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PROFESSOR: Today, our goal
is to discuss some concepts

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00:00:25,290 --> 00:00:28,670
in this question of a kind of
population or ecosystem level

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00:00:28,670 --> 00:00:32,996
stability, resilience, and
associated sudden transitions

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00:00:32,996 --> 00:00:33,620
in populations.

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00:00:40,070 --> 00:00:42,640
And something about
diversity of populations.

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00:00:42,640 --> 00:00:45,540
We will revisit this
question of diversity

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00:00:45,540 --> 00:00:47,861
more on the last
class, where we're

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00:00:47,861 --> 00:00:49,360
going to discuss
some of these ideas

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00:00:49,360 --> 00:00:52,090
from the neutral
theory in ecology,

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00:00:52,090 --> 00:00:55,350
but it'll be a bit
of a preview today.

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00:00:59,027 --> 00:01:00,610
So what we're going
to do is start out

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00:01:00,610 --> 00:01:05,030
by just considering the dynamics
of just a single population,

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00:01:05,030 --> 00:01:07,810
where you have maybe a
one population that's

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00:01:07,810 --> 00:01:11,600
growing either logistically,
or via some cooperative kind

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00:01:11,600 --> 00:01:12,820
of dynamic.

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00:01:12,820 --> 00:01:17,380
So, single population.

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00:01:17,380 --> 00:01:19,530
And in here, we're going
to understand something

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00:01:19,530 --> 00:01:21,550
about this idea of
either logistic growth

26
00:01:21,550 --> 00:01:24,030
as compared to something called
the Allee effect, whereby

27
00:01:24,030 --> 00:01:26,720
the population has some
cooperative type dynamic.

28
00:01:26,720 --> 00:01:30,920
So this is maybe logistic
versus the Allee effect.

29
00:01:30,920 --> 00:01:34,290
And try to understand
how this Allee effect can

30
00:01:34,290 --> 00:01:37,010
lead to the sudden transitions
that you read about

31
00:01:37,010 --> 00:01:39,390
in the review.

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00:01:39,390 --> 00:01:42,760
And then we'll say something
about the Lotka-Volterra

33
00:01:42,760 --> 00:01:44,899
competition model.

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00:01:44,899 --> 00:01:46,440
So this is a slightly
different model

35
00:01:46,440 --> 00:01:48,231
from the Lotka-Volterra
predator-prey model

36
00:01:48,231 --> 00:01:51,590
that we've all been
thinking about recently

37
00:01:51,590 --> 00:01:54,000
over the last week or so.

38
00:01:54,000 --> 00:01:56,390
But it's again, kind
of the simplest model

39
00:01:56,390 --> 00:01:58,000
you might write
down that captures

40
00:01:58,000 --> 00:02:00,550
this idea of interactions
between species.

41
00:02:00,550 --> 00:02:02,930
In particular,
competitive interactions.

42
00:02:06,880 --> 00:02:09,580
Then at the end, and depending
on how much time we have,

43
00:02:09,580 --> 00:02:10,830
we'll either finish it or not.

44
00:02:10,830 --> 00:02:13,163
We'll talk about some these
non transitive interactions.

45
00:02:15,220 --> 00:02:18,640
The so-called rock,
paper, scissors,

46
00:02:18,640 --> 00:02:22,380
RPS type interactions.

47
00:02:22,380 --> 00:02:24,380
And some of the
measurements that

48
00:02:24,380 --> 00:02:28,120
have been made both in male
mating strategies in lizards

49
00:02:28,120 --> 00:02:31,900
in the California mountains, as
well as kind of a rock, paper,

50
00:02:31,900 --> 00:02:34,670
scissors type dynamic that
was explored experimentally

51
00:02:34,670 --> 00:02:39,050
by Benjamin Kerr in the context
of bacterial toxin production,

52
00:02:39,050 --> 00:02:41,160
or chemical warfare bacteria.

53
00:02:41,160 --> 00:02:44,800
And this subject
does come in a bit

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00:02:44,800 --> 00:02:47,060
into this question
of spatial structure,

55
00:02:47,060 --> 00:02:50,960
and how spatial structure
may help facilitate

56
00:02:50,960 --> 00:02:52,455
the stability of populations.

57
00:02:52,455 --> 00:02:54,830
So there's this idea of that
non-transitive interactions,

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00:02:54,830 --> 00:02:57,430
whether you have Rock, Paper,
Scissor type interactions,

59
00:02:57,430 --> 00:03:01,810
may facilitate coexistence of
either genotypes or species.

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00:03:01,810 --> 00:03:03,460
But in some of these
studies, there's

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00:03:03,460 --> 00:03:06,899
an argument that sometimes
this non-transitive interaction

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00:03:06,899 --> 00:03:08,690
may not be enough, but
then in the presence

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00:03:08,690 --> 00:03:11,190
of spatial structure,
maybe it does help.

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00:03:11,190 --> 00:03:13,640
On Thursday, the primary
topic of the class

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00:03:13,640 --> 00:03:16,650
will be trying to understand
the dynamics of populations

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00:03:16,650 --> 00:03:19,590
when their spatially extended.

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00:03:19,590 --> 00:03:21,537
So this will
transition naturally

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00:03:21,537 --> 00:03:22,745
into the subject on Thursday.

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00:03:26,070 --> 00:03:30,350
Are there any questions
about where we are here?

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00:03:30,350 --> 00:03:32,930
Or administrative type issues?

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00:03:35,597 --> 00:03:37,930
All right, so I just want to
make sure that we're first,

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00:03:37,930 --> 00:03:38,830
all on the same page.

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00:03:38,830 --> 00:03:40,490
We talk about logistic growth.

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00:03:45,350 --> 00:03:47,350
So we can think
about-- Well, what

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00:03:47,350 --> 00:03:51,910
are the simplest ways that we
might consider the population

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00:03:51,910 --> 00:03:54,570
dynamics of just a
single population.

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00:03:54,570 --> 00:03:57,660
Well, the logistic growth
is maybe the simplest model

78
00:03:57,660 --> 00:04:03,000
you might write down, where the
population is at least bounded.

79
00:04:03,000 --> 00:04:08,520
And I might even-- Maybe
I'll even just first draw

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00:04:08,520 --> 00:04:10,960
exponential growth,
just so that we can--

81
00:04:13,820 --> 00:04:15,320
The very simplest
thing you might do

82
00:04:15,320 --> 00:04:18,240
is you might just
say that N dot might

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00:04:18,240 --> 00:04:20,290
be equal to some r times n.

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00:04:22,800 --> 00:04:27,760
In this case, if
we plotted gamma,

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00:04:27,760 --> 00:04:33,620
the per capita rate
of population growth,

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00:04:33,620 --> 00:04:37,280
this is just some line at r.

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00:04:37,280 --> 00:04:41,130
In this case, the population
grows exponentially.

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00:04:41,130 --> 00:04:44,490
And that that means
that if you plot

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00:04:44,490 --> 00:04:46,904
say the number as a
function of time, well

90
00:04:46,904 --> 00:04:48,320
it doesn't matter
where you start,

91
00:04:48,320 --> 00:04:49,690
you always go to infinity.

92
00:04:49,690 --> 00:04:50,460
Right?

93
00:04:50,460 --> 00:04:58,870
Now we can compare this to the
logistic growth case, where

94
00:04:58,870 --> 00:05:02,600
we just assume that
at low density,

95
00:05:02,600 --> 00:05:06,190
indeed, we grow
exponentially at this rate r,

96
00:05:06,190 --> 00:05:08,135
but then we have
a linear decrease

97
00:05:08,135 --> 00:05:09,810
in this per capita
rate of growth

98
00:05:09,810 --> 00:05:11,890
as a function of density.

99
00:05:11,890 --> 00:05:22,390
So we might write this as
something that looks like this.

100
00:05:22,390 --> 00:05:25,756
In which case, if we write
this per capita growth

101
00:05:25,756 --> 00:05:28,130
rate of the population, as a
function of population size,

102
00:05:28,130 --> 00:05:32,710
it starts out at r,
but then it goes to 0.

103
00:05:32,710 --> 00:05:34,904
And indeed you, might
even if you'd like,

104
00:05:34,904 --> 00:05:37,070
you could say that it goes
down to a negative value.

105
00:05:37,070 --> 00:05:39,069
So in that case, you could
start at a population

106
00:05:39,069 --> 00:05:42,500
size above the
carrying capacity k,

107
00:05:42,500 --> 00:05:44,395
and you still come back down it.

108
00:05:51,132 --> 00:05:53,340
One of things that we want
to make sure we understand

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00:05:53,340 --> 00:05:58,460
is what the bifurcation
structure of these equations

110
00:05:58,460 --> 00:05:59,520
is going to look like.

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00:05:59,520 --> 00:06:01,360
In particular, we
can think about what

112
00:06:01,360 --> 00:06:04,575
happens if we add some sort of
death rate to the population.

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00:06:07,890 --> 00:06:11,960
Right now, in this
case, a death rate

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00:06:11,960 --> 00:06:16,880
corresponds to is
something-- OK, we're looking

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00:06:16,880 --> 00:06:20,189
at the bifurcation diagram.

116
00:06:20,189 --> 00:06:21,730
What we're going to
do is we're going

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00:06:21,730 --> 00:06:23,750
to think about what
happens if you look

118
00:06:23,750 --> 00:06:25,850
at the size of the population.

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00:06:25,850 --> 00:06:27,975
In the bifurcation diagram,
what we typically do is

120
00:06:27,975 --> 00:06:29,920
we look at the fixed points.

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00:06:29,920 --> 00:06:32,940
The fixed points
in n, as a function

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00:06:32,940 --> 00:06:34,430
of some external parameter.

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00:06:34,430 --> 00:06:36,950
And here, we might just
think about delta here

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00:06:36,950 --> 00:06:38,590
which is just some death rate.

125
00:06:43,420 --> 00:06:45,905
The simplest way to
include this here would

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00:06:45,905 --> 00:06:50,210
be to have a minus delta n.

127
00:06:50,210 --> 00:06:52,575
So the idea is that
there's this population.

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00:06:52,575 --> 00:06:56,054
It grows exponentially
at small sizes.

129
00:06:56,054 --> 00:06:57,970
As the population size
grows, it might run out

130
00:06:57,970 --> 00:07:01,150
of nutrients or so, and that
limits its overall population

131
00:07:01,150 --> 00:07:02,140
size.

132
00:07:02,140 --> 00:07:04,300
Then, we can add
another term here

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00:07:04,300 --> 00:07:05,959
which is corresponding
to something

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00:07:05,959 --> 00:07:07,500
about the quality
of the environment.

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00:07:07,500 --> 00:07:09,999
So this could be the amount of
hunting that is taking place,

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00:07:09,999 --> 00:07:13,620
or it could be a
reflection of pollutants

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00:07:13,620 --> 00:07:15,069
in the environment or so.

138
00:07:15,069 --> 00:07:16,610
And what we want to
do is try to make

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00:07:16,610 --> 00:07:20,220
sure we understand how
the size of the population

140
00:07:20,220 --> 00:07:22,910
will respond to some death rate.

141
00:07:26,930 --> 00:07:30,980
Well maybe we'll go ahead
and do a verbal vote.

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00:07:30,980 --> 00:07:34,960
So in the review
that you guys read,

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00:07:34,960 --> 00:07:37,650
this was talking about
in these early warning

144
00:07:37,650 --> 00:07:39,800
indicators in the context
of sudden transitions

145
00:07:39,800 --> 00:07:41,950
and populations.

146
00:07:41,950 --> 00:07:43,690
So this was saying
that whether you're

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00:07:43,690 --> 00:07:47,830
looking at a population, or
an ecosystem, or maybe even

148
00:07:47,830 --> 00:07:49,900
other complex system,
such as climate regime

149
00:07:49,900 --> 00:07:52,700
shifts, epileptic seizures,
and so forth, the authors were

150
00:07:52,700 --> 00:07:55,840
arguing that in
response to a slowly

151
00:07:55,840 --> 00:07:57,620
changing environmental
knob, the system

152
00:07:57,620 --> 00:08:01,730
can experience a sudden
transition in the state.

153
00:08:01,730 --> 00:08:04,910
Now, the question is here.

154
00:08:04,910 --> 00:08:08,140
And the sudden transitions
that they were describing,

155
00:08:08,140 --> 00:08:13,110
those were fold bifurcations,
or saddle-node bifurcations.

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00:08:13,110 --> 00:08:14,560
Where there's a
sudden transition

157
00:08:14,560 --> 00:08:16,040
in say, the equilibrium.

158
00:08:16,040 --> 00:08:17,540
Size of the population
is a function

159
00:08:17,540 --> 00:08:21,050
of some external knob describing
the quality of the environment.

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00:08:21,050 --> 00:08:25,530
So the question is
whether this population

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00:08:25,530 --> 00:08:29,212
experiences such a full
bifurcation at least

162
00:08:29,212 --> 00:08:30,170
to a sudden transition.

163
00:08:34,169 --> 00:08:36,890
Do understand the question?

164
00:08:36,890 --> 00:08:42,085
I'll say fold bifurcation.

165
00:08:46,700 --> 00:08:51,812
I'll let you guys vote, just
so you can use your cards.

166
00:08:51,812 --> 00:08:54,270
And this is a full bifurcation
as a function of this delta,

167
00:08:54,270 --> 00:08:55,770
which is death rate.

168
00:08:55,770 --> 00:08:57,110
Ready?

169
00:08:57,110 --> 00:08:59,895
Three, two, one.

170
00:09:03,170 --> 00:09:05,520
We have a majority
now that are saying

171
00:09:05,520 --> 00:09:07,510
the answer is no,
this does not actually

172
00:09:07,510 --> 00:09:11,500
have a fold bifurcation.

173
00:09:11,500 --> 00:09:13,270
And the reason for
that is that we

174
00:09:13,270 --> 00:09:17,580
can find the-- In the
absence of a death rate, what

175
00:09:17,580 --> 00:09:20,050
is the equilibrium
population size?

176
00:09:22,650 --> 00:09:23,650
k, right?

177
00:09:25,912 --> 00:09:28,370
Indeed, what's going to happen
is that as we add this death

178
00:09:28,370 --> 00:09:33,170
rate, we can just figure out,
OK well, the fixed points

179
00:09:33,170 --> 00:09:35,350
occur when N dots equal to zero.

180
00:09:35,350 --> 00:09:36,210
All of these things.

181
00:09:36,210 --> 00:09:38,491
There's an n here, we
can divide by that.

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00:09:38,491 --> 00:09:40,490
Well we could just do it,
just so that we're all

183
00:09:40,490 --> 00:09:42,210
on the same page.

184
00:09:42,210 --> 00:09:43,210
So the fixed point.

185
00:09:46,874 --> 00:09:50,910
What we want to do is
set N dot equal to zero.

186
00:09:50,910 --> 00:09:59,320
That tells us that n times
r1 minus n over k minus delta

187
00:09:59,320 --> 00:10:00,740
is equal to 0.

