930 Or plate them on something with colicin. 1069 00:59:16,930 --> 00:59:20,780 This is a system that has been very influential, 1070 00:59:20,780 --> 00:59:24,100 but it's really not yet or has not 1071 00:59:24,100 --> 00:59:27,400 been a domesticated kind of model system in the sense 1072 00:59:27,400 --> 00:59:31,460 that they are not nice, fluorescent proteins. 1073 00:59:31,460 --> 00:59:34,480 This is not on a nice cloning plasmid. 1074 00:59:34,480 --> 00:59:37,230 It was a natural plasmid. 1075 00:59:37,230 --> 00:59:41,390 And this resistant strain, the way that they get it is, 1076 00:59:41,390 --> 00:59:46,060 basically, they take some sensitive cells, 1077 00:59:46,060 --> 00:59:47,370 and they add colicin. 1078 00:59:47,370 --> 00:59:49,060 Let's say they supernatant from here, and they ask, 1079 00:59:49,060 --> 00:59:51,075 which cells grow> and then that's a resistant cell. 1080 00:59:51,075 --> 00:59:52,430 And it's genetically resistant. 1081 00:59:52,430 --> 00:59:55,170 But then each one's going to different. 1082 00:59:55,170 --> 00:59:57,730 They have done sequencing. 1083 00:59:57,730 --> 01:00:00,660 It's a surface receptor that the colicin maybe uses to go in 1084 01:00:00,660 --> 01:00:01,560 or other things. 1085 01:00:01,560 --> 01:00:05,290 But indeed, we actually did some experiments 1086 01:00:05,290 --> 01:00:07,570 with some of these strains. 1087 01:00:07,570 --> 01:00:09,675 And it's a little bit messy. 1088 01:00:09,675 --> 01:00:12,710 And I think that there's a sense that the field maybe 1089 01:00:12,710 --> 01:00:17,722 needs to just make some nice plasmids, with nice colors, 1090 01:00:17,722 --> 01:00:19,430 so that we can really distinguish things. 1091 01:00:19,430 --> 01:00:24,020 Because I think that this system, despite 500 citations 1092 01:00:24,020 --> 01:00:26,410 or so, almost none of those citations 1093 01:00:26,410 --> 01:00:30,250 are experimental papers really exploring this thing. 1094 01:00:30,250 --> 01:00:34,640 Because I think that it still is just a little bit messy. 1095 01:00:34,640 --> 01:00:37,820 And I think that it's also a very pretty system 1096 01:00:37,820 --> 01:00:38,908 to explore these ideas. 1097 01:00:46,376 --> 01:00:46,876 Yeah? 1098 01:00:46,876 --> 01:00:47,792 AUDIENCE: [INAUDIBLE]? 1099 01:00:53,367 --> 01:00:55,200 PROFESSOR: So the question is, why can't you 1100 01:00:55,200 --> 01:00:56,190 distinguish these? 1101 01:00:56,190 --> 01:00:59,450 And they say it's because they're similar densities. 1102 01:00:59,450 --> 01:01:02,790 I don't even know if that's even. 1103 01:01:02,790 --> 01:01:09,650 Of course, well, if you added sensitive cells here 1104 01:01:09,650 --> 01:01:12,200 and sensitive cells here and they grew up, 1105 01:01:12,200 --> 01:01:15,790 it's likely you're not going to see a boundary. 1106 01:01:15,790 --> 01:01:18,780 And just more generally, if you take 1107 01:01:18,780 --> 01:01:21,090 similar strains-- and these are rather similar strains, 1108 01:01:21,090 --> 01:01:23,340 this just has some mutation relative to this-- then 1109 01:01:23,340 --> 01:01:26,810 you won't, often, see a boundary when the colonies kind of grow 1110 01:01:26,810 --> 01:01:28,732 up. 1111 01:01:28,732 --> 01:01:31,212 AUDIENCE: So they just didn't see boundaries. 1112 01:01:31,212 --> 01:01:34,024 It's not like they didn't see-- 1113 01:01:34,024 --> 01:01:35,940 PROFESSOR: So they saw there were cells there, 1114 01:01:35,940 --> 01:01:39,894 but they couldn't necessarily say that it was true 1115 01:01:39,894 --> 01:01:41,310 that the sensitive cells were kind 1116 01:01:41,310 --> 01:01:43,615 of spreading into the resistant cell region, because they 1117 01:01:43,615 --> 01:01:44,698 couldn't see the boundary. 