1
00:00:04,091 --> 00:00:07,186
now let's turn to exploring the dynamics
at the logistic model
2
00:00:08,086 --> 00:00:11,094
so you might remember that
3
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n of sub (t+1) is the population
that time t + 1
4
00:00:15,369 --> 00:00:19,510
and that was equal to the birthrate – the death rate times the population at time t minus the numbers individuals who died
5
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times the population that time
6
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T minus the
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numbers individuals who died due to
overcrowding which is the population
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a time T-square divided by maximum
population or the carrying capacity
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I'm going to start by reading this
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00:00:37,057 --> 00:00:40,866
in the simplest format.
First I'm going to let
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R represents the birth rate minus the
death rate
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00:00:44,929 --> 00:00:47,985
and I'm going to let K equal
13
00:00:48,489 --> 00:00:51,555
the maximum population. now I can write
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this top equation using these new
symbols
15
00:00:55,999 --> 00:01:01,420
now I'm going to do
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a little bit algebra so if you don't
like algebra
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then just try and follow along and if
you don't understand it doesn't really
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matter it's only the endpoint
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that you need to know. So I'm gonna do is
divide
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both sides of this equation by K
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the carrying capacity. And now I'm gonna
define
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one more new symbol and that's
going to be
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X sub t it equals to
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00:01:35,082 --> 00:01:39,134
n sub t over K. So now I can rewrite
25
00:01:40,034 --> 00:01:43,040
this equation using my new symbol
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00:01:43,004 --> 00:01:48,025
X
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This equation represents the fraction
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the current population is of the
carrying capacity
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at a given time and that's equal to
30
00:02:01,053 --> 00:02:05,055
R times the fraction that the previous
time step
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minus that same fraction square and this
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00:02:09,094 --> 00:02:12,155
is known as the logistic
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00:02:13,055 --> 00:02:18,139
map. And this turns out to be
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the most famous equation in the field of
chaos theory.
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00:02:23,069 --> 00:02:26,710
Let's rewrite the logistic map here for
clarity
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00:02:26,071 --> 00:02:30,129
Really simple huh
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00:02:31,029 --> 00:02:35,108
as it happens though it's more
interesting than it appears
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00:02:36,008 --> 00:02:41,010
many people have studied this equation
in depth since Verhulst proposed
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00:02:41,028 --> 00:02:44,103
Two prominent examples have people who
have studied it
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00:02:45,003 --> 00:02:49,542
our Lord Robert May a theoretical
biologist who had a very influential
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00:02:49,569 --> 00:02:52,470
paper about sequestering in the 1970s
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00:02:52,047 --> 00:02:55,050
and Michelle Feigenbaum a theoretical
physicist
43
00:02:55,005 --> 00:02:58,007
who worked extensively on this equation
in the nineteen eighties
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00:02:58,007 --> 00:03:01,045
as probably the person most commonly
associated with it in the scientific
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community
46
00:03:03,005 --> 00:03:06,066
note that X
47
00:03:06,066 --> 00:03:09,067
is the population at time
48
00:03:09,076 --> 00:03:12,120
sometime divided by the carrying
capacity
49
00:03:13,002 --> 00:03:18,059
for the maximum population so X always
is a real number between 0&1
50
00:03:18,077 --> 00:03:22,094
this is why the question is called a map
that is it takes
51
00:03:22,094 --> 00:03:26,140
in this on this side a current value of
x between 0&1
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00:03:27,004 --> 00:03:31,050
and maps it into a new value of x which
is also between 0&1
53
00:03:31,086 --> 00:03:35,125
so let's look at an example let's let
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00:03:36,025 --> 00:03:40,028
are equal 2 at our initial
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00:03:40,055 --> 00:03:43,073
population over carrying capacity X 0
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00:03:43,073 --> 00:03:46,134
equal 0.2 that is our
population is 20 percent
57
00:03:47,034 --> 00:03:50,080
at at the carrying capacity. Now i can
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00:03:50,008 --> 00:03:54,023
iterate this map so get out a calculator
let's calculate
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00:03:54,095 --> 00:03:57,117
x1. That's gonna be equal to 2.