188
00:10:00,740 --> 00:10:01,720
OK.

189
00:10:01,720 --> 00:10:04,610
How many fixed points are there
going to be in the system?

190
00:10:04,610 --> 00:10:05,220
Two.

191
00:10:05,220 --> 00:10:09,540
Right, so indeed, this is N dot.

192
00:10:09,540 --> 00:10:14,059
So n equal to zero is going
to be one fixed point,

193
00:10:14,059 --> 00:10:15,600
but the other one
is going to be when

194
00:10:15,600 --> 00:10:17,660
this thing is equal to zero.

195
00:10:17,660 --> 00:10:20,910
And so then we end up
with 1 minus n over k,

196
00:10:20,910 --> 00:10:23,190
is going to be equal
to some delta over r.

197
00:10:36,890 --> 00:10:39,100
So we can see that
this stable fixed point

198
00:10:39,100 --> 00:10:42,660
is going to decrease linearly
with this death rate delta.

199
00:10:42,660 --> 00:10:44,510
And it's just going to
go smoothly to zero.

200
00:10:44,510 --> 00:10:47,900
That's this key feature of
this bifurcation digraph.

201
00:10:47,900 --> 00:10:54,190
So what happens is that we come
down like this, and indeed,

202
00:10:54,190 --> 00:10:57,430
we could even continue this and
the bifurcation occurs here.

203
00:10:59,940 --> 00:11:02,410
We want to draw--
This corresponds

204
00:11:02,410 --> 00:11:03,640
to a stable fixed point.

205
00:11:06,580 --> 00:11:08,215
And a dashed line is unstable.

206
00:11:14,580 --> 00:11:18,550
In this case, this is the
stable and this is the unstable.

207
00:11:23,194 --> 00:11:24,610
So this thing here
is what's known

208
00:11:24,610 --> 00:11:26,095
as a trans-critical bifurcation.

209
00:11:28,630 --> 00:11:31,650
And it's a bifurcation because
the fixed points do something.

210
00:11:31,650 --> 00:11:34,030
In this case, they
exchange stability.

211
00:11:34,030 --> 00:11:38,610
So what happens is that
this thing becomes unstable,

212
00:11:38,610 --> 00:11:40,600
whereas the fixed
point at n equals

213
00:11:40,600 --> 00:11:43,524
0 becomes a stable fixed point.

214
00:11:48,940 --> 00:11:52,630
So there's no tipping
point or sudden transition

215
00:11:52,630 --> 00:11:57,380
in this logistic growth as
you change the death rate.

216
00:11:57,380 --> 00:11:59,270
All right, you can think
of it as a situation

217
00:11:59,270 --> 00:12:01,190
here where the death
rate just kind of starts

218
00:12:01,190 --> 00:12:02,190
pulling this thing down.

219
00:12:06,590 --> 00:12:10,090
And this bifurcation occurs
when you pull it down--

220
00:12:10,090 --> 00:12:11,225
when delta is equal to r.

221
00:12:18,910 --> 00:12:24,050
And what this means is
that if you plot say,

222
00:12:24,050 --> 00:12:32,400
the number as a function
of time, for low delta,

223
00:12:32,400 --> 00:12:34,400
in particular, for delta
equal to zero then this

224
00:12:34,400 --> 00:12:45,240
is your-- You come to
this equilibrium k.

225
00:12:45,240 --> 00:12:48,350
Where as delta increases,
eventually you just

226
00:12:48,350 --> 00:12:51,660
get to a situation where all of,
regardless of where you start,

227
00:12:51,660 --> 00:12:52,300
you go extinct.

228
00:12:55,270 --> 00:12:58,770
Now, in many cases, we don't
draw what happens down here

229
00:12:58,770 --> 00:13:02,620
because n being less than
0 is not a physical thing.

230
00:13:02,620 --> 00:13:05,185
But it's useful to think
about it mathematically,

231
00:13:05,185 --> 00:13:07,685
because that's what tells you
that that was the bifurcation.

232
00:13:14,310 --> 00:13:16,750
Are there any questions
about where we are here?

233
00:13:21,530 --> 00:13:24,580
One thing that's valuable in
the case of-- well, in general--

234
00:13:24,580 --> 00:13:29,610
is to switch between the
algebraic characterization

235
00:13:29,610 --> 00:13:32,094
of the dynamics a system
and the graphical dynamics,

236
00:13:32,094 --> 00:13:34,260
because you end up seeing
different things depending

237
00:13:34,260 --> 00:13:35,866
on how you do it.

238
00:13:35,866 --> 00:13:37,240
For the case of
the Allee effect,

239
00:13:37,240 --> 00:13:40,300
we're just going to look at
it and analyze it graphically,

240
00:13:40,300 --> 00:13:43,829
just because the equation
is a little bit cumbersome,

241
00:13:43,829 --> 00:13:45,745
and I think don't provide
very much intuition.

242
00:13:51,490 --> 00:13:55,530
What you can see is that
for the logistic growth,

243
00:13:55,530 --> 00:13:57,920
other individuals
in the population

244
00:13:57,920 --> 00:14:01,590
always decrease your
happiness as an individual.

245
00:14:01,590 --> 00:14:05,000
So if you imagine that you are
some member of this population,

246
00:14:05,000 --> 00:14:06,770
and having more
members the population

247
00:14:06,770 --> 00:14:08,270
there is always
kind of bad for you,

248
00:14:08,270 --> 00:14:10,940
because it decreases
your growth rate.

249
00:14:10,940 --> 00:14:14,434
And to think about the
happiness of an individual, what

250
00:14:14,434 --> 00:14:16,100
you typically want
to do is divide by n,

251
00:14:16,100 --> 00:14:18,475
because you're thinking about
the per capita growth rate.

252
00:14:18,475 --> 00:14:20,880
So that's telling you
about the ability hat

253
00:14:20,880 --> 00:14:22,380
you will have to reproduce.

254
00:14:22,380 --> 00:14:24,440
So what you did we do
is just divide by n.

255
00:14:24,440 --> 00:14:30,160
What you can see is that
per capita division, gamma,

256
00:14:30,160 --> 00:14:32,910
is a monotonically decreasing
function of the population

257
00:14:32,910 --> 00:14:33,870
size.

258
00:14:33,870 --> 00:14:36,550
This is saying that more
individuals in the population

259
00:14:36,550 --> 00:14:39,410
just take up space,
resources, mate, something.

260
00:14:39,410 --> 00:14:42,510
And so you as an
individual never benefit

261
00:14:42,510 --> 00:14:44,970
from the presence of
other individuals.

262
00:14:44,970 --> 00:14:47,440
Whereas, when you have the
Allee effect, what that's saying

263
00:14:47,440 --> 00:14:50,560
is that over some range of
population size or densities,

264
00:14:50,560 --> 00:14:52,785
there is a positive effect
of other individuals

265
00:14:52,785 --> 00:14:53,580
in the population.

266
00:15:01,000 --> 00:15:07,740
This is saying that if we
plot gamma, in particular,

267
00:15:07,740 --> 00:15:14,670
the derivative of gamma with
respect to n is greater than 0

268
00:15:14,670 --> 00:15:19,220
for some n.

269
00:15:19,220 --> 00:15:24,010
This is just saying that if we
plot this per capita division

270
00:15:24,010 --> 00:15:26,750
rate as a function of the
size of the population,

271
00:15:26,750 --> 00:15:30,710
there's some region in which
this thing has positive slope.

272
00:15:30,710 --> 00:15:32,870
And often, people
distinguish between

273
00:15:32,870 --> 00:15:35,280
the so-called strong Allee
effect and the weak Allee

274
00:15:35,280 --> 00:15:36,500
effect.

275
00:15:36,500 --> 00:15:38,700
The strong Allee effect
corresponds to the situation

276
00:15:38,700 --> 00:15:43,740
where at n equal to zero,
you start out with negative

277
00:15:43,740 --> 00:15:44,810
per capita growth rates.

278
00:15:47,900 --> 00:15:50,140
So this is just some
generic curve here,

279
00:15:50,140 --> 00:15:51,530
that describes the Allee effect.

280
00:15:51,530 --> 00:15:56,230
The important thing is that
it's coming up here at low n.

281
00:15:56,230 --> 00:15:59,710
And what this
tells us is that we

282
00:15:59,710 --> 00:16:01,860
will have some sort
of different dynamics

283
00:16:01,860 --> 00:16:05,780
where this is going to be the
stable size of the population

284
00:16:05,780 --> 00:16:07,380
here.

285
00:16:07,380 --> 00:16:10,140
But there's also a minimal
size required for survival.

286
00:16:10,140 --> 00:16:13,310
So if you start out below
this n, then you come down.

287
00:16:17,695 --> 00:16:19,320
If you want you want,
you could call it

288
00:16:19,320 --> 00:16:27,960
some-- This could be some
k, and this could be n-min.

289
00:16:27,960 --> 00:16:32,450
This is saying if we plot
n as a function of time

290
00:16:32,450 --> 00:16:36,980
starting with different
sizes of the population,

291
00:16:36,980 --> 00:16:40,960
then indeed, this thing
is a stable state.

292
00:16:40,960 --> 00:16:44,904
But the key thing is that
there's this other minimal size

293
00:16:44,904 --> 00:16:45,820
required for survival.

294
00:16:45,820 --> 00:16:49,360
So if you start out around here,
then just above it you come up,

295
00:16:49,360 --> 00:16:51,382
but just below
it, you come down.

296
00:16:57,154 --> 00:16:57,820
What do you say?

297
00:16:57,820 --> 00:16:59,760
What you see is
that this population

298
00:16:59,760 --> 00:17:02,130
that experiences the
strong Allee effect

299
00:17:02,130 --> 00:17:03,670
has bistable fates.

300
00:17:06,290 --> 00:17:07,190
So it's bistable.

301
00:17:07,190 --> 00:17:09,730
This is bistable depending
on the starting size

302
00:17:09,730 --> 00:17:10,564
n of the population.

303
00:17:14,359 --> 00:17:20,310
Now, can somebody suggest
possible explanations

304
00:17:20,310 --> 00:17:22,240
for why there might
an Allee effect?

305
00:17:41,960 --> 00:17:43,789
AUDIENCE: Sexual reproduction?

306
00:17:43,789 --> 00:17:45,080
PROFESSOR: Sexual reproduction.

307
00:17:45,080 --> 00:17:47,583
And can you explain
that a little bit more?

308
00:17:47,583 --> 00:17:48,458
AUDIENCE: [INAUDIBLE]

309
00:17:55,920 --> 00:17:56,920
PROFESSOR: That's right.

310
00:17:56,920 --> 00:17:59,360
And basically, just
the need to find mates.

311
00:18:10,440 --> 00:18:15,600
So this implies that
at some density,

312
00:18:15,600 --> 00:18:17,287
sexual reproducing
species are expected

313
00:18:17,287 --> 00:18:18,370
to have this Allee effect.

314
00:18:18,370 --> 00:18:20,510
It doesn't mean
that the Allee fact

315
00:18:20,510 --> 00:18:23,980
is super strong at
moderate sizes, right?

316
00:18:23,980 --> 00:18:25,940
I think that this
statement is true,

317
00:18:25,940 --> 00:18:29,679
but it might lead us to conclude
that these extreme dynamics

318
00:18:29,679 --> 00:18:32,220
that we're talking about here
are relevant for every sexually

319
00:18:32,220 --> 00:18:33,053
reproducing species.

320
00:18:33,053 --> 00:18:38,670
But ultimately, the question
is maybe how big is this n-min?

321
00:18:38,670 --> 00:18:43,850
If n-min is two, then it's
not a severe constraint

322
00:18:43,850 --> 00:18:45,700
on the population.

323
00:18:45,700 --> 00:18:50,850
Whereas if n-min is-- if you
need to have 100 California

324
00:18:50,850 --> 00:18:53,110
condors in order to have
a viable population,

325
00:18:53,110 --> 00:18:54,580
then that's a problem.

326
00:18:54,580 --> 00:18:56,990
So there really is a
quantitative question

327
00:18:56,990 --> 00:19:01,790
of how big is this minimal size.

328
00:19:01,790 --> 00:19:05,474
Other suggestions of why it is
you might have an Allee effect?

329
00:19:05,474 --> 00:19:08,210
AUDIENCE: The animal
species in groups?

330
00:19:08,210 --> 00:19:09,210
PROFESSOR: That's right.

331
00:19:09,210 --> 00:19:10,350
Group hunting.

332
00:19:10,350 --> 00:19:15,106
And are you aware of any cases
that have group hunting ?

333
00:19:15,106 --> 00:19:17,299
AUDIENCE: Wolves?

334
00:19:17,299 --> 00:19:18,840
PROFESSOR: That's
right, so, exactly.

335
00:19:21,670 --> 00:19:24,751
Right, so wolves for
example, can take down

336
00:19:24,751 --> 00:19:26,750
like bison, of large
animals they would never be

337
00:19:26,750 --> 00:19:28,980
able to take down on their own.

338
00:19:28,980 --> 00:19:33,920
So this could be this could be
wolves, it could be primates.

339
00:19:33,920 --> 00:19:37,350
We historically did-- I guess
we still have some groups.

340
00:19:37,350 --> 00:19:41,970
Societies will have
things like this.

341
00:19:41,970 --> 00:19:44,240
But even at the
microscopic realm,

342
00:19:44,240 --> 00:19:47,500
if you look at just bacteria
or yeast and so forth,

343
00:19:47,500 --> 00:19:51,480
there are often cases where
food is generated or broken

344
00:19:51,480 --> 00:19:53,230
down outside of the cell.

345
00:19:53,230 --> 00:19:54,810
So any of these
cases where you have

346
00:19:54,810 --> 00:19:57,000
secreted enzymatic breakdown.

347
00:20:07,190 --> 00:20:07,840
That was an i.

348
00:20:12,440 --> 00:20:13,100
Other thoughts?

349
00:20:20,777 --> 00:20:21,277
Yes?

350
00:20:21,277 --> 00:20:22,777
AUDIENCE: Protection
from predators?

351
00:20:22,777 --> 00:20:24,000
PROFESSOR: Right, protection.

352
00:20:24,000 --> 00:20:26,050
And so this is the
flip side of this,

353
00:20:26,050 --> 00:20:28,300
so this would be
predator avoidance.

354
00:20:28,300 --> 00:20:32,490
Or an example of
that would be what?

355
00:20:32,490 --> 00:20:34,430
AUDIENCE: Antelopes.

356
00:20:34,430 --> 00:20:37,093
PROFESSOR: What do antelopes do?

357
00:20:37,093 --> 00:20:38,980
AUDIENCE: They run in groups.

358
00:20:38,980 --> 00:20:39,980
PROFESSOR: That's right.