1118 01:01:52,070 --> 01:01:54,960 So in the last 20 minutes here, 15, 20, 1119 01:01:54,960 --> 01:01:58,240 I just want to say a few things about these population waves. 1120 01:02:08,270 --> 01:02:11,380 For many, many purposes, there are strong arguments 1121 01:02:11,380 --> 01:02:14,620 to be made that, in the context of evolution or ecology, 1122 01:02:14,620 --> 01:02:16,960 spatial dynamics matter. 1123 01:02:16,960 --> 01:02:20,030 And the way that we often think about the spatial dynamics 1124 01:02:20,030 --> 01:02:22,857 is via some effective diffusive process. 1125 01:02:22,857 --> 01:02:24,440 And that's convenient, because we know 1126 01:02:24,440 --> 01:02:25,700 a lot about how to model it. 1127 01:02:25,700 --> 01:02:28,280 But it's also maybe a reasonable description 1128 01:02:28,280 --> 01:02:32,340 of the motion of animals and other living things over length 1129 01:02:32,340 --> 01:02:34,590 scales that are large compared to the movements 1130 01:02:34,590 --> 01:02:36,700 of the animals. 1131 01:02:36,700 --> 01:02:40,600 So what we do is we take some equation, such as, 1132 01:02:40,600 --> 01:02:44,260 well, we often have, say, dN/dt. 1133 01:02:44,260 --> 01:02:46,220 So this is going to be about population waves. 1134 01:02:49,700 --> 01:02:52,570 So we take our standard thing where we say, oh, here's 1135 01:02:52,570 --> 01:02:58,180 r N 1 minus N/K. 1136 01:02:58,180 --> 01:03:00,830 And then we just want to add some spatial dynamic. 1137 01:03:00,830 --> 01:03:01,840 So what we're going to do is we're going to say, 1138 01:03:01,840 --> 01:03:03,506 now, the derivative, now it's a density. 1139 01:03:03,506 --> 01:03:05,540 So we're going to use a little n, just for fun. 1140 01:03:09,280 --> 01:03:14,120 And we might still use a K just for simple. 1141 01:03:14,120 --> 01:03:16,550 But then we have to add some diffusive term. 1142 01:03:22,097 --> 01:03:24,430 So there still is going to be a local carrying capacity. 1143 01:03:24,430 --> 01:03:27,100 Now this is in terms of density of the organism. 1144 01:03:27,100 --> 01:03:29,850 And we're going to assume that there is some diffusive type 1145 01:03:29,850 --> 01:03:31,970 motion of the organism. 1146 01:03:31,970 --> 01:03:34,600 Now this is this is, of course, going over the life-scale 1147 01:03:34,600 --> 01:03:38,750 over which the animals are actually doing things 1148 01:03:38,750 --> 01:03:39,880 over shorter time scales. 1149 01:03:39,880 --> 01:03:42,390 So it could be that different organisms 1150 01:03:42,390 --> 01:03:45,040 have very different modes of motility. 1151 01:03:45,040 --> 01:03:47,350 In some cases they walk or swim, in some cases 1152 01:03:47,350 --> 01:03:49,860 they just get picked up by a passing deer. 1153 01:03:49,860 --> 01:03:51,940 So there are a wide range of ways 1154 01:03:51,940 --> 01:03:54,100 in which organisms move around. 1155 01:03:54,100 --> 01:03:57,630 But if you look at it over kind of longer length time scales, 1156 01:03:57,630 --> 01:04:01,890 then maybe it doesn't matter. 1157 01:04:01,890 --> 01:04:05,100 And certainly, if it's an unbiased kind of motion, 1158 01:04:05,100 --> 01:04:09,950 with motility being well-behaved, 1159 01:04:09,950 --> 01:04:14,400 i.e., if the probability distribution of these steps, 1160 01:04:14,400 --> 01:04:17,512 as long as it's not long-tailed. 