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00:03:58,017 --> 00:04:02,026
that's our value times 0.2
61
00:04:02,026 --> 00:04:06,040
0.2^2 and
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00:04:06,004 --> 00:04:09,007
quite my calculator that's equal to 0.32
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00:04:09,007 --> 00:04:13,071
so we've gone from twenty percent are
carrying capacity to 32 percent
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00:04:14,034 --> 00:04:18,040
now what happens the next year. At the
next year we have
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00:04:18,094 --> 00:04:22,823
2 times well now we gotta take this
value for
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00:04:23,669 --> 00:04:28,960
our previous generation 0.32 minus
0.32^2
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00:04:28,096 --> 00:04:32,138
and that's equal 0.4352
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00:04:33,038 --> 00:04:36,051
(writing)
69
00:04:36,051 --> 00:04:42,096
okay let's keep going but a little
faster now
70
00:04:42,096 --> 00:04:49,096
(writing)
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00:05:01,082 --> 00:05:03,131
(writing)
72
00:05:04,031 --> 00:05:08,037
(writing)
73
00:05:08,037 --> 00:05:15,037
(writing)
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00:05:21,024 --> 00:05:24,233
and forever after will get .5 is our
answer.
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00:05:24,449 --> 00:05:27,545
This means that if your growth rate that
is the
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00:05:28,409 --> 00:05:32,300
birthrate months the death rate or R
is equal to 2
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00:05:32,003 --> 00:05:35,272
and you start out at twenty percent of
the carrying capacity
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00:05:35,569 --> 00:05:40,240
under this model the population what
always end up at fifty percent of the
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carrying capacity.
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00:05:41,379 --> 00:05:44,383
There are two things I should note here
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00:05:44,779 --> 00:05:48,150
first I am using the term model
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00:05:48,015 --> 00:05:52,030
here to refer to a mathematical equation
that is
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00:05:52,003 --> 00:05:55,025
the logistic map. This is the model.
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00:05:55,052 --> 00:05:59,821
It's called the model because its
simplified representation
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00:06:00,289 --> 00:06:05,960
of real phenomenon of population growth,
also refer to the computer programs we
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00:06:05,096 --> 00:06:05,105
write or use
87
00:06:06,086 --> 00:06:11,087
in NetLogo as models since they're also
simplified representations of real
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00:06:11,087 --> 00:06:11,806
phenomena.
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00:06:12,589 --> 00:06:16,656
the word model is very general term in
science for any simplified
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00:06:17,259 --> 00:06:18,830
representation of nature
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00:06:18,083 --> 00:06:22,502
whether it be an equation, computer
program, a drawing or what have you.
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00:06:23,249 --> 00:06:26,250
The second thing to note
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00:06:26,025 --> 00:06:30,624
is that this value of .5 is called
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00:06:30,849 --> 00:06:37,520
an attractor.
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00:06:37,052 --> 00:06:40,083
It's an attractor for the system because
the system
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00:06:40,083 --> 00:06:45,084
is in some sense attracted to it. It
turns out that even if we had started
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00:06:45,093 --> 00:06:45,101
with
98
00:06:46,073 --> 00:06:49,074
different initial population say
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00:06:49,083 --> 00:06:54,084
x0 equals 0.8 the system would still
end up with a value
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00:06:54,093 --> 00:06:58,129
0.5 after some number of steps. When the
system and sub
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00:06:59,029 --> 00:07:03,075
a single value like 0.5 this value is
called
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a fixed point.
103
00:07:09,064 --> 00:07:12,077
since the value with the point stays
fixed
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00:07:12,077 --> 00:07:15,086
Thus for this system 0.5
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00:07:15,086 --> 00:07:18,160
is called a fixed-point attractor and by
this
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00:07:19,006 --> 00:07:22,047
system i mean again this equation with
R
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00:07:23,001 --> 00:07:27,050
equals to 2. Often the terms model and
system
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00:07:27,005 --> 00:07:31,009
are used to synonymously I hope this
doesn't get too confusing.
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00:07:31,054 --> 00:07:35,060
In any case we'll see some other kinds of
attractor in the next subunit.
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00:07:36,014 --> 00:07:41,046
Finally let's look at a different way of
visualizing the dynamics at the logistic
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00:07:41,046 --> 00:07:41,100
map
112
00:07:42,000 --> 00:07:46,096
that is how it changes as you iterate it.