359
00:20:39,980 --> 00:20:41,880
So the standard
example here is kind

360
00:20:41,880 --> 00:20:45,130
of what herding behavior
of land animals, schooling

361
00:20:45,130 --> 00:20:47,450
behavior of fish.

362
00:20:47,450 --> 00:20:51,370
Of course, researchers
argue about to what degree

363
00:20:51,370 --> 00:20:54,750
the benefits are primarily
this, versus something else.

364
00:20:54,750 --> 00:20:56,940
I am not going to weigh
in on that debate.

365
00:20:56,940 --> 00:21:04,680
But, I'd be surprised
if this were never true.

366
00:21:04,680 --> 00:21:09,560
In all these examples, there's
clearly something cooperative

367
00:21:09,560 --> 00:21:10,779
about the dynamics.

368
00:21:10,779 --> 00:21:12,320
In that sense, it
kind of makes sense

369
00:21:12,320 --> 00:21:14,920
to think there's some element
of cooperative growth,

370
00:21:14,920 --> 00:21:17,750
whether it's on the hunting
side, or the protection side.

371
00:21:20,560 --> 00:21:24,440
So if you look at
all these cases

372
00:21:24,440 --> 00:21:26,424
these would all be very
naturally described

373
00:21:26,424 --> 00:21:27,590
as some sort of cooperation.

374
00:21:30,057 --> 00:21:32,390
I do want to highlight though,
that the Allee effect can

375
00:21:32,390 --> 00:21:34,640
be caused by things that
lead to what you might

376
00:21:34,640 --> 00:21:36,100
call effective cooperation.

377
00:21:36,100 --> 00:21:38,145
So it's not like
obviously cooperative,

378
00:21:38,145 --> 00:21:39,520
or not intentionally
cooperative,

379
00:21:39,520 --> 00:21:42,220
but still has the same effect.

380
00:21:42,220 --> 00:21:45,075
A nice example of this is what
we call predator satiation.

381
00:21:51,130 --> 00:21:53,692
Can somebody guess
what this might mean?

382
00:21:53,692 --> 00:21:54,900
Somebody's laughing so it's--

383
00:21:57,265 --> 00:21:59,139
AUDIENCE: If there are
enough animals around,

384
00:21:59,139 --> 00:22:00,361
the predator will be fed.

385
00:22:00,361 --> 00:22:01,360
PROFESSOR: That's right.

386
00:22:01,360 --> 00:22:03,630
If the predators
gets full, then you

387
00:22:03,630 --> 00:22:05,660
can end up with something
that's like this.

388
00:22:05,660 --> 00:22:09,620
And the way I think about this
is just in a group like this,

389
00:22:09,620 --> 00:22:15,180
if 20 bears come in
and eat 20 of us,

390
00:22:15,180 --> 00:22:18,350
then we hope that there
are 20 other people that

391
00:22:18,350 --> 00:22:21,070
will get eaten first, so that
you know that the leftovers can

392
00:22:21,070 --> 00:22:21,570
run off.

393
00:22:21,570 --> 00:22:22,070
Right?

394
00:22:25,180 --> 00:22:28,170
This is embodied in that joke
that people tell when they're

395
00:22:28,170 --> 00:22:32,004
the two hikers in the woods, and
they see the mother bear that's

396
00:22:32,004 --> 00:22:33,670
angry and one of the
hikers reaches down

397
00:22:33,670 --> 00:22:36,350
and starts tying his shoelaces.

398
00:22:36,350 --> 00:22:38,990
And the other thing the hiking
partner says why are you

399
00:22:38,990 --> 00:22:40,230
tying your shoelaces?

400
00:22:40,230 --> 00:22:41,632
You can't outrun a bear, right?

401
00:22:41,632 --> 00:22:43,340
And the guys says,
well yeah I know that,

402
00:22:43,340 --> 00:22:46,370
but I only have to outrun you.

403
00:22:46,370 --> 00:22:48,680
That's predator satiation.

404
00:22:48,680 --> 00:22:52,500
So if the predator
gets full, then you

405
00:22:52,500 --> 00:22:54,670
can get something
that looks like this.

406
00:22:54,670 --> 00:22:57,350
It's below some size,
below this n-min size,

407
00:22:57,350 --> 00:22:59,100
they all get eaten by that bear.

408
00:22:59,100 --> 00:23:03,960
And above it, the population
may be able to grow.

409
00:23:03,960 --> 00:23:06,425
So this is a good
example of just

410
00:23:06,425 --> 00:23:08,240
that it's an effective
form of cooperation.

411
00:23:14,580 --> 00:23:21,140
Now can somebody explain--
Well, first of all,

412
00:23:21,140 --> 00:23:23,530
does this lead to
a full bifurcation

413
00:23:23,530 --> 00:23:24,730
if we add a death rate?

414
00:23:27,424 --> 00:23:29,590
We're going to think about
this first seven seconds,

415
00:23:29,590 --> 00:23:30,760
then we'll vote.

416
00:23:30,760 --> 00:23:34,430
The question: if we add
a death rate to this,

417
00:23:34,430 --> 00:23:36,645
does that lead to a fold
or saddle-node bifurcation?

418
00:23:42,497 --> 00:23:44,830
I'm going to give you more
than seven seconds, because I

419
00:23:44,830 --> 00:23:46,620
see a lot of brain activity.

420
00:23:53,310 --> 00:23:55,018
Does this one lead to
a full bifurcation?

421
00:24:04,000 --> 00:24:04,690
Ready?

422
00:24:04,690 --> 00:24:07,040
Three, two, one.

423
00:24:13,980 --> 00:24:15,420
The answer is, in
this case, yes.

424
00:24:19,180 --> 00:24:23,384
So we'll say Allee effect is in
general the kind of thing that

425
00:24:23,384 --> 00:24:24,550
leads to a full bifurcation.

426
00:24:24,550 --> 00:24:26,010
In this case, it
definitely does.

427
00:24:28,920 --> 00:24:31,100
So it's really very
important to be

428
00:24:31,100 --> 00:24:34,040
able to take curves like
this, and figure out

429
00:24:34,040 --> 00:24:37,750
what the bifurcation
diagram looks like.

430
00:24:37,750 --> 00:24:41,380
And just remember that the
solid and dashed lines here

431
00:24:41,380 --> 00:24:43,640
are very useful for
getting intuition

432
00:24:43,640 --> 00:24:45,950
on what's going on, because
stable is telling us

433
00:24:45,950 --> 00:24:49,932
that any perturbation away
from this point, it goes away.

434
00:24:49,932 --> 00:24:51,390
So this is why we
draw arrows here.

435
00:24:54,590 --> 00:24:57,790
Arrows come here.

436
00:24:57,790 --> 00:25:04,325
So this is saying that starting
at 0, from a differential

437
00:25:04,325 --> 00:25:05,700
equation standpoint
at least, you

438
00:25:05,700 --> 00:25:09,895
can add a single individual
n, and that individual

439
00:25:09,895 --> 00:25:12,320
will reproduce and gets to
whatever the effective carrying

440
00:25:12,320 --> 00:25:14,050
capacity is at that delta.

441
00:25:19,080 --> 00:25:21,370
So what we can do is we can
draw the same bifurcation

442
00:25:21,370 --> 00:25:23,351
diagram here.

443
00:25:23,351 --> 00:25:29,440
Where we look at the population
size as a function of delta.

444
00:25:29,440 --> 00:25:31,810
And now it's actually somewhat
more interesting looking,

445
00:25:31,810 --> 00:25:35,560
because in the absence
of this death rate,

446
00:25:35,560 --> 00:25:37,109
we have it start at k.

447
00:25:37,109 --> 00:25:39,525
So what's going to happen is
it's going to come like this.

448
00:25:43,897 --> 00:25:45,730
And the reason we call
it a full bifurcation

449
00:25:45,730 --> 00:25:50,860
is because these fixed points
fold over on themselves.

450
00:25:50,860 --> 00:25:53,980
Now this is a bifurcation where
as a function is a control

451
00:25:53,980 --> 00:25:56,650
parameter, it's not that
these fixed points exchange

452
00:25:56,650 --> 00:25:59,070
stability, but rather that
the fixed points collide

453
00:25:59,070 --> 00:26:00,820
and annihilate.

454
00:26:00,820 --> 00:26:03,710
So in general, a bifurcation
is where something qualitative

455
00:26:03,710 --> 00:26:06,670
happens to the fixed points.

456
00:26:06,670 --> 00:26:08,550
In this case, the
number of fixed points

457
00:26:08,550 --> 00:26:11,200
did not change as a function
of the control parameter.

458
00:26:11,200 --> 00:26:14,400
Whereas in this
case, it does change.

459
00:26:14,400 --> 00:26:17,600
But there are only certain
characteristic ways

460
00:26:17,600 --> 00:26:19,610
in which these six points
are allowed to change.

461
00:26:28,520 --> 00:26:30,670
So from this, we
can draw our arrows.

462
00:26:34,610 --> 00:26:38,510
n equal to 0 is still a
stable point down here.

463
00:26:38,510 --> 00:26:41,547
So we can draw.

464
00:26:41,547 --> 00:26:43,380
This is where colored
chalk would be useful.

465
00:26:51,602 --> 00:26:52,560
We can draw our arrows.

466
00:27:01,804 --> 00:27:02,720
What do I do out here?

467
00:27:07,430 --> 00:27:11,690
Now, the reason that we say
that this population experiences

468
00:27:11,690 --> 00:27:14,170
a sudden transition,
or a tipping point,

469
00:27:14,170 --> 00:27:16,920
is because you can imagine
that this death rate being

470
00:27:16,920 --> 00:27:18,794
a slowly changing parameter.

471
00:27:18,794 --> 00:27:20,210
So you could imagine
that it could

472
00:27:20,210 --> 00:27:22,470
be a slow acidification
of the oceans,

473
00:27:22,470 --> 00:27:25,560
or a slow increase in
the amount of fishing

474
00:27:25,560 --> 00:27:26,790
that takes place here.

475
00:27:26,790 --> 00:27:29,730
So then what can happen is that
the size of the population,

476
00:27:29,730 --> 00:27:31,550
it's very healthy here.

477
00:27:31,550 --> 00:27:32,940
It's pretty happy, happy.

478
00:27:32,940 --> 00:27:36,020
Here it's maybe decreasing
some, but you might not

479
00:27:36,020 --> 00:27:36,810
sound an alarm.

480
00:27:36,810 --> 00:27:39,950
But what can happen is
that at some critical level

481
00:27:39,950 --> 00:27:42,540
of environmental conditions,
some critical level of fishing,

482
00:27:42,540 --> 00:27:45,380
for example, you can get
this catastrophic collapse

483
00:27:45,380 --> 00:27:46,340
of the population.

484
00:27:46,340 --> 00:27:50,162
Where over short periods of
time, it can suddenly collapse.

485
00:27:50,162 --> 00:27:51,620
Indeed, this sort
of thing has been

486
00:27:51,620 --> 00:27:54,560
observed in a number of
fisheries around the world.

487
00:27:54,560 --> 00:27:57,480
So if you go to, for example,
the Monterey Bay Aquarium

488
00:27:57,480 --> 00:28:03,110
in Monterey, California,
they have a very nice display

489
00:28:03,110 --> 00:28:06,050
explaining how that space
ended up becoming and aquarium.

490
00:28:06,050 --> 00:28:08,380
Because it used to
be a sardine fishery,

491
00:28:08,380 --> 00:28:12,230
and was a very productive
fishery in the years leading up

492
00:28:12,230 --> 00:28:13,590
to World War II.

493
00:28:13,590 --> 00:28:15,815
But if you look at the
number of fish that

494
00:28:15,815 --> 00:28:18,010
were caught as a function
of time, it goes up,

495
00:28:18,010 --> 00:28:19,970
up, because there's
fishing more and more, then

496
00:28:19,970 --> 00:28:22,200
all of the sudden, they
just presumably fish

497
00:28:22,200 --> 00:28:25,369
too much and over a time
span of a few years,

498
00:28:25,369 --> 00:28:26,910
the fishery collapsed,
and there were

499
00:28:26,910 --> 00:28:30,570
just no more sardines to catch.

500
00:28:30,570 --> 00:28:33,630
That whole street there
used to be canneries.

501
00:28:33,630 --> 00:28:35,330
But then it all went
out of business.

502
00:28:35,330 --> 00:28:38,890
And those buildings were
largely unused for many years.

503
00:28:38,890 --> 00:28:41,107
But then eventually,
the aquarium

504
00:28:41,107 --> 00:28:43,190
went and bought some of
them and refurbished them,

505
00:28:43,190 --> 00:28:44,606
and now it's a
beautiful aquarium.

506
00:28:44,606 --> 00:28:48,250
So I encourage you
to check it out.

507
00:28:48,250 --> 00:28:50,370
But that being
said, the collapse

508
00:28:50,370 --> 00:28:52,740
is probably still
bad, even though it

509
00:28:52,740 --> 00:28:54,750
led to this nice
outcome eventually.

510
00:28:54,750 --> 00:28:59,100
And what you can see is that not
only are these tipping points

511
00:28:59,100 --> 00:29:03,020
or the sudden transitions
maybe undesirable in the case

512
00:29:03,020 --> 00:29:06,380
of fishery, but it also would
be very difficult to reverse.

513
00:29:06,380 --> 00:29:08,590
Because it's not that--
You couldn't imagined

514
00:29:08,590 --> 00:29:10,750
that the bifurcation diagram
look rather different.

515
00:29:10,750 --> 00:29:12,041
It could have looked like this.

516
00:29:16,605 --> 00:29:17,980
What kind of
bifurcation is this?

517
00:29:21,640 --> 00:29:23,290
So this is still
a transcritical.

518
00:29:23,290 --> 00:29:25,475
It doesn't have to
be a line throughout,

519
00:29:25,475 --> 00:29:29,390
because in principle there was
this unstable thing down here.

520
00:29:29,390 --> 00:29:31,960
This is again, and, as a
function of some parameter,

521
00:29:31,960 --> 00:29:33,610
we'll say delta.

522
00:29:33,610 --> 00:29:37,400
And so you'd say, oh this is
a rather sudden transition.

523
00:29:37,400 --> 00:29:39,590
In the sense that a
modest change in delta

524
00:29:39,590 --> 00:29:43,170
will lead to a dramatic change
in the size of the population.

525
00:29:43,170 --> 00:29:45,622
But there's something that's
very qualitatively different

526
00:29:45,622 --> 00:29:46,580
about these situations.

527
00:29:46,580 --> 00:29:49,430
Which is that, here you can
imagine that if you improve

528
00:29:49,430 --> 00:29:52,130
the quality of the environment,
and then you reintroduce fish,

529
00:29:52,130 --> 00:29:54,730
or maybe you don't even
have to introduce fish,

530
00:29:54,730 --> 00:29:56,719
they can come from
a different area.

531
00:29:56,719 --> 00:29:59,010
Then, just by improving the
quality of the environment,

532
00:29:59,010 --> 00:30:01,180
you get recovery.