1161 01:04:17,512 --> 01:04:19,470 Similarly, Oskar Hallatschek, over at Berkeley, 1162 01:04:19,470 --> 01:04:21,136 has been doing a lot of fun work looking 1163 01:04:21,136 --> 01:04:23,070 at epidemic spread in cases where 1164 01:04:23,070 --> 01:04:26,110 this kernel, the kind of step size distribution, 1165 01:04:26,110 --> 01:04:27,710 has long tails. 1166 01:04:27,710 --> 01:04:31,510 It leads to qualitatively different behaviors. 1167 01:04:31,510 --> 01:04:34,200 So if you're curious about such things, check out Oskar's work. 1168 01:04:34,200 --> 01:04:38,387 But for normal kind of the step size distributions, 1169 01:04:38,387 --> 01:04:40,470 then the central limit theorem type considerations 1170 01:04:40,470 --> 01:04:42,920 just tell you that you can maybe just look 1171 01:04:42,920 --> 01:04:47,737 at it like this over time spatial scales that are bigger. 1172 01:04:47,737 --> 01:04:48,237 Yeah? 1173 01:04:48,237 --> 01:04:51,576 AUDIENCE: But is it a unique correlation in your step? 1174 01:04:51,576 --> 01:04:54,676 Because the central limit theorem, as long 1175 01:04:54,676 --> 01:04:56,475 as the variance-- I guess, you're saying? 1176 01:04:56,475 --> 01:04:58,350 PROFESSOR: That's why I'm saying long-tailed. 1177 01:04:58,350 --> 01:05:00,302 AUDIENCE: If the variance is infinite? 1178 01:05:00,302 --> 01:05:02,510 PROFESSOR: Exactly Yeah, so that's what I was saying. 1179 01:05:02,510 --> 01:05:06,130 And indeed, people argue, in the case of disease spread, 1180 01:05:06,130 --> 01:05:08,580 with modern air travel and so forth, 1181 01:05:08,580 --> 01:05:10,290 that the probability distribution 1182 01:05:10,290 --> 01:05:13,440 of kind of step sizes for infected individuals, 1183 01:05:13,440 --> 01:05:16,950 over the next week, is long-tailed, right? 1184 01:05:16,950 --> 01:05:19,130 Because there's a fair chance that you're just 1185 01:05:19,130 --> 01:05:20,730 going to stay around your local neighborhood, 1186 01:05:20,730 --> 01:05:21,890 but there's a smaller chance you're 1187 01:05:21,890 --> 01:05:23,473 going to go to the other side of town. 1188 01:05:23,473 --> 01:05:27,540 But you might go to a business trip over in DC. 1189 01:05:27,540 --> 01:05:30,780 At some small right, though, you also fly to South Africa 1190 01:05:30,780 --> 01:05:33,010 and go to a conference. 1191 01:05:33,010 --> 01:05:36,090 So all of these things have reasonable probabilities, 1192 01:05:36,090 --> 01:05:39,240 and so there are arguments that this is kind of some power law 1193 01:05:39,240 --> 01:05:40,200 type distribution. 1194 01:05:40,200 --> 01:05:42,540 And in those cases, you don't necessarily 1195 01:05:42,540 --> 01:05:46,612 have finite variance, and so then you can't just 1196 01:05:46,612 --> 01:05:47,820 put everything under the rug. 1197 01:05:51,237 --> 01:05:53,320 But we first want to understand what happens here, 1198 01:05:53,320 --> 01:05:54,810 and then we can-- well, we're not 1199 01:05:54,810 --> 01:05:56,310 going to-- but then other people can 1200 01:05:56,310 --> 01:05:58,768 think more deeply about what happens in fancier situations. 1201 01:06:02,830 --> 01:06:04,960 It is worth saying, though, that this approach, 1202 01:06:04,960 --> 01:06:08,420 I mean it looks very physics-y, in the sense that 1203 01:06:08,420 --> 01:06:13,020 physicists like simple equations where we add diffusion 1204 01:06:13,020 --> 01:06:13,610 and so forth. 1205 01:06:13,610 --> 01:06:17,872 But this is not what a physicist came up with. 1206 01:06:17,872 --> 01:06:21,980 These are classic ideas in evolution and ecology. 1207 01:06:21,980 --> 01:06:26,780 The solution to this was originally done by Fisher. 