533
00:30:01,180 --> 00:30:03,720
Whereas here, that's
very difficult to do,

534
00:30:03,720 --> 00:30:06,135
because the system is
hysteretic, or has memory.

535
00:30:09,012 --> 00:30:10,470
That means that
even if you improve

536
00:30:10,470 --> 00:30:11,969
the quality of the
environment here,

537
00:30:11,969 --> 00:30:15,302
it may be very difficult to get
back to your previous state.

538
00:30:15,302 --> 00:30:16,760
People think that
they've seen this

539
00:30:16,760 --> 00:30:20,040
in the context of
these transitions

540
00:30:20,040 --> 00:30:23,520
in lake ecosystems, so-called
eutrophication transition,

541
00:30:23,520 --> 00:30:26,570
where you get as
a result of maybe

542
00:30:26,570 --> 00:30:29,880
runoff from agricultural uses.

543
00:30:29,880 --> 00:30:32,370
You can get the sun
transitions of not

544
00:30:32,370 --> 00:30:34,990
just a single population,
but of an entire ecosystem.

545
00:30:34,990 --> 00:30:36,700
So the lake can
go from this thing

546
00:30:36,700 --> 00:30:40,330
where it's nice and clear and
beautiful summer vacations

547
00:30:40,330 --> 00:30:45,580
spot, to this really green
kind of algal take over.

548
00:30:45,580 --> 00:30:49,060
And it can be very
difficult to get recovery.

549
00:30:49,060 --> 00:30:50,716
There's was a question?

550
00:30:50,716 --> 00:30:51,216
No?

551
00:30:58,960 --> 00:31:02,430
And so the basic--
We'll say something

552
00:31:02,430 --> 00:31:04,450
about the early
warning indicators.

553
00:31:04,450 --> 00:31:06,620
Well maybe I'll say it now.

554
00:31:06,620 --> 00:31:08,660
So based on the
review that you read,

555
00:31:08,660 --> 00:31:11,270
there's this phenomenon
of critical slowing down

556
00:31:11,270 --> 00:31:16,220
that tells us that
in principle, you

557
00:31:16,220 --> 00:31:19,050
can anticipate the transitions
about to take place.

558
00:31:22,350 --> 00:31:25,590
And the statement was that
the dominant eigenvalue

559
00:31:25,590 --> 00:31:27,120
described in dynamics,
a system went

560
00:31:27,120 --> 00:31:29,150
to 0 at one of these points.

561
00:31:35,690 --> 00:31:37,450
Now, the systems they
were talking about

562
00:31:37,450 --> 00:31:40,690
were presumably
systems like this.

563
00:31:40,690 --> 00:31:44,000
This is a so-called zero
eigenvalue bifurcation,

564
00:31:44,000 --> 00:31:49,300
were the eigenvalue goes to
0 at this point right here.

565
00:31:54,110 --> 00:31:55,420
So here's a question for you.

566
00:31:57,950 --> 00:32:00,630
Does the eigenvalue describe
the dynamics of a system

567
00:32:00,630 --> 00:32:02,470
go to 0 at this point?

568
00:32:12,030 --> 00:32:13,090
At transcritical.

569
00:32:15,675 --> 00:32:17,130
In a transcritical bifurcation.

570
00:32:22,449 --> 00:32:23,740
Do you understand the question?

571
00:32:30,709 --> 00:32:32,500
I'll let you think
about it for 20 seconds,

572
00:32:32,500 --> 00:32:35,180
because this is maybe
not totally obvious.

573
00:32:55,580 --> 00:32:57,350
Do you need more time?

574
00:32:57,350 --> 00:32:59,080
I'm wondering.

575
00:32:59,080 --> 00:33:02,500
Maybe people are sufficiently
lost or confused.

576
00:33:02,500 --> 00:33:10,580
They're not sure what--
Nod if you want more time.

577
00:33:10,580 --> 00:33:11,170
OK.

578
00:33:11,170 --> 00:33:11,670
Ready?

579
00:33:14,410 --> 00:33:16,305
Three, two, one.

580
00:33:20,200 --> 00:33:22,590
I'd say it's a kind
of a 50-50 thing.

581
00:33:22,590 --> 00:33:24,120
So go ahead and
turn to you neighbor

582
00:33:24,120 --> 00:33:27,460
and try to figure out, using
some combination of math

583
00:33:27,460 --> 00:33:30,300
and graphical analysis,
whether the bifurcate--

584
00:33:30,300 --> 00:33:32,870
Whether at this
transcritical bifurcation

585
00:33:32,870 --> 00:33:34,357
the eigenvalue goes to 0.

586
00:33:37,339 --> 00:34:50,830
[SIDE CONVERSATIONS]

587
00:34:50,830 --> 00:34:53,730
PROFESSOR: It sounds like
the discussion is maybe

588
00:34:53,730 --> 00:34:56,830
gone to completion.

589
00:34:56,830 --> 00:34:58,220
Let's go ahead and vote again.

590
00:34:58,220 --> 00:35:00,930
The question is, does
the dominant eigenvalue

591
00:35:00,930 --> 00:35:03,140
of this system go to
0 at this point here?

592
00:35:03,140 --> 00:35:04,940
The trans critical bifurcation.

593
00:35:04,940 --> 00:35:05,700
Ready?

594
00:35:05,700 --> 00:35:07,885
Three, two, one.

595
00:35:10,450 --> 00:35:10,950
Alright.

596
00:35:13,440 --> 00:35:15,440
It doesn't seem like it's
had much of an effect.

597
00:35:15,440 --> 00:35:18,194
I see that one person has
been convinced of something.

598
00:35:18,194 --> 00:35:19,860
I don't know if he's
actually convinced,

599
00:35:19,860 --> 00:35:21,805
of if he was just
surrounded and bullied.

600
00:35:25,500 --> 00:35:27,530
All right, so what are
some different ways

601
00:35:27,530 --> 00:35:28,780
of thinking about this?

602
00:35:32,410 --> 00:35:33,210
Yes?

603
00:35:33,210 --> 00:35:37,674
AUDIENCE: Well,
one of my neighbors

604
00:35:37,674 --> 00:35:40,154
decided to actually
just lineralize

605
00:35:40,154 --> 00:35:41,445
that equation at that point.

606
00:35:41,445 --> 00:35:42,320
PROFESSOR: All right.

607
00:35:46,040 --> 00:35:48,920
AUDIENCE: If you were
completely into you intuition,

608
00:35:48,920 --> 00:35:50,357
this would be a way to check.

609
00:35:50,357 --> 00:35:51,190
PROFESSOR: Right OK.

610
00:35:51,190 --> 00:35:54,240
And now in general, in
life, I'm a big believer

611
00:35:54,240 --> 00:35:55,990
when you're confronted
with some question,

612
00:35:55,990 --> 00:35:59,930
you should first
think about it, what

613
00:35:59,930 --> 00:36:02,390
you should guess based
on whatever intuition

614
00:36:02,390 --> 00:36:04,280
might be available to you.

615
00:36:04,280 --> 00:36:07,080
And then after you have made
a guess based on intuition,

616
00:36:07,080 --> 00:36:09,110
then you go and you
do the calculation.

617
00:36:09,110 --> 00:36:11,980
Which in this case,
linearizing around.

618
00:36:11,980 --> 00:36:13,230
And where do linearize around?

619
00:36:16,640 --> 00:36:17,620
Around the fixed point.

620
00:36:17,620 --> 00:36:20,380
I would say that you
really want to linearize

621
00:36:20,380 --> 00:36:24,069
maybe not around-- You get the
same answer if you lineralize

622
00:36:24,069 --> 00:36:26,360
around 0, but you really want
to, I think conceptually,

623
00:36:26,360 --> 00:36:30,611
you want to linearize around
this stable fixed point.

624
00:36:30,611 --> 00:36:31,110
Right?

625
00:36:31,110 --> 00:36:33,680
Because, in some ways, it's
the stable fixed point that

626
00:36:33,680 --> 00:36:36,340
the system is sitting in, and
you want to know how is it that

627
00:36:36,340 --> 00:36:38,673
you're-- How is it is you're
going to react when you get

628
00:36:38,673 --> 00:36:40,650
perturbed away from
that fixed point?

629
00:36:40,650 --> 00:36:44,520
And you do get the same answer.

630
00:36:44,520 --> 00:36:48,430
The thing is that it's true
that the eigenvalue describing

631
00:36:48,430 --> 00:36:50,560
the dynamics around this
unstable fixed point

632
00:36:50,560 --> 00:36:53,030
also goes to zero.

633
00:36:53,030 --> 00:36:55,660
But that's because, I think
the rules of mathematics

634
00:36:55,660 --> 00:36:57,785
somehow say that the fixed
points are going to have

635
00:36:57,785 --> 00:36:59,180
to do something similar.

636
00:36:59,180 --> 00:37:02,185
So I think you do also
find the eigenvalue here

637
00:37:02,185 --> 00:37:03,560
as it goes to 0,
but conceptually

638
00:37:03,560 --> 00:37:05,351
that's not the eigenvalue
that you actually

639
00:37:05,351 --> 00:37:06,791
want to know about.

640
00:37:06,791 --> 00:37:08,280
Do you see why I'm saying that?

641
00:37:11,290 --> 00:37:14,860
Again, the eigenvalue
here is going to 0.

642
00:37:14,860 --> 00:37:16,660
We can try to explore
this a little bit,

643
00:37:16,660 --> 00:37:20,950
but I just want to kind of
tie this together and say--

644
00:37:20,950 --> 00:37:23,980
These local bifurcations
that you have maybe

645
00:37:23,980 --> 00:37:27,070
heard about, which is that
this fold bifurcation,

646
00:37:27,070 --> 00:37:31,250
the transcritical bifurcation,
and also the Hopf bifurcation.

647
00:37:31,250 --> 00:37:34,020
So the Hopf is one
that looks where

648
00:37:34,020 --> 00:37:35,820
you have a stable
fixed point that goes--

649
00:37:35,820 --> 00:37:37,910
And in terms of
an unstable line,

650
00:37:37,910 --> 00:37:39,394
you end up getting oscillations.

651
00:37:39,394 --> 00:37:40,810
So that's where
you have something

652
00:37:40,810 --> 00:37:49,340
that that's going to look like--
but now that I'm drawing this,

653
00:37:49,340 --> 00:37:51,970
I'm a little bit--
Is there a difference

654
00:37:51,970 --> 00:37:53,809
between a Hopf and a
pitchfork bifurcation?

655
00:37:53,809 --> 00:37:55,975
AUDIENCE: It's whether you
have oscillations or not.

656
00:37:55,975 --> 00:37:57,597
You can't really show that.

657
00:37:57,597 --> 00:37:58,180
PROFESSOR: OK.

658
00:37:58,180 --> 00:38:02,912
So as drawn, it's
either or both, or?

659
00:38:02,912 --> 00:38:04,060
AUDIENCE: It's both.

660
00:38:04,060 --> 00:38:05,870
PROFESSOR: OK well.

661
00:38:05,870 --> 00:38:10,670
I'll say that this thing also
the eigenvalue goes to 0 here.

662
00:38:10,670 --> 00:38:13,610
Because I guess,
in this-- You don't

663
00:38:13,610 --> 00:38:15,690
have to have
oscillations in order--

664
00:38:15,690 --> 00:38:18,400
And it depends on the
higher dynamic systems.

665
00:38:18,400 --> 00:38:20,985
But again, this is another
0 eigenvalue bifurcation.

666
00:38:24,160 --> 00:38:27,740
And so the statement is
that in principle, you

667
00:38:27,740 --> 00:38:30,500
can use this fact that
the eigenvalue goes to 0,

668
00:38:30,500 --> 00:38:32,540
this thing called
critical slowing down,

669
00:38:32,540 --> 00:38:36,840
to measure some changes in
the dynamics of the system

670
00:38:36,840 --> 00:38:39,230
before, in this case,
you get collapse,

671
00:38:39,230 --> 00:38:43,120
or before, in this case,
you get extinction.

672
00:38:43,120 --> 00:38:47,344
The issue maybe is that
in this transition,

673
00:38:47,344 --> 00:38:48,760
there really is
something big that

674
00:38:48,760 --> 00:38:50,359
happened at the bifurcation.

675
00:38:50,359 --> 00:38:52,150
So you really want to
have an early warning

676
00:38:52,150 --> 00:38:55,450
signal because there's
a dramatic change,

677
00:38:55,450 --> 00:38:56,920
and it may be
difficult to reverse.

678
00:38:56,920 --> 00:39:00,830
Whereas here, the population
size is smoothly going to 0.

679
00:39:00,830 --> 00:39:04,347
And once you get down to your
last two California condors,

680
00:39:04,347 --> 00:39:05,680
you know that you're in trouble.

681
00:39:05,680 --> 00:39:08,510
You don't need any sort
of fancy early warning

682
00:39:08,510 --> 00:39:11,420
indicator based on fluctuations
or return time, and so forth.

683
00:39:11,420 --> 00:39:15,400
Instead, just the
number is maybe

684
00:39:15,400 --> 00:39:20,640
your best measure for the
health of the population.

685
00:39:20,640 --> 00:39:24,140
So I think that yeah, maybe
I won't do the linearization,

686
00:39:24,140 --> 00:39:26,410
but I encourage you to do it.

687
00:39:26,410 --> 00:39:30,270
Because, that's what's
going to be relevant.

688
00:39:30,270 --> 00:39:31,810
In particular, what
you want to know

689
00:39:31,810 --> 00:39:36,280
is if there's some-- If you
linearize around there some,

690
00:39:36,280 --> 00:39:41,330
just be clear, there's
some n equilibrium.

691
00:39:41,330 --> 00:39:44,250
OK there's some equilibrium,
and then you want to ask,

692
00:39:44,250 --> 00:39:49,240
well OK now you want to say
if your n is equal to some n

693
00:39:49,240 --> 00:39:52,330
equilibrium, plus some epsilon.

694
00:39:52,330 --> 00:39:56,910
Then, if you plug it
into the N dot function,

695
00:39:56,910 --> 00:39:58,730
you should get an
equation that is

696
00:39:58,730 --> 00:40:01,319
something where
this epsilon dot is

697
00:40:01,319 --> 00:40:02,610
going to be equal to something.

698
00:40:02,610 --> 00:40:03,735
What should it be equal to?

699
00:40:07,580 --> 00:40:10,240
Something times epsilon.

700
00:40:10,240 --> 00:40:11,270
And what something?

701
00:40:14,002 --> 00:40:15,710
AUDIENCE: What we've
been calling lambda?

702
00:40:15,710 --> 00:40:18,400
PROFESSOR: What we've
been calling labmda.

703
00:40:18,400 --> 00:40:21,210
So this is saying that if
you go a little bit away

704
00:40:21,210 --> 00:40:25,351
from the equilibrium,
so small epsilon here,

705
00:40:25,351 --> 00:40:27,600
that epsilon-- The solution
to this is an exponential,

706
00:40:27,600 --> 00:40:29,340
it's going to get
exponential decay,

707
00:40:29,340 --> 00:40:32,120
assuming that lambda is what?

708
00:40:32,120 --> 00:40:32,750
Negative.

709
00:40:32,750 --> 00:40:37,512
And that's the
definition of-- is

710
00:40:37,512 --> 00:40:39,220
When we say this is
a stable fixed point,

711
00:40:39,220 --> 00:40:40,760
this is unstable, that's
equivalent to saying

712
00:40:40,760 --> 00:40:42,980
that the lambda here is
negative and the lambda here

713
00:40:42,980 --> 00:40:43,480
is positive.

714
00:40:49,020 --> 00:40:52,440
AUDIENCE: That's a
very easy argument.

715
00:40:52,440 --> 00:40:55,250
They exchange stability.

716
00:40:55,250 --> 00:40:57,270
So you know that lambda
has to be completely 0.

717
00:40:57,270 --> 00:40:59,410
PROFESSOR: Yes,
this is the point.

718
00:40:59,410 --> 00:41:03,679
So we know that lambda here is
positive lambda-- I'm sorry,

719
00:41:03,679 --> 00:41:05,720
lambda here is negative,
lambda here is positive.

720
00:41:05,720 --> 00:41:07,099
These fixed points
at this point,

721
00:41:07,099 --> 00:41:08,890
they're going to exchange
stability so that

722
00:41:08,890 --> 00:41:10,989
means they had to be equal.

723
00:41:10,989 --> 00:41:12,780
And where they exchange
is when it crosses.

724
00:41:12,780 --> 00:41:16,520
So they both-- Well, I
guess independently this

725
00:41:16,520 --> 00:41:18,790
went-- We should be
able to draw this,

726
00:41:18,790 --> 00:41:20,600
although I'm a
little bit worried

727
00:41:20,600 --> 00:41:22,270
that once I try
to draw something

728
00:41:22,270 --> 00:41:23,353
I'm going to get confused.

729
00:41:23,353 --> 00:41:25,355
But this is a useful exercise.

730
00:41:29,240 --> 00:41:34,360
In particular, if we plot
as a function of delta.

731
00:41:34,360 --> 00:41:38,890
So here, we have the lambdas,
and we have this delta.

732
00:41:38,890 --> 00:41:42,350
and a delta equal to r
what's going to happen

733
00:41:42,350 --> 00:41:45,974
is that we're going
to get the one

734
00:41:45,974 --> 00:41:47,140
guy's going to go like this.

735
00:41:55,220 --> 00:41:58,080
Something like that.

736
00:41:58,080 --> 00:42:01,800
Given, that the fixed points
went from stable to unstable,

737
00:42:01,800 --> 00:42:03,757
unstable to stable, so
they had to cross 0.

738
00:42:03,757 --> 00:42:05,340
So if they both cross
0 at that point,

739
00:42:05,340 --> 00:42:09,970
they're going to
be equal n zero.

740
00:42:09,970 --> 00:42:11,866
You guys agree?

741
00:42:11,866 --> 00:42:15,642
AUDIENCE: Do we know that the
eigenvalues are going to change

742
00:42:15,642 --> 00:42:18,207
linearly with the death rate?

743
00:42:18,207 --> 00:42:18,790
PROFESSOR: No.

744
00:42:18,790 --> 00:42:20,130
So in general, they don't.

745
00:42:25,670 --> 00:42:29,970
I'm trying to think if
close to the bifurcation,

746
00:42:29,970 --> 00:42:33,115
whether I can say.

747
00:42:33,115 --> 00:42:35,440
The mathematicians I'm
sure can say something.

748
00:42:35,440 --> 00:42:38,934
I'm not going to.

749
00:42:38,934 --> 00:42:40,725
AUDIENCE: Does this
also mean that whenever

750
00:42:40,725 --> 00:42:44,250
we're on a steady state,
approaching bifurcation,

751
00:42:44,250 --> 00:42:48,800
we'll always see the
critical slowing down?

752
00:42:48,800 --> 00:42:53,250
If it's a stable fixed
point at a critical value.

753
00:42:53,250 --> 00:42:58,050
PROFESSOR: I think that the
statement is that in principle,

754
00:42:58,050 --> 00:42:59,974
critical slowing down occurs.

755
00:42:59,974 --> 00:43:01,890
But that does not mean
that you'll necessarily

756
00:43:01,890 --> 00:43:03,000
be able to see it.

757
00:43:03,000 --> 00:43:05,416
Because it could be, there's
just too much noise, or this,

758
00:43:05,416 --> 00:43:07,620
or that.

759
00:43:07,620 --> 00:43:11,540
AUDIENCE: So something that
I was thinking about related

760
00:43:11,540 --> 00:43:15,210
to that is-- So in this case,
it seems like everything

761
00:43:15,210 --> 00:43:20,970
about the system, the
fluctuations, or anything,

762
00:43:20,970 --> 00:43:26,282
should be getting very
small, as you approach that.

763
00:43:26,282 --> 00:43:27,990
PROFESSOR: In principle,
the fluctuations

764
00:43:27,990 --> 00:43:29,710
are supposed to grow.

765
00:43:29,710 --> 00:43:32,510
Although this is assuming
that the strength of the noise

766
00:43:32,510 --> 00:43:33,950
is constant.

767
00:43:33,950 --> 00:43:36,310
And in this case, the
strength of the noise

768
00:43:36,310 --> 00:43:39,220
may not be constant, because
if it's demographic noise,

769
00:43:39,220 --> 00:43:41,620
and the size of your
population is going down.

770
00:43:41,620 --> 00:43:43,850
This is actually
subtle then, I think.

771
00:43:47,260 --> 00:43:49,820
And when we typically
talk about these dynamics,

772
00:43:49,820 --> 00:43:52,480
we're assuming that there's
an added noise source that

773
00:43:52,480 --> 00:43:56,030
is independent of the variable.

774
00:43:56,030 --> 00:43:57,790
But if it's demographic
fluctuations

775
00:43:57,790 --> 00:43:59,706
you're talking about,
that that won't be true.

776
00:44:01,970 --> 00:44:04,190
I think that it's
also useful to draw,

777
00:44:04,190 --> 00:44:06,060
to get some intuition
around this,

778
00:44:06,060 --> 00:44:08,840
based on this idea of
an effective potential.

779
00:44:08,840 --> 00:44:10,810
So these are
one-dimensional systems,

780
00:44:10,810 --> 00:44:14,920
which means you can always write
down an effective potential.

781
00:44:14,920 --> 00:44:21,020
And in this case, what it's
going to look like is--

782
00:44:21,020 --> 00:44:24,930
This is some u effective as
a function of the population

783
00:44:24,930 --> 00:44:26,150
size.

784
00:44:26,150 --> 00:44:32,080
And up here, we have
some stable state here

785
00:44:32,080 --> 00:44:38,250
that corresponds to something
that looks like this.

786
00:44:38,250 --> 00:44:40,070
So it starts out maybe at k.

787
00:44:40,070 --> 00:44:42,720
What happens is, as the
death rate increases,

788
00:44:42,720 --> 00:44:53,110
this thing kind of turns-- Or,
maybe I didn't quite make it.

789
00:44:53,110 --> 00:44:57,120
So what you see is
that as the death rate

790
00:44:57,120 --> 00:45:00,920
here-- So, as we go down
the death rate is going up.

791
00:45:04,190 --> 00:45:06,670
So that what happens is
the effective potential,

792
00:45:06,670 --> 00:45:10,110
describing the dynamics of the
population near the equilibrium

793
00:45:10,110 --> 00:45:12,702
is broadening out.

794
00:45:12,702 --> 00:45:14,160
So you can think
about the dynamics

795
00:45:14,160 --> 00:45:17,930
as being equivalent to
an overdamped particle

796
00:45:17,930 --> 00:45:19,710
in effective potential.

797
00:45:19,710 --> 00:45:22,700
So for example, for those of
you that have studied single

798
00:45:22,700 --> 00:45:24,530
molecule biophysics
kinds of things,

799
00:45:24,530 --> 00:45:28,760
this could be thought of as
a [? poly-centered ?] bead

800
00:45:28,760 --> 00:45:30,860
that's trapped in
a the laser trap.

801
00:45:30,860 --> 00:45:33,500
And as you turn down
the power of your laser,

802
00:45:33,500 --> 00:45:36,660
then the spring constant
describing the dynamics

803
00:45:36,660 --> 00:45:40,290
of that B in the trap, it
gets weaker and weaker.

804
00:45:40,290 --> 00:45:42,160
So the potential
broadens, and that

805
00:45:42,160 --> 00:45:45,680
means that for fixed injection
noise-- fixed temperature-- you

806
00:45:45,680 --> 00:45:50,000
get these effects, where the
fluctuations will increase

807
00:45:50,000 --> 00:45:52,610
in magnitude, and
you get an increase

808
00:45:52,610 --> 00:45:54,610
in this autocorrelation time.

809
00:45:54,610 --> 00:45:56,660
Because for example,
the bead in the trap,

810
00:45:56,660 --> 00:45:58,180
the autocorrelation
time is indeed

811
00:45:58,180 --> 00:46:06,360
just equal to the relaxation
time of the bead in the trap.

812
00:46:06,360 --> 00:46:09,060
And the bifurcation
occurs right at this point

813
00:46:09,060 --> 00:46:13,230
here, where you see that
this local potential is,

814
00:46:13,230 --> 00:46:15,580
or the local minimum,
is disappearing.

815
00:46:15,580 --> 00:46:19,410
That's when this stable
fixed point is gone.

816
00:46:19,410 --> 00:46:21,440
And then you just
kind of fall off here.

817
00:46:21,440 --> 00:46:26,970
And that's what leads
to this collapse, which

818
00:46:26,970 --> 00:46:27,865
I have covered up.

819
00:46:27,865 --> 00:46:29,115
That leads to the bifurcation.

820
00:46:35,340 --> 00:46:38,320
And you should, for
example, be able to draw

821
00:46:38,320 --> 00:46:40,095
an effective potential
here as well.

822
00:46:42,890 --> 00:46:45,900
But I'll let you play with it.

823
00:46:49,496 --> 00:46:51,620
But you can see the location
of this unstable fixed

824
00:46:51,620 --> 00:46:52,700
point shifting as well.

825
00:46:56,856 --> 00:47:00,780
I'm not sure how
good my drawing is.

826
00:47:00,780 --> 00:47:03,020
Because in principal,
this unstable fixed point

827
00:47:03,020 --> 00:47:05,661
should have to come together.

828
00:47:05,661 --> 00:47:07,660
I'm not sure if that's
very clear in my drawing.

829
00:47:11,950 --> 00:47:15,900
Just to summarize
this discussion,

830
00:47:15,900 --> 00:47:18,800
I think there are
a few ways that you

831
00:47:18,800 --> 00:47:21,480
can think of these early
warning indicators.

832
00:47:21,480 --> 00:47:24,700
And there's a diagram
that I like to make,

833
00:47:24,700 --> 00:47:27,640
that I think makes things
more clear, for me, at least.

834
00:47:27,640 --> 00:47:34,980
Which is that if you
look at this population

835
00:47:34,980 --> 00:47:39,030
as a function of time, it goes.

836
00:47:39,030 --> 00:47:44,390
And if there's an environment
quality as a function of time.

837
00:47:44,390 --> 00:47:47,950
If the environment
has a perturbation,

838
00:47:47,950 --> 00:47:52,180
then the population will shrink,
and then you'll get recovery.

839
00:47:52,180 --> 00:47:55,822
And this thing here tells you
about the time for recovery.

840
00:47:55,822 --> 00:47:57,780
And this basic phenomenon
critical slowing down

841
00:47:57,780 --> 00:48:00,910
tells you that as the
tipping point approaches,

842
00:48:00,910 --> 00:48:03,176
as you approach the
bifurcation, that time

843
00:48:03,176 --> 00:48:05,300
to recover from this
perturbation is going to grow.

844
00:48:08,660 --> 00:48:10,470
Of course, the other
thing that you can say

845
00:48:10,470 --> 00:48:14,370
is that even in the absence
of a defied perturbation, even

846
00:48:14,370 --> 00:48:16,760
if the environment is
constant over time,

847
00:48:16,760 --> 00:48:20,050
there could just a natural
noise in the system that

848
00:48:20,050 --> 00:48:21,780
will fluctuate, just
like temperature,

849
00:48:21,780 --> 00:48:23,100
for the bead in the trap.

850
00:48:23,100 --> 00:48:25,050
And principle, then
you can measure

851
00:48:25,050 --> 00:48:27,870
the size of the
fluctuations, the variance,

852
00:48:27,870 --> 00:48:31,010
as well as the
autocorrelation time

853
00:48:31,010 --> 00:48:34,540
tau, which is, for civil
systems, equal to t.

854
00:48:34,540 --> 00:48:39,323
And this also grows as you
approach the bifurcation

855
00:48:39,323 --> 00:48:42,269
AUDIENCE: Is it [? clear ?]
that-- Because the perturbation

856
00:48:42,269 --> 00:48:46,540
[INAUDIBLE] that you're talking
about is I think n [INAUDIBLE]

857
00:48:46,540 --> 00:48:49,160
perturbed n into
epsilon, and then it

858
00:48:49,160 --> 00:48:50,360
relaxes the steady state.

859
00:48:50,360 --> 00:48:53,002
Is it [INAUDIBLE] to
perturb the environment?

860
00:48:53,002 --> 00:48:53,960
PROFESSOR: Yeah, right.

861
00:48:53,960 --> 00:48:55,540
So that's a good question.

862
00:48:55,540 --> 00:48:57,680
What I've been talking
about is a situation

863
00:48:57,680 --> 00:49:00,457
where you really literally pull
it away, and then you let go.

864
00:49:00,457 --> 00:49:02,540
And that's equivalent for
in the bead in the trap,

865
00:49:02,540 --> 00:49:04,581
that you pull the bead
away, and then you let go,

866
00:49:04,581 --> 00:49:05,717
and you watch it come back.

867
00:49:05,717 --> 00:49:07,550
The situation in the
case of the environment

868
00:49:07,550 --> 00:49:11,270
is that, if it's a kind of
a sudden shift-- It could

869
00:49:11,270 --> 00:49:15,890
be anything, it could be a brief
change in-- Well it could just

870
00:49:15,890 --> 00:49:16,490
be delta.

871
00:49:16,490 --> 00:49:19,249
So it could be that over
some period of time,

872
00:49:19,249 --> 00:49:21,290
the death rate increases,
or some period of time,

873
00:49:21,290 --> 00:49:23,430
the growth rate decreases.

874
00:49:23,430 --> 00:49:26,105
Or it could be that for--
The perturbations don't

875
00:49:26,105 --> 00:49:27,860
have to be bad, they
can also be good,

876
00:49:27,860 --> 00:49:29,450
and you actually get the same.