1208 01:06:26,780 --> 01:06:28,920 So this was in the 1920s or so. 1209 01:06:28,920 --> 01:06:30,560 So a long time ago, originally to try 1210 01:06:30,560 --> 01:06:32,440 to understand not the spread of a population 1211 01:06:32,440 --> 01:06:36,260 but the spread of a beneficial allele in a spatial population. 1212 01:06:36,260 --> 01:06:39,290 And once again, this highlights the deep connections 1213 01:06:39,290 --> 01:06:42,090 between evolution and ecology. 1214 01:06:42,090 --> 01:06:44,840 You can have a genetic wave in space 1215 01:06:44,840 --> 01:06:47,460 of a of a beneficial mutant spreading, 1216 01:06:47,460 --> 01:06:50,446 or you can have a population wave of an invasive species 1217 01:06:50,446 --> 01:06:52,821 or whatnot, and you end up getting very similar dynamics. 1218 01:07:00,596 --> 01:07:01,970 So the basic idea, here, is that, 1219 01:07:01,970 --> 01:07:10,730 if you look at the density as a function of position. 1220 01:07:10,730 --> 01:07:14,180 If start with, there's one individual. 1221 01:07:14,180 --> 01:07:15,671 What's going to happen? 1222 01:07:19,150 --> 01:07:22,450 It's going to start dividing, right? 1223 01:07:22,450 --> 01:07:24,337 So we kind of get up. 1224 01:07:24,337 --> 01:07:25,420 And it's going to come up. 1225 01:07:25,420 --> 01:07:27,590 And eventually it's going to saturate 1226 01:07:27,590 --> 01:07:32,900 at this carrying capacity, K. 1227 01:07:32,900 --> 01:07:35,360 And then you end up getting these spreading population 1228 01:07:35,360 --> 01:07:38,467 waves that look like this. 1229 01:07:38,467 --> 01:07:40,050 And the reason we're calling it a wave 1230 01:07:40,050 --> 01:07:44,000 is because the shape of this front is the same over time. 1231 01:07:47,330 --> 01:07:59,450 So it can really to be described as some function x minus vt. 1232 01:07:59,450 --> 01:08:01,840 And we are going to, maybe, typically assume 1233 01:08:01,840 --> 01:08:03,910 that we're in a situation where we don't have 1234 01:08:03,910 --> 01:08:04,810 to think about the left and the right, 1235 01:08:04,810 --> 01:08:06,510 because it's just too complicated. 1236 01:08:06,510 --> 01:08:09,070 So we'll just imagine it being that it's at saturation here, 1237 01:08:09,070 --> 01:08:10,861 and then we're looking at some front that's 1238 01:08:10,861 --> 01:08:12,860 moving to the right. 1239 01:08:12,860 --> 01:08:17,988 Now, by dimensional analysis, we should 1240 01:08:17,988 --> 01:08:20,279 be able to figure out what the velocity is going to be. 1241 01:08:22,707 --> 01:08:24,665 Remember how much we liked dimensional analysis 1242 01:08:24,665 --> 01:08:27,600 in this class? 1243 01:08:27,600 --> 01:08:29,060 Yes. 1244 01:08:29,060 --> 01:08:31,910 So what we're going to do is I'll 1245 01:08:31,910 --> 01:08:33,950 give you some characters that you're 1246 01:08:33,950 --> 01:08:35,765 going to be able to use in your quest. 1247 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 1248 01:08:48,069 --> 01:08:50,950 find it useful. 1249 01:08:50,950 --> 01:08:59,600 And you can raise something to the second power as well. 1250 01:08:59,600 --> 01:09:02,080 So what you can do is you're going to set up your cards 1251 01:09:02,080 --> 01:09:05,330 so that when I look at it, from the left to the right, 1252 01:09:05,330 --> 01:09:08,380 it will describe the velocity of this wave. 1253 01:09:12,988 --> 01:09:14,029 I'll give you 30 seconds. 1254 01:09:17,069 --> 01:09:24,049 So this is dimensional analysis for this wave velocity. 1255 01:10:06,890 --> 01:10:08,310 All right, do you need more time? 1256 01:10:08,310 --> 01:10:08,810 Yes? 1257 01:10:08,810 --> 01:10:10,000 OK, that's fine. 1258 01:11:06,650 --> 01:11:08,800 All right, let's go ahead and vote. 1259 01:11:13,310 --> 01:11:16,730 Construct your answer, remember, from me, from left to right. 