877
00:49:29,450 --> 00:49:31,420
The principal for small
perturbations to go,

878
00:49:31,420 --> 00:49:32,545
it's the same thing anyway.

879
00:49:37,680 --> 00:49:39,790
So these are early
warning indicators

880
00:49:39,790 --> 00:49:41,990
of an impending
transition that are

881
00:49:41,990 --> 00:49:43,205
based on temporal indicators.

882
00:49:45,950 --> 00:49:48,990
And one of the things that we've
been excited about in my group

883
00:49:48,990 --> 00:49:51,080
is actually just trying
to measure these things

884
00:49:51,080 --> 00:49:53,300
in a well controlled
laboratory population.

885
00:49:53,300 --> 00:49:55,097
Since in our case,
we're using yeast

886
00:49:55,097 --> 00:49:57,430
that are engaging in what you
might call a group hunting

887
00:49:57,430 --> 00:50:00,060
behavior, where they secrete an
enzyme that breaks down sugar,

888
00:50:00,060 --> 00:50:02,143
and that kind of leads to
this cooperative growth.

889
00:50:02,143 --> 00:50:04,270
And at least in that case,
we can experimentally

890
00:50:04,270 --> 00:50:06,230
measure an increase
in all of these things

891
00:50:06,230 --> 00:50:07,563
as we approach this bifurcation.

892
00:50:10,549 --> 00:50:12,090
The other thing you
might think about

893
00:50:12,090 --> 00:50:14,620
is what happens for spatially
extended populations?

894
00:50:14,620 --> 00:50:17,640
We'll talk more about spatial
populations on Thursday,

895
00:50:17,640 --> 00:50:20,180
but just while we're here,
it's useful to think about it.

896
00:50:20,180 --> 00:50:22,596
So instead of thinking about
just n as a function of time,

897
00:50:22,596 --> 00:50:24,690
now you want to think about
density of population.

898
00:50:24,690 --> 00:50:26,240
So one thing that
people talked about

899
00:50:26,240 --> 00:50:29,300
is that-- Sorry, this is
not a function of time,

900
00:50:29,300 --> 00:50:31,230
it's now a function
of position x.

901
00:50:31,230 --> 00:50:33,259
Density is a
function of position.

902
00:50:33,259 --> 00:50:35,050
Now, environment is a
function of position.

903
00:50:35,050 --> 00:50:37,980
If you have a uniform
environment over position

904
00:50:37,980 --> 00:50:39,442
or space, then in
principle you can

905
00:50:39,442 --> 00:50:40,650
look at density fluctuations.

906
00:50:40,650 --> 00:50:43,400
And this is something that
people have talked about.

907
00:50:43,400 --> 00:50:45,850
One of things that
we have argued

908
00:50:45,850 --> 00:50:49,680
is that there should be a
spatial analog to this recovery

909
00:50:49,680 --> 00:50:50,614
time.

910
00:50:50,614 --> 00:50:52,030
So that corresponds
to a situation

911
00:50:52,030 --> 00:50:54,239
where you have environment
as a function of position.

912
00:50:54,239 --> 00:50:56,363
Like for example, we have
a sharp boundary, or just

913
00:50:56,363 --> 00:50:57,460
a region of poor quality.

914
00:50:57,460 --> 00:51:01,120
And in that case, you
can look at the density

915
00:51:01,120 --> 00:51:05,240
as a function of position, and
you get some linked scale here.

916
00:51:05,240 --> 00:51:07,862
So the statement here is
that, just because you're

917
00:51:07,862 --> 00:51:09,320
in a region of high
quality doesn't

918
00:51:09,320 --> 00:51:10,990
mean that you're at your
equilibrium density,

919
00:51:10,990 --> 00:51:12,698
because if you're
close to a poor region,

920
00:51:12,698 --> 00:51:14,480
so if you're close
to hunting grounds,

921
00:51:14,480 --> 00:51:17,724
then you'll get local
depletion of the population.

922
00:51:17,724 --> 00:51:19,890
You have to get some distance
away from a bad region

923
00:51:19,890 --> 00:51:21,223
before you get your equilibrium.

924
00:51:21,223 --> 00:51:23,960
That distance or
recovery length tells you

925
00:51:23,960 --> 00:51:26,500
about the quality
of the environment.

926
00:51:26,500 --> 00:51:29,120
And it's the quality of the
environment at this region,

927
00:51:29,120 --> 00:51:30,430
the good region.

928
00:51:30,430 --> 00:51:31,632
Where you are.

929
00:51:31,632 --> 00:51:33,090
At least in the
laboratory, this is

930
00:51:33,090 --> 00:51:34,120
something that's
actually much easier

931
00:51:34,120 --> 00:51:35,350
to measure than other things.

932
00:51:35,350 --> 00:51:38,670
Because, this is a
deterministic phenomenon.

933
00:51:38,670 --> 00:51:41,670
You don't have to measure
fluctuations over time

934
00:51:41,670 --> 00:51:43,110
with a high quality time series.

935
00:51:43,110 --> 00:51:45,200
You don't have to wait for
a perturbation in time,

936
00:51:45,200 --> 00:51:46,870
like a drought, and
look at recovery.

937
00:51:46,870 --> 00:51:49,870
Instead, you just take
advantage of natural variations

938
00:51:49,870 --> 00:51:53,240
in quality of the environment
over space, position, and then

939
00:51:53,240 --> 00:51:54,860
you measure profiles.

940
00:51:54,860 --> 00:51:58,370
We have a collaboration
with a professor

941
00:51:58,370 --> 00:52:01,270
at the University of Pisa who
does field ecology experiments

942
00:52:01,270 --> 00:52:03,830
with these algal mats on
intertidal communities,

943
00:52:03,830 --> 00:52:05,265
on islands in the Mediterranean.

944
00:52:08,490 --> 00:52:10,410
And we have a
[? pronates ?] that we

945
00:52:10,410 --> 00:52:13,540
can measure this in those
island communities as well.

946
00:52:17,094 --> 00:52:18,760
Are there any questions
about this stuff

947
00:52:18,760 --> 00:52:20,652
before we switch gears?

948
00:52:20,652 --> 00:52:21,152
Yes?

949
00:52:24,068 --> 00:52:27,660
AUDIENCE: In that example, is
it-- How much control can you

950
00:52:27,660 --> 00:52:31,224
have over actually [? saying ?]
the quality to the environment?

951
00:52:33,950 --> 00:52:37,190
PROFESSOR: So in that
manipulation, what they did was

952
00:52:37,190 --> 00:52:41,870
they basically go in and
they-- So there these,

953
00:52:41,870 --> 00:52:43,820
they're like miniature
forest somehow.

954
00:52:43,820 --> 00:52:47,460
So they're little-- They have
alternative stable states.

955
00:52:47,460 --> 00:52:50,300
What they actually do, they
go and they like chip away

956
00:52:50,300 --> 00:52:52,390
at the rock to
remove the things.

957
00:52:52,390 --> 00:52:54,950
And they do it over some range.

958
00:52:54,950 --> 00:52:57,670
So they experimentally
basically make it a

959
00:52:57,670 --> 00:53:00,360
challenging environment.

960
00:53:00,360 --> 00:53:03,380
And apparently, they have
permission to do this.

961
00:53:03,380 --> 00:53:06,690
So don't try that at
home without asking

962
00:53:06,690 --> 00:53:08,940
the proper authorities.

963
00:53:08,940 --> 00:53:11,850
Any other questions about this?

964
00:53:16,350 --> 00:53:19,030
So I think that the nice
thing about studying

965
00:53:19,030 --> 00:53:21,350
all these dynamics just
for a single population,

966
00:53:21,350 --> 00:53:25,350
is that it really clarifies the
essential ingredients, in order

967
00:53:25,350 --> 00:53:27,280
to get things like
sudden transitions.

968
00:53:27,280 --> 00:53:29,950
Of course, in
natural populations,

969
00:53:29,950 --> 00:53:31,660
we have many, many
species and tracking

970
00:53:31,660 --> 00:53:32,826
in lots of complicated ways.

971
00:53:32,826 --> 00:53:35,030
But we would like to
understand those things,

972
00:53:35,030 --> 00:53:36,710
but I think since
it's so complicated,

973
00:53:36,710 --> 00:53:38,830
you kind of like, oh
anything can happen,

974
00:53:38,830 --> 00:53:40,790
and you get a little
bit discouraged

975
00:53:40,790 --> 00:53:42,337
from thinking deeply about it.

976
00:53:42,337 --> 00:53:44,670
Because, you just think it's
going to be too complicated

977
00:53:44,670 --> 00:53:45,760
to try to understand.

978
00:53:45,760 --> 00:53:48,920
I very much like the idea
of trying to really hone

979
00:53:48,920 --> 00:53:51,480
your intuition on the simplest
possible situation like this,

980
00:53:51,480 --> 00:53:56,510
and then bringing that intuition
to more complicated or complex

981
00:53:56,510 --> 00:53:57,460
situations.

982
00:53:57,460 --> 00:53:58,140
Yeah?

983
00:53:58,140 --> 00:53:59,806
AUDIENCE: Can you
partially [? adjust ?]

984
00:53:59,806 --> 00:54:01,900
But what I want to ask
you is whether any of this

985
00:54:01,900 --> 00:54:04,250
can be applied to human society?

986
00:54:04,250 --> 00:54:05,620
PROFESSOR: Oh, yeah.

987
00:54:05,620 --> 00:54:08,270
So whether this could be
applied to human society.

988
00:54:08,270 --> 00:54:09,660
It's a good question.

989
00:54:09,660 --> 00:54:11,360
People certainly try to.

990
00:54:11,360 --> 00:54:16,330
I think you'll always have to
decide what we mean by apply.

991
00:54:16,330 --> 00:54:18,570
I would say that this
basic idea that there

992
00:54:18,570 --> 00:54:20,280
could be these
feedback loops that

993
00:54:20,280 --> 00:54:22,630
can lead to sudden transitions
in complex systems.

994
00:54:22,630 --> 00:54:26,044
I think this is a very robust
phenomenon, in the sense

995
00:54:26,044 --> 00:54:28,210
that I think if you have
strong enough interactions,

996
00:54:28,210 --> 00:54:31,750
then I think you kind
of expect it to be true.

997
00:54:31,750 --> 00:54:36,780
Another question is whether you
could quantitatively predict

998
00:54:36,780 --> 00:54:38,960
when that's going to happen.

999
00:54:38,960 --> 00:54:42,040
There was a recent article
written in science or nature

1000
00:54:42,040 --> 00:54:47,540
about potential tipping
points in human society

1001
00:54:47,540 --> 00:54:49,000
on a global scale.

1002
00:54:49,000 --> 00:54:51,620
Things in terms of
productivity of crops.

1003
00:54:51,620 --> 00:54:54,160
And so they have a
question mark about,

1004
00:54:54,160 --> 00:55:00,150
they say 2050, question
mark, collapse maybe.

1005
00:55:00,150 --> 00:55:03,770
I say it's very important to be
thinking about these things in.

1006
00:55:03,770 --> 00:55:07,710
But the question is what to do,
whether given the uncertainties

1007
00:55:07,710 --> 00:55:10,050
and your knowledge of where
these tipping points might

1008
00:55:10,050 --> 00:55:12,700
occur, it's always hard to
know whether you could make

1009
00:55:12,700 --> 00:55:16,330
a strong argument saying, oh
we have to stop fishing here

1010
00:55:16,330 --> 00:55:17,780
because it's going to collapse.

1011
00:55:20,520 --> 00:55:22,880
There's always uncertainty
in your decision making.

1012
00:55:22,880 --> 00:55:25,130
But certainly in the context
of climate regime shifts,

1013
00:55:25,130 --> 00:55:26,730
people are worried
about these sorts

1014
00:55:26,730 --> 00:55:29,399
of feedback loops and the
North Atlantic Oscillation

1015
00:55:29,399 --> 00:55:29,940
and so forth.

1016
00:55:29,940 --> 00:55:31,523
And I'm not at all
an expert for that,

1017
00:55:31,523 --> 00:55:33,710
so I don't know whether
we should be worried.

1018
00:55:33,710 --> 00:55:38,390
But it's at least
good to remember

1019
00:55:38,390 --> 00:55:44,750
that systems can respond in
dramatic ways to small changes.

1020
00:55:44,750 --> 00:55:47,650
And then, you have to decide
what to do with that knowledge,

1021
00:55:47,650 --> 00:55:50,034
and I think that's more
of a judgement call.

1022
00:55:52,998 --> 00:55:56,950
AUDIENCE: In the time
example, you had death,

1023
00:55:56,950 --> 00:55:59,895
it was a death rate.

1024
00:55:59,895 --> 00:56:02,557
Is that in a [INAUDIBLE] space?

1025
00:56:02,557 --> 00:56:03,140
PROFESSOR: OK.

1026
00:56:03,140 --> 00:56:05,280
So I want to be a little
bit maybe more clear.

1027
00:56:05,280 --> 00:56:09,390
The claim is that-- Like
the death, the delta there

1028
00:56:09,390 --> 00:56:11,380
that we're thinking
about, that didn't

1029
00:56:11,380 --> 00:56:12,870
have to be the perturbation.

1030
00:56:12,870 --> 00:56:15,467
It could be, but it
didn't have to be.

1031
00:56:15,467 --> 00:56:17,300
When we're plotting the
bifurcation diagram,

1032
00:56:17,300 --> 00:56:19,216
we're assuming implicitly
somehow that there's

1033
00:56:19,216 --> 00:56:21,614
a separation of time scales,
such that this delta might

1034
00:56:21,614 --> 00:56:23,530
be changing very slowly,
and then other things

1035
00:56:23,530 --> 00:56:25,480
are changing more rapidly.

1036
00:56:25,480 --> 00:56:27,050
So this bifurcation
diagram, it's

1037
00:56:27,050 --> 00:56:29,630
really that you're tracking it.

1038
00:56:29,630 --> 00:56:33,380
So in the context of this
n as a function of delta,

1039
00:56:33,380 --> 00:56:34,620
this thing goes like this.

1040
00:56:34,620 --> 00:56:37,290
And the idea is that
oh, slowly you're

1041
00:56:37,290 --> 00:56:39,200
getting more and more
agricultural runoff,

1042
00:56:39,200 --> 00:56:40,920
or the temperature
is increasing.

1043
00:56:40,920 --> 00:56:42,794
Doesn't have to be human
induced, by the way,

1044
00:56:42,794 --> 00:56:45,070
it could be something else.

1045
00:56:45,070 --> 00:56:47,810
So the idea is that the
population is slowly

1046
00:56:47,810 --> 00:56:49,200
changing like this.

1047
00:56:49,200 --> 00:56:53,480
And then eventually maybe
you get this collapse.

1048
00:56:53,480 --> 00:56:57,100
Now, a perturbation could
be something that could just

1049
00:56:57,100 --> 00:57:00,290
be a drought, or something that
is independent of this delta.