1260 01:11:19,530 --> 01:11:22,950 Ready, three, two, one. 1261 01:11:27,160 --> 01:11:28,170 All right. 1262 01:11:28,170 --> 01:11:30,700 We have trouble. 1263 01:11:30,700 --> 01:11:32,850 A key skill is being able to imagine yourself 1264 01:11:32,850 --> 01:11:34,680 in someone else's shoes. 1265 01:11:34,680 --> 01:11:36,800 So if I'm viewing-- but that's OK. 1266 01:11:41,630 --> 01:11:47,250 So we have, here, this is kind of units of 1 over time. 1267 01:11:47,250 --> 01:11:51,850 This is what length squared over time. 1268 01:11:51,850 --> 01:11:54,515 Whereas this is a density. 1269 01:11:57,090 --> 01:12:01,190 If we want something that is a length over time, 1270 01:12:01,190 --> 01:12:04,190 then we're going to end up having to take r times D 1271 01:12:04,190 --> 01:12:06,230 and take a square root. 1272 01:12:06,230 --> 01:12:14,230 So this should be DAC, which is a square root 1273 01:12:14,230 --> 01:12:16,689 of-- of course, it depends on how you're entering it 1274 01:12:16,689 --> 01:12:18,980 into your calculator, if you have scientific calculator 1275 01:12:18,980 --> 01:12:21,920 or something else, maybe. 1276 01:12:21,920 --> 01:12:24,290 And indeed, it ends up, there's a 2 here. 1277 01:12:24,290 --> 01:12:26,100 So this is the velocity. 1278 01:12:31,080 --> 01:12:31,658 Yes? 1279 01:12:31,658 --> 01:12:32,574 AUDIENCE: [INAUDIBLE]. 1280 01:12:47,970 --> 01:12:49,470 PROFESSOR: I see what you're saying. 1281 01:12:49,470 --> 01:12:51,840 But it ends up not being true. 1282 01:12:51,840 --> 01:12:57,030 These derivative signs don't have any units. 1283 01:12:57,030 --> 01:13:00,210 So this still has units of a density 1284 01:13:00,210 --> 01:13:02,150 divided by a length squared. 1285 01:13:02,150 --> 01:13:06,096 So for unit purposes, you just look at this thing and at this. 1286 01:13:06,096 --> 01:13:09,155 This squared does mean there's a length squared 1287 01:13:09,155 --> 01:13:11,530 in the denominator, but this squared doesn't do anything. 1288 01:13:15,140 --> 01:13:17,210 So D is still a length squared over time. 1289 01:13:20,140 --> 01:13:22,380 So this is the famous Fisher velocity. 1290 01:13:25,590 --> 01:13:27,185 There are some mathematical subtleties 1291 01:13:27,185 --> 01:13:29,705 to all of this that we're not really going to get into. 1292 01:13:29,705 --> 01:13:30,630 I don't know. 1293 01:13:30,630 --> 01:13:31,880 I'm having trouble-- velocity. 1294 01:13:34,540 --> 01:13:38,350 There are a few features to highlight. 1295 01:13:38,350 --> 01:13:42,497 If the organism grows faster, the wave 1296 01:13:42,497 --> 01:13:43,580 is going to spread faster. 1297 01:13:43,580 --> 01:13:44,860 That make sense? 1298 01:13:44,860 --> 01:13:48,660 If it has a larger mobility, it also moves faster. 1299 01:13:48,660 --> 01:13:50,000 That makes sense. 1300 01:13:50,000 --> 01:13:54,260 Of course, the velocity is given by both of those things. 1301 01:13:54,260 --> 01:13:59,440 So this wave coming out really is a population level property. 1302 01:13:59,440 --> 01:14:01,130 Because it's not just growth. 1303 01:14:01,130 --> 01:14:03,430 It's not just motion. 1304 01:14:03,430 --> 01:14:08,406 It's a result of the coupled division and diffusion 1305 01:14:08,406 --> 01:14:10,280 that leads to this population wave spreading. 1306 01:14:15,330 --> 01:14:17,440 Importantly, to first order, it doesn't 1307 01:14:17,440 --> 01:14:22,030 depend on the carrying capacity, at least 1308 01:14:22,030 --> 01:14:23,470 within the deterministic regime. 1309 01:14:27,350 --> 01:14:31,957 AUDIENCE: It's interesting that, if the reproductive rate goes 1310 01:14:31,957 --> 01:14:35,110 to 0, suddenly the population stops spreading. 1311 01:14:39,226 --> 01:14:40,350 PROFESSOR: So first of all. 1312 01:14:40,350 --> 01:14:42,890 You'd say, oh, it would be sort of surprising if it really 1313 01:14:42,890 --> 01:14:44,910 kept on spreading in the absence of growth. 1314 01:14:44,910 --> 01:14:46,470 But what you're pointing out is that, 1315 01:14:46,470 --> 01:14:49,232 if you just, at one moment, turn off division, 1316 01:14:49,232 --> 01:14:50,690 then there will still be diffusion. 1317 01:14:50,690 --> 01:14:52,330 It'll still keep on going. 1318 01:14:52,330 --> 01:14:55,820 But this is the velocity of a wave when it's a wave. 1319 01:14:55,820 --> 01:15:00,630 When it's described by a function like this. 1320 01:15:00,630 --> 01:15:03,400 So it's true that you could turn off division, 1321 01:15:03,400 --> 01:15:04,456 and it'll still diffuse. 1322 01:15:04,456 --> 01:15:06,080 But then the shape is changing as well. 1323 01:15:12,840 --> 01:15:18,450 I'm going to draw a few lines describing 1324 01:15:18,450 --> 01:15:22,310 possible populations. 1325 01:15:22,310 --> 01:15:27,850 Now, let's assume that they have the same motion, diffusion. 1326 01:15:27,850 --> 01:15:30,650 I want to know, which one has the largest velocity? 1327 01:15:44,530 --> 01:15:51,110 Is it A, B, C, D? 1328 01:16:09,030 --> 01:16:10,030 It's the same diffusion. 1329 01:16:17,040 --> 01:16:19,700 Per capita growth rate as a function of the density 1330 01:16:19,700 --> 01:16:21,810 for three different organisms. 1331 01:16:44,470 --> 01:16:47,865 Ready, three, two, one. 1332 01:16:50,820 --> 01:16:53,870 All right, so I'd say we have a fair number of B's, D's. 1333 01:16:57,080 --> 01:17:03,670 It seems like it's B versus D. Now, this is tricky because D, 1334 01:17:03,670 --> 01:17:07,950 we have not explicitly considered here. 1335 01:17:07,950 --> 01:17:12,232 But it turns out that the answer is B. Certainly, 1336 01:17:12,232 --> 01:17:13,940 between these three, these are all really 1337 01:17:13,940 --> 01:17:16,500 logistic growth functions. 1338 01:17:16,500 --> 01:17:20,140 And so from the standpoint of here, it's just this r. 1339 01:17:20,140 --> 01:17:25,017 And r is the division rate at 0 cell density. 1340 01:17:25,017 --> 01:17:26,850 The per capita growth rate at 0 cell density 1341 01:17:26,850 --> 01:17:30,050 is what determines the velocity in a Fisher wave. 1342 01:17:30,050 --> 01:17:32,400 And indeed, that's true even if there's 1343 01:17:32,400 --> 01:17:35,583 no decrease in the growth up right until you 1344 01:17:35,583 --> 01:17:37,480 get to some carrying capacity. 1345 01:17:37,480 --> 01:17:41,944 And indeed, all of these cases, the division rate 1346 01:17:41,944 --> 01:17:44,110 and the growth rate that's relevant for the velocity 1347 01:17:44,110 --> 01:17:49,445 is when it hits this axis. 1348 01:17:49,445 --> 01:17:53,100 And indeed, all of these waves are described as Fisher 1349 01:17:53,100 --> 01:17:54,000 or pulled waves. 1350 01:17:56,630 --> 01:17:59,810 Because there's a sense that the entire wave 1351 01:17:59,810 --> 01:18:02,480 is determined by the front of the wave. 1352 01:18:02,480 --> 01:18:05,240 So we drew this profile. 1353 01:18:05,240 --> 01:18:06,522 I didn't do that very well. 1354 01:18:06,522 --> 01:18:07,730 Here, this is an exponential. 