1050
00:57:00,290 --> 00:57:05,367
But the statement is
that, over here-- Now I'm

1051
00:57:05,367 --> 00:57:06,450
going to be mixing things.

1052
00:57:09,570 --> 00:57:13,170
At low delta, the claim is that
you should get a dip and then

1053
00:57:13,170 --> 00:57:14,910
rapid recovery.

1054
00:57:14,910 --> 00:57:17,252
As the delta gets bigger,
you get maybe even

1055
00:57:17,252 --> 00:57:19,710
a larger-- The same perturbation
could [? principally ?] do

1056
00:57:19,710 --> 00:57:22,650
a larger dip, and it takes
longer to get recovery.

1057
00:57:32,357 --> 00:57:33,940
This actually brings
up another point,

1058
00:57:33,940 --> 00:57:39,030
which is that these could all
be the same perturbations.

1059
00:57:39,030 --> 00:57:45,810
So this is a case where delta
is increasing as we go down.

1060
00:57:45,810 --> 00:57:48,420
So they could be the
same perturbation here.

1061
00:57:48,420 --> 00:57:50,830
Here you survive,
here you survive.

1062
00:57:50,830 --> 00:57:53,130
But then that same perturbation
that was survived here

1063
00:57:53,130 --> 00:57:56,290
could be the same
magnitude drought, just as

1064
00:57:56,290 --> 00:57:59,351
delta increases
you expect this--

1065
00:57:59,351 --> 00:58:01,720
This may be able
to push the system

1066
00:58:01,720 --> 00:58:03,530
past the unstable
fixed point, because

1067
00:58:03,530 --> 00:58:04,956
of a loss of resilience.

1068
00:58:04,956 --> 00:58:06,330
This is something
that we've seen

1069
00:58:06,330 --> 00:58:07,788
in a variety of
different contexts.

1070
00:58:07,788 --> 00:58:10,280
I think it's a very
general phenomenon.

1071
00:58:10,280 --> 00:58:11,759
Just because the
separation here is

1072
00:58:11,759 --> 00:58:13,800
some measure of the
resilience of the population,

1073
00:58:13,800 --> 00:58:16,670
or the ability of the population
to withstand perturbations.

1074
00:58:16,670 --> 00:58:19,980
We can see that the
resilience shrinks as we

1075
00:58:19,980 --> 00:58:21,230
get close to this bifurcation.

1076
00:58:30,380 --> 00:58:32,090
Did that answer your question?

1077
00:58:32,090 --> 00:58:32,820
Sort of?

1078
00:58:32,820 --> 00:58:33,640
Not really.

1079
00:58:33,640 --> 00:58:35,000
AUDIENCE: No, but it
answered another question.

1080
00:58:35,000 --> 00:58:35,550
PROFESSOR: It answered
another question?

1081
00:58:35,550 --> 00:58:36,442
Oh good.

1082
00:58:36,442 --> 00:58:37,900
I'm glad that I
answered something.

1083
00:58:37,900 --> 00:58:40,460
But what was your--
Because you were saying

1084
00:58:40,460 --> 00:58:43,955
is it the death rate that is
causing this perturbation?

1085
00:58:43,955 --> 00:58:44,945
AUDIENCE: Well, no.

1086
00:58:44,945 --> 00:58:47,327
I was wondering for
[? per ?] space.

1087
00:58:47,327 --> 00:58:47,910
PROFESSOR: OK.

1088
00:58:47,910 --> 00:58:49,180
Oh yes, I forgot.

1089
00:58:49,180 --> 00:58:53,100
Yes, I do remember
that you did ask that.

1090
00:58:53,100 --> 00:58:58,570
So the idea here is that this
could be a region that we fish,

1091
00:58:58,570 --> 00:59:01,680
and this could be region that
we don't fish, for example.

1092
00:59:01,680 --> 00:59:04,980
Or it could be
that, here there's

1093
00:59:04,980 --> 00:59:07,860
just not as much food for the
organism as there are here.

1094
00:59:07,860 --> 00:59:09,730
So just the idea
is that you would

1095
00:59:09,730 --> 00:59:12,870
need kind of a sudden sharp
boundary between regions

1096
00:59:12,870 --> 00:59:13,760
of different quality.

1097
00:59:13,760 --> 00:59:15,343
And that's kind of
the situation where

1098
00:59:15,343 --> 00:59:18,315
you would be able to measure
this recovery length.

1099
00:59:18,315 --> 00:59:19,690
And the basic
reason for that, is

1100
00:59:19,690 --> 00:59:22,510
that if the environmental
quality is something

1101
00:59:22,510 --> 00:59:26,370
that changes very slowly,
then the population density

1102
00:59:26,370 --> 00:59:29,060
will just track that.

1103
00:59:29,060 --> 00:59:33,680
So then you can't use that to
measure this recovery length.

1104
00:59:33,680 --> 00:59:36,390
Of course, the same
statement is true here.

1105
00:59:36,390 --> 00:59:39,120
That if you have environmental
perturbations that

1106
00:59:39,120 --> 00:59:40,870
are changing slowly
over time, then

1107
00:59:40,870 --> 00:59:42,681
that also can
complicate this picture.

1108
00:59:48,940 --> 00:59:51,630
Any other questions
about that before we

1109
00:59:51,630 --> 00:59:53,890
move towards
multispecies ecosystems?

1110
00:59:53,890 --> 00:59:55,970
We're going to start
with two species.

1111
01:00:07,265 --> 01:00:08,640
So what I'm going
to do is I want

1112
01:00:08,640 --> 01:00:10,850
to spend the rest
of the class talking

1113
01:00:10,850 --> 01:00:12,910
about Lotka-Volterra
populations.

1114
01:00:12,910 --> 01:00:15,400
And we're going to
move the rock, paper,

1115
01:00:15,400 --> 01:00:18,120
scissors discussion to
Thursday, because it's

1116
01:00:18,120 --> 01:00:20,570
a nice spatial example anyways.

1117
01:00:20,570 --> 01:00:23,810
So let's think about
Lotka-Volterra.

1118
01:00:23,810 --> 01:00:27,380
Now, there's a lot.

1119
01:00:27,380 --> 01:00:28,980
Lotka-Volterra
competition model.

1120
01:00:33,329 --> 01:00:34,870
So the assumption
is that we're going

1121
01:00:34,870 --> 01:00:38,880
to make here is that we
have these two species that

1122
01:00:38,880 --> 01:00:40,950
going to be
interacting, but they're

1123
01:00:40,950 --> 01:00:44,722
going to interact in
kind of the simplest way

1124
01:00:44,722 --> 01:00:45,680
that you might imagine.

1125
01:00:45,680 --> 01:00:50,614
Which is that, if we plot the
derivative of population size

1126
01:00:50,614 --> 01:00:53,280
n1 as a function of time, we get
something that looks like this.

1127
01:01:12,846 --> 01:01:14,220
Now the first
thing that you want

1128
01:01:14,220 --> 01:01:16,470
to do when you see something
like this is just to make

1129
01:01:16,470 --> 01:01:17,970
sense the basic equation.

1130
01:01:26,170 --> 01:01:33,780
In particular, in the
absence of the other species,

1131
01:01:33,780 --> 01:01:36,235
what happens to
these populations?

1132
01:01:38,810 --> 01:01:45,670
We're going to assume here maybe
first is that r1 and r2 are

1133
01:01:45,670 --> 01:01:46,670
both greater than 0.

1134
01:01:49,752 --> 01:01:51,460
It's in the absence
of the other species.

1135
01:01:56,100 --> 01:02:00,260
Do these populations
survive or not?

1136
01:02:00,260 --> 01:02:00,990
Ready?

1137
01:02:00,990 --> 01:02:06,210
Three, we're going do it
A, yes, B, no, our typical.

1138
01:02:06,210 --> 01:02:07,710
All right, absence of a species.

1139
01:02:11,570 --> 01:02:12,070
ready?

1140
01:02:12,070 --> 01:02:14,580
Three, two, one, survival?

1141
01:02:14,580 --> 01:02:15,550
Yeah, they survive.

1142
01:02:15,550 --> 01:02:18,490
For sure, because I told
you that these guys are

1143
01:02:18,490 --> 01:02:21,360
greater than 0, that means--
We can think about n1.

1144
01:02:21,360 --> 01:02:27,100
In the absence n2,
what is this equation?

1145
01:02:27,100 --> 01:02:28,920
Logistic.

1146
01:02:28,920 --> 01:02:33,210
Now, if we wanted to think
about these as describing

1147
01:02:33,210 --> 01:02:36,720
competitive interactions,
what does that tell us

1148
01:02:36,720 --> 01:02:40,590
about the sign of the betas?

1149
01:02:40,590 --> 01:02:45,940
Are the betas greater than
0, or are they less than 0?

1150
01:02:48,842 --> 01:02:50,300
I'll think about
it for 10 seconds.

1151
01:03:02,990 --> 01:03:04,998
All right.

1152
01:03:04,998 --> 01:03:07,230
Do you need more time?

1153
01:03:07,230 --> 01:03:07,760
Ready?

1154
01:03:07,760 --> 01:03:12,080
Three, two, one.

1155
01:03:12,080 --> 01:03:16,330
Competitive means that the
betas are greater than 0.

1156
01:03:16,330 --> 01:03:20,420
And what is the assumption
somehow that's going into this?

1157
01:03:23,940 --> 01:03:28,630
And we should remember that this
is a minus everything up here.

1158
01:03:32,039 --> 01:03:34,961
AUDIENCE: I guess that
somehow these two things

1159
01:03:34,961 --> 01:03:39,344
are-- there's a fixed
amount of some resource,

1160
01:03:39,344 --> 01:03:41,497
that both of these things need.

1161
01:03:41,497 --> 01:03:43,705
PROFESSOR: OK, so there
could be some fixed resource.

1162
01:03:50,157 --> 01:03:54,450
AUDIENCE: It's basically like
an effective carrying capacity.

1163
01:03:54,450 --> 01:03:56,769
PROFESSOR: That's
right, so somehow it's

1164
01:03:56,769 --> 01:03:58,560
modulating the effect
of carrying capacity.

1165
01:04:01,210 --> 01:04:05,430
And how is it that you
would describe these betas?

1166
01:04:09,010 --> 01:04:12,970
AUDIENCE: Sort of seems like
you expect that the beta should

1167
01:04:12,970 --> 01:04:16,373
be probably less than one.

1168
01:04:16,373 --> 01:04:21,100
[INAUDIBLE] I guess it doesn't
have to be less than one.

1169
01:04:23,660 --> 01:04:29,350
Compared to how well
n1-- and individual of n1

1170
01:04:29,350 --> 01:04:31,780
takes up some of this.

1171
01:04:31,780 --> 01:04:35,250
This is how much [INAUDIBLE]

1172
01:04:35,250 --> 01:04:38,230
PROFESSOR: That's right.

1173
01:04:38,230 --> 01:04:39,200
I think that's right.

1174
01:04:39,200 --> 01:04:41,950
Now I think we--
There's a question

1175
01:04:41,950 --> 01:04:44,690
about how explicitly to be
thinking about these resources.

1176
01:04:44,690 --> 01:04:47,040
Because it could be, it doesn't
have to be one resource.

1177
01:04:47,040 --> 01:04:50,530
And this is of course, a
very phenomenological model.

1178
01:04:50,530 --> 01:04:52,700
It's lumping all the
interact-- All of the ways

1179
01:04:52,700 --> 01:04:57,340
in which a species interacts in
just a single parameter beta.

1180
01:04:57,340 --> 01:04:59,820
But it's somehow, the betas
are telling us something

1181
01:04:59,820 --> 01:05:04,920
about how much does a
member of the other species

1182
01:05:04,920 --> 01:05:08,820
inhibit my growth, as compared
to a member of my own species?

1183
01:05:08,820 --> 01:05:13,950
Because we have this n1 here,
and just this species one

1184
01:05:13,950 --> 01:05:16,330
will naturally lead to
a carrying capacity k1,

1185
01:05:16,330 --> 01:05:17,910
you only have species one.

1186
01:05:17,910 --> 01:05:21,190
Now, the betas are telling
you something about

1187
01:05:21,190 --> 01:05:23,280
how much overlap they
have in terms of maybe

1188
01:05:23,280 --> 01:05:24,870
[? niche ?] or so.

1189
01:05:24,870 --> 01:05:27,340
Now, if the betas
are small, it means

1190
01:05:27,340 --> 01:05:31,270
that they're not competing
with each other very much.

1191
01:05:31,270 --> 01:05:33,890
The betas could be larger than
one, and what that's saying

1192
01:05:33,890 --> 01:05:36,360
is that a member of
this other species

1193
01:05:36,360 --> 01:05:38,720
is inhibiting my growth
more than a member

1194
01:05:38,720 --> 01:05:39,485
of my own species.

1195
01:05:51,170 --> 01:05:54,640
So the standard way to
analyze these equations

1196
01:05:54,640 --> 01:05:57,630
is to look at these isoclines.

1197
01:05:57,630 --> 01:06:01,960
So basically, if you say, OK
well n1 dot is equal to zero,

1198
01:06:01,960 --> 01:06:03,950
what does that mean?

1199
01:06:03,950 --> 01:06:07,060
Well, we can just see that
that's equivalent to saying

1200
01:06:07,060 --> 01:06:17,500
that it's n1 plus beta1,2
n2 is equal to k1.

1201
01:06:21,850 --> 01:06:27,430
So this is giving us a
relationship between n1 and n2.

1202
01:06:27,430 --> 01:06:33,210
What do these look like on a
plane, n1 versus n2 drawings?

1203
01:06:42,220 --> 01:06:47,220
So the parabola, it's a line.

1204
01:06:47,220 --> 01:06:49,810
And indeed, we can think about
what happens when each of these

1205
01:06:49,810 --> 01:06:52,350
is equal to one
thing or another.

1206
01:06:52,350 --> 01:06:57,010
So if n1-- If n2 is
equal to 0, this thing

1207
01:06:57,010 --> 01:06:58,060
is going across at k1.

1208
01:07:02,820 --> 01:07:05,340
And that makes sense,
because we already

1209
01:07:05,340 --> 01:07:07,730
decided that in
the absence of n2,

1210
01:07:07,730 --> 01:07:09,700
this thing is just
following logistic growth,

1211
01:07:09,700 --> 01:07:13,550
where the species one
will grow and go to k1.

1212
01:07:13,550 --> 01:07:16,800
And that's a stable
fixed point that n1 dot

1213
01:07:16,800 --> 01:07:18,185
has to be equal to zero.

1214
01:07:18,185 --> 01:07:21,019
It's a fixed point, so n1
dot has to be equal to 0.

1215
01:07:21,019 --> 01:07:22,810
And it's going to cross
at this other point

1216
01:07:22,810 --> 01:07:33,479
here where n2 is going to
be some k1 over beta 1, 2.