1355 01:18:11,540 --> 01:18:16,029 And the exponential actually is what's pulling this wave. 1356 01:18:16,029 --> 01:18:17,820 There ends up being a characteristic length 1357 01:18:17,820 --> 01:18:23,160 scale here that is the square root of D/r. 1358 01:18:23,160 --> 01:18:26,230 So this is the length scale of the exponential. 1359 01:18:26,230 --> 01:18:30,000 And the velocity and the length scale 1360 01:18:30,000 --> 01:18:34,630 are only functions of the division 1361 01:18:34,630 --> 01:18:36,770 rate in the limit below cell density 1362 01:18:36,770 --> 01:18:38,040 or low density of organism. 1363 01:18:42,860 --> 01:18:46,187 The shape of what goes on here changes, indeed, 1364 01:18:46,187 --> 01:18:48,020 the bulk properties of the wave, but doesn't 1365 01:18:48,020 --> 01:18:52,910 change the velocity. 1366 01:18:52,910 --> 01:18:56,370 And I just want to make one comparison of all this 1367 01:18:56,370 --> 01:18:59,020 to-- because there's another qualitatively different kind 1368 01:18:59,020 --> 01:19:02,660 of wave, which is a so-called pushed wave. 1369 01:19:07,680 --> 01:19:10,760 And that's what happens if you have an Allee 1370 01:19:10,760 --> 01:19:13,060 effect, particularly like a strong Allee effect. 1371 01:19:13,060 --> 01:19:18,610 If this thing looks-- like this is certainly possible. 1372 01:19:21,840 --> 01:19:25,230 This is an Allee effect. 1373 01:19:25,230 --> 01:19:27,939 Now, if you just said, oh, the only thing that matters is 1374 01:19:27,939 --> 01:19:30,230 the growth rate at low cell density, you would say, oh, 1375 01:19:30,230 --> 01:19:32,300 this thing cannot possibly expand. 1376 01:19:32,300 --> 01:19:35,130 Although it turns out that it still is possible. 1377 01:19:35,130 --> 01:19:37,770 And in this situation, it would be 1378 01:19:37,770 --> 01:19:41,200 called a push wave, where your profile somehow 1379 01:19:41,200 --> 01:19:42,500 maybe looks kind of similar. 1380 01:19:42,500 --> 01:19:43,910 But instead of it being the front 1381 01:19:43,910 --> 01:19:45,920 of the wave that's pulling the wave, 1382 01:19:45,920 --> 01:19:47,890 it's diffusion around the bulk. 1383 01:19:47,890 --> 01:19:49,430 Because the bulk is the part that 1384 01:19:49,430 --> 01:19:51,840 is actually happily growing. 1385 01:19:51,840 --> 01:19:55,350 Because the front, here, in this case, is dying. 1386 01:19:55,350 --> 01:19:59,610 Yet it still is possible to have a positive velocity. 1387 01:19:59,610 --> 01:20:03,330 And so this is, then, a qualitatively different kind 1388 01:20:03,330 --> 01:20:05,681 of population expansion. 1389 01:20:05,681 --> 01:20:07,180 So cooperatively growing populations 1390 01:20:07,180 --> 01:20:10,280 expand very differently from logistically growing 1391 01:20:10,280 --> 01:20:11,200 populations. 1392 01:20:11,200 --> 01:20:14,200 And one of things that the reading in Physics Today 1393 01:20:14,200 --> 01:20:16,000 talked about is these different rates 1394 01:20:16,000 --> 01:20:20,340 of loss of heterozygocity and so forth in different populations. 1395 01:20:20,340 --> 01:20:23,990 And as you might expect, the pulled waves 1396 01:20:23,990 --> 01:20:26,570 have a smaller effective population size 1397 01:20:26,570 --> 01:20:31,137 than the pushed waves, because, here, the relevant population 1398 01:20:31,137 --> 01:20:32,720 is at the front if it's a low density. 1399 01:20:32,720 --> 01:20:34,261 Whereas here, the relevant population 1400 01:20:34,261 --> 01:20:37,144 is the bulk that's at high density. 1401 01:20:37,144 --> 01:20:38,560 With that, I think we should quit. 1402 01:20:38,560 --> 01:20:39,955 But I will see you on Tuesday. 1403 01:20:39,955 --> 01:20:42,500 And we'll talk about this neutral theory in ecology. 1404 01:20:42,500 --> 01:20:44,050 Thanks.