1217
01:07:33,479 --> 01:07:34,770
And then we end up with a line.

1218
01:07:43,690 --> 01:07:47,540
Now if we go, and we ask
what n2 dot is equal to,

1219
01:07:47,540 --> 01:07:49,820
and this is equal
to 0, we're going

1220
01:07:49,820 --> 01:07:52,390
to end up with something
that looks very similar.

1221
01:07:52,390 --> 01:07:58,767
But now it's going to be n2 plus
beta 2, 1, n1 is equal to k2.

1222
01:07:58,767 --> 01:08:00,100
This is also going to be a line.

1223
01:08:04,620 --> 01:08:10,030
Given what I have said, how many
different qualitative outcomes

1224
01:08:10,030 --> 01:08:12,460
can you get in this model?

1225
01:08:12,460 --> 01:08:15,414
Where we just assume--
Where we have beta-- Overall

1226
01:08:15,414 --> 01:08:17,664
I think we're thinking about
competitive interactions.

1227
01:08:23,420 --> 01:08:27,029
How many cases do
we need to consider?

1228
01:08:33,990 --> 01:08:36,914
Alright, it's going to be a, 0.

1229
01:08:49,385 --> 01:08:52,140
Do you guys understand
what I mean by cases?

1230
01:08:58,720 --> 01:09:00,479
No?

1231
01:09:00,479 --> 01:09:03,069
I mean, how many different
kind of qualitative outcomes

1232
01:09:03,069 --> 01:09:12,660
can there be in terms of species
one or species two winning,

1233
01:09:12,660 --> 01:09:14,420
or can you get coexistence?

1234
01:09:14,420 --> 01:09:17,890
Or how many different kind
of qualitative outcomes

1235
01:09:17,890 --> 01:09:18,834
can there be?

1236
01:09:23,180 --> 01:09:26,890
And if you're confused, you
can not vote, or give me

1237
01:09:26,890 --> 01:09:27,920
an unhappy face.

1238
01:09:27,920 --> 01:09:29,040
OK, ready?

1239
01:09:29,040 --> 01:09:30,940
Three, two, one.

1240
01:09:33,050 --> 01:09:33,550
So

1241
01:09:33,550 --> 01:09:35,454
We have many Es.

1242
01:09:38,060 --> 01:09:40,640
And can somebody
explain why it that this

1243
01:09:40,640 --> 01:09:45,071
is the case, without invoking
that you've already studied

1244
01:09:45,071 --> 01:09:46,279
this and you know the answer?

1245
01:09:51,019 --> 01:09:55,470
AUDIENCE: If you add another
line, completely on top,

1246
01:09:55,470 --> 01:09:57,920
or under, or it can cross too.

1247
01:09:57,920 --> 01:09:59,730
PROFESSOR: That's
right, that's right.

1248
01:09:59,730 --> 01:10:04,320
So the next thing I was about
to do is draw another line.

1249
01:10:04,320 --> 01:10:07,190
And the reason I stopped is
because there are actually

1250
01:10:07,190 --> 01:10:11,720
four different ways in which
this other line can be drawn.

1251
01:10:11,720 --> 01:10:17,560
And basically, you can have
another line that does one.

1252
01:10:17,560 --> 01:10:19,306
You can have two.

1253
01:10:19,306 --> 01:10:21,910
You can have three.

1254
01:10:21,910 --> 01:10:23,470
And you can have four.

1255
01:10:26,630 --> 01:10:27,130
OK?

1256
01:10:30,100 --> 01:10:33,875
These two cases-- So
this is we'll say one.

1257
01:10:33,875 --> 01:10:35,250
I don't know what
order i did it.

1258
01:10:35,250 --> 01:10:39,190
Two, three, and four.

1259
01:10:39,190 --> 01:10:44,280
Two and three, in both cases we
have the crossing of the lines.

1260
01:10:44,280 --> 01:10:48,070
So maybe does that mean
they're the same outcome,

1261
01:10:48,070 --> 01:10:51,640
the same qualitative outcome?

1262
01:10:51,640 --> 01:10:53,860
No.

1263
01:10:53,860 --> 01:10:57,980
And what will end up
happening here is that we're

1264
01:10:57,980 --> 01:11:02,650
going to get cases of--

1265
01:11:08,680 --> 01:11:10,450
So you basically
can get that species

1266
01:11:10,450 --> 01:11:14,310
one dominates independent
of starting condition,

1267
01:11:14,310 --> 01:11:16,920
assuming that both are
present at the beginning.

1268
01:11:16,920 --> 01:11:18,620
Species two could dominate.

1269
01:11:22,420 --> 01:11:29,010
They could coexist, or
you get bistability.

1270
01:11:29,010 --> 01:11:31,945
History dependent,
mutual exclusion.

1271
01:11:38,260 --> 01:11:40,210
I just want to draw.

1272
01:11:40,210 --> 01:11:45,820
So this one here was the-- We
had a dashed line for n1 dot

1273
01:11:45,820 --> 01:11:48,850
and we had the
solid line for this.

1274
01:11:48,850 --> 01:11:49,574
Yes?

1275
01:11:49,574 --> 01:11:51,615
AUDIENCE: Where is
[? the breaking ?] [INAUDIBLE]

1276
01:12:03,780 --> 01:12:08,670
PROFESSOR: So I guess
if you swap the labels,

1277
01:12:08,670 --> 01:12:09,590
or so, you're saying?

1278
01:12:09,590 --> 01:12:12,100
I guess in the case
that if one of them

1279
01:12:12,100 --> 01:12:17,324
is just above the other one,
then you can distinguish them.

1280
01:12:17,324 --> 01:12:18,740
So if you have a
dashed line here,

1281
01:12:18,740 --> 01:12:22,264
you have a solid line here,
that means that they--

1282
01:12:22,264 --> 01:12:24,724
AUDIENCE: How do I
distinguish case 2 and case 3?

1283
01:12:28,190 --> 01:12:28,690
[INAUDIBLE]

1284
01:12:34,290 --> 01:12:38,840
PROFESSOR: Maybe we
should try to figure out

1285
01:12:38,840 --> 01:12:44,740
which one's which, and then we
can figure it out from there.

1286
01:12:44,740 --> 01:12:47,670
Can somebody remind us,
what are these lines again?

1287
01:12:52,070 --> 01:12:53,988
Why are we drawing them,
or what do they mean?

1288
01:12:58,260 --> 01:13:03,400
So they're not actually
quite a fixed point.

1289
01:13:03,400 --> 01:13:04,170
It's a no incline.

1290
01:13:04,170 --> 01:13:05,961
What's the difference
between these things?

1291
01:13:12,260 --> 01:13:14,500
If we have a fixed point
here, what we're saying

1292
01:13:14,500 --> 01:13:18,080
is that both n1 dot and
n2 dot are equal to 0.

1293
01:13:18,080 --> 01:13:20,080
Fixed point means that
if you start right there,

1294
01:13:20,080 --> 01:13:21,080
you'll stay right there.

1295
01:13:21,080 --> 01:13:23,600
We're not yet say
anything about stability.

1296
01:13:23,600 --> 01:13:28,130
And this is going to be very
relevant for Sam's question.

1297
01:13:28,130 --> 01:13:29,005
AUDIENCE: [INAUDIBLE]

1298
01:13:35,800 --> 01:13:37,780
PROFESSOR: The axes
are definitely labeled.

1299
01:13:37,780 --> 01:13:45,320
I think I may agree, but I'm
a little worried that I'm-- If

1300
01:13:45,320 --> 01:13:47,130
you're happy I'm happy.

1301
01:13:47,130 --> 01:13:55,530
OK let's-- A fixed point of this
pair of equations will be when

1302
01:13:55,530 --> 01:13:58,420
both of the derivatives
are equal 0.

1303
01:13:58,420 --> 01:14:00,820
And that's not the
line that we've drawn.

1304
01:14:00,820 --> 01:14:03,930
The lines that we've drawn are
just when one or the other one

1305
01:14:03,930 --> 01:14:04,610
is equal to 0.

1306
01:14:08,770 --> 01:14:12,080
So what does that mean
about in case four,

1307
01:14:12,080 --> 01:14:15,940
is it possible well
to have coexistence?

1308
01:14:15,940 --> 01:14:16,440
No.

1309
01:14:16,440 --> 01:14:18,980
Because there's no
fixed one in the middle.

1310
01:14:18,980 --> 01:14:21,140
The only case it's
possible to get coexistence

1311
01:14:21,140 --> 01:14:23,600
is either line
two or line three.

1312
01:14:26,970 --> 01:14:29,520
But that tells us
there's a fixed point,

1313
01:14:29,520 --> 01:14:33,794
it doesn't tell us about the
stability of the fixed point,

1314
01:14:33,794 --> 01:14:35,170
though.

1315
01:14:35,170 --> 01:14:37,752
What we'll find is
that in one case,

1316
01:14:37,752 --> 01:14:39,710
it's a stable fixed point,
you get coexistence.

1317
01:14:39,710 --> 01:14:41,811
In the other case, it's
an unstable fixed point,

1318
01:14:41,811 --> 01:14:42,810
and you get bistability.

1319
01:14:46,972 --> 01:14:48,430
What we're going
to do now is we're

1320
01:14:48,430 --> 01:14:51,180
going to ask and try to
figure out in which case,

1321
01:14:51,180 --> 01:14:54,060
how do we label these things?

1322
01:14:54,060 --> 01:15:04,530
So what we'll do is ask,
first of all in line one,

1323
01:15:04,530 --> 01:15:07,130
we want to figure out,
is that a situation

1324
01:15:07,130 --> 01:15:09,970
where we know it's
not going to be

1325
01:15:09,970 --> 01:15:12,650
coexistence are bistability,
but is it going to a situation

1326
01:15:12,650 --> 01:15:16,530
where a species one wins,
or when species two wins?

1327
01:15:16,530 --> 01:15:18,830
But, if you want, you can
vote for one of the others.

1328
01:15:18,830 --> 01:15:20,750
I don't want to
constrain your choices.

1329
01:15:20,750 --> 01:15:29,730
So it's going to be a is one
wins, B, two wins, not three

1330
01:15:29,730 --> 01:15:30,230
wins.

1331
01:15:42,402 --> 01:15:43,360
This is something else.

1332
01:15:46,060 --> 01:15:50,380
I'm going to give you a
minute to think about what

1333
01:15:50,380 --> 01:15:52,110
might be going on here.

1334
01:15:52,110 --> 01:15:54,480
So the question is
put in situation one.

1335
01:16:26,830 --> 01:16:29,230
Oh no, I did that wrong.

1336
01:16:29,230 --> 01:16:31,280
In this.

1337
01:16:31,280 --> 01:16:34,400
For the solid line,
this is what we're

1338
01:16:34,400 --> 01:16:35,690
asking for n1 is equal to 0.

1339
01:16:35,690 --> 01:16:38,125
This is just equal to k2.

1340
01:16:38,125 --> 01:16:39,462
AUDIENCE: [INAUDIBLE]

1341
01:16:39,462 --> 01:16:40,420
PROFESSOR: What's that?

1342
01:16:43,228 --> 01:16:45,980
AUDIENCE: There's
other [? null ?] lines.

1343
01:16:45,980 --> 01:16:47,730
PROFESSOR: You want
more [? null lines? ?]

1344
01:16:47,730 --> 01:16:50,994
I feel like there are lots
of lines up there already.

1345
01:16:50,994 --> 01:16:55,467
AUDIENCE: Where n equals
0 [? null ?] lines.

1346
01:16:55,467 --> 01:16:57,952
The axes are actual
[? null ?] lines.

1347
01:17:02,162 --> 01:17:03,370
PROFESSOR: Yes, that's right.

1348
01:17:03,370 --> 01:17:05,292
Yeah, that's true.

1349
01:17:05,292 --> 01:17:09,450
AUDIENCE: That helps-- I'm
confused between labels.

1350
01:17:09,450 --> 01:17:10,590
PROFESSOR: Oh, I see.

1351
01:17:13,680 --> 01:17:16,370
OK I'm hesitant to draw
anything more up on the board.

1352
01:17:16,370 --> 01:17:20,950
But it is true that the axes
are also [? null ?] lines.

1353
01:17:20,950 --> 01:17:22,540
Because we have n's up there.

1354
01:17:26,736 --> 01:17:28,360
I'll give you another
20 seconds to try

1355
01:17:28,360 --> 01:17:29,526
to figure out this case one.

1356
01:17:34,650 --> 01:17:36,887
Which of the species
is going to win?

1357
01:18:22,587 --> 01:18:24,670
And let's go ahead and
vote, and see where we are.

1358
01:18:24,670 --> 01:18:25,170
Ready?

1359
01:18:25,170 --> 01:18:27,960
Three, two, one.

1360
01:18:32,960 --> 01:18:35,231
So there's a slight
majority that

1361
01:18:35,231 --> 01:18:36,980
are agreeing that in
this case, it's going

1362
01:18:36,980 --> 01:18:40,320
to be that one is going to win.

1363
01:18:40,320 --> 01:18:46,700
And I think that it's a little
bit hard to figure it out

1364
01:18:46,700 --> 01:18:50,550
completely, but what I will say
is that in the case where we're

1365
01:18:50,550 --> 01:18:54,940
down here, what this means is
that k1 divided nu beta 1,2

1366
01:18:54,940 --> 01:18:58,020
is larger than k2.

1367
01:18:58,020 --> 01:18:59,500
Now the question
says, what happens

1368
01:18:59,500 --> 01:19:05,520
in the limit of-- If species
two does not hurt species one?

1369
01:19:05,520 --> 01:19:10,330
Well that's when
beta 1,2 goes to 0.

1370
01:19:10,330 --> 01:19:14,380
And that's the situation
where this goes up.

1371
01:19:14,380 --> 01:19:17,360
So this is saying
that if the beta 1,2--

1372
01:19:17,360 --> 01:19:19,290
And then we can figure
out where this line is

1373
01:19:19,290 --> 01:19:25,370
as well, because this thing
is K2 divided by beta 2,1.

1374
01:19:25,370 --> 01:19:28,150
So if species to
doesn't hurt to see one,

1375
01:19:28,150 --> 01:19:31,597
but species one really
hurts species two,

1376
01:19:31,597 --> 01:19:33,513
that's going to be the
limit where species one

1377
01:19:33,513 --> 01:19:34,731
is going to win.

1378
01:19:42,530 --> 01:19:45,870
We are out of time.

1379
01:19:45,870 --> 01:19:52,000
Maybe what we'll do is
start class on Thursday

1380
01:19:52,000 --> 01:19:54,760
by-- I'll start with
this on the board,

1381
01:19:54,760 --> 01:20:00,017
and then we'll complete
these four possibilities,

1382
01:20:00,017 --> 01:20:01,600
to try to get some
intuition about it.

1383
01:20:01,600 --> 01:20:04,050
And also draw these options.