I'm going to draw
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00:07:46,096 --> 00:07:53,096
a plot logistic map equation for R
equals 2
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00:07:55,006 --> 00:07:58,052
now I am plotting down here is X sub
t
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00:07:58,052 --> 00:08:01,137
and I am plotting over here
116
00:08:02,037 --> 00:08:05,090
X sub t+1. OK, so
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00:08:05,009 --> 00:08:10,778
not a great drawing but this is roughly
what it looks like it's a parabola
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00:08:11,669 --> 00:08:15,630
and it goes between 0&1
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00:08:15,063 --> 00:08:18,132
here and here for E equals 2
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00:08:19,032 --> 00:08:22,032
right that goes between
121
00:08:22,032 --> 00:08:28,130
0 and 0.5 just the maximum. Okay let's
put 0.5 here on the
122
00:08:29,003 --> 00:08:32,010
x-axis and then we can follow
123
00:08:32,037 --> 00:08:35,101
the steps we took before in calculating
the
124
00:08:36,001 --> 00:08:39,470
values taken. Okay so our first value if
you remember
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00:08:39,479 --> 00:08:43,690
x1 was point 0.32 thats x1
126
00:08:43,068 --> 00:08:46,118
we find this on the .32 to its about
right here
127
00:08:47,018 --> 00:08:51,044
and the y value for that on the parabola
128
00:08:51,044 --> 00:08:54,090
was point 0.4352, that was x2
129
00:08:54,009 --> 00:08:57,108
okay so this was the point
130
00:08:58,089 --> 00:09:02,136
(X1,X2)
131
00:09:03,036 --> 00:09:06,080
then we take our value for X2
132
00:09:06,008 --> 00:09:09,011
and we find it down here on that
133
00:09:09,083 --> 00:09:13,098
x-axis here because we're gonna
calculate the next value
134
00:09:13,098 --> 00:09:16,130
of the function so
135
00:09:17,003 --> 00:09:20,037
0.4352 you down here is
136
00:09:20,064 --> 00:09:23,883
around here and that corresponds up to
137
00:09:24,459 --> 00:09:28,350
this point on the problem that's X3
138
00:09:28,035 --> 00:09:31,083
.49160192
139
00:09:31,083 --> 00:09:34,089
okay this point here is
140
00:09:34,089 --> 00:09:37,158
(X2,X3)
141
00:09:37,959 --> 00:09:42,470
Okay and we take our X3
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00:09:42,047 --> 00:09:45,054
which was .49160192
143
00:09:45,054 --> 00:09:48,075
We find that on the x-axis
144
00:09:48,075 --> 00:09:53,108
to make up here to find X4 which was .4
999999...
145
00:09:54,008 --> 00:09:57,087
et cetera okay I keep doing
that
146
00:09:57,087 --> 00:10:01,093
and finally we get to you exactly .5
147
00:10:01,093 --> 00:10:04,179
and .5 and once these are both .5
148
00:10:05,079 --> 00:10:08,125
the system doesn't go anywhere
149
00:10:09,025 --> 00:10:12,093
you know it just stays at this point.
So you can think of hopping
150
00:10:12,093 --> 00:10:15,125
from one point to another along
151
00:10:16,025 --> 00:10:20,031
this parabola as an example of the
dynamics of
152
00:10:20,085 --> 00:10:23,114
the system with this r-value and
153
00:10:24,014 --> 00:10:27,018
are starting point and
154
00:10:27,054 --> 00:10:33,152
that is called a trajectory. Now it's
time for our next quiz.
155
00:10:34,052 --> 00:10:38,117
You need a calculator for this quiz. Said
R
156
00:10:39,017 --> 00:10:43,044
equals 2.5 and X0 equals to that 0.2.
157
00:10:43,044 --> 00:10:47,077
then you see equation for the logistic
map filling in
158
00:10:47,077 --> 00:10:52,146
2.5 for R and starting out with X0 of
0.2
159
00:10:52,839 --> 00:10:58,250
To calculate X1 X2 X3 and so on until
you reach a fixed-point.
160
00:10:58,025 --> 00:11:02,050
What is that fixed-point? and recall
161
00:11:02,005 --> 00:11:05,005
the fixed-point is the value of X
162
00:11:05,005 --> 00:11:08,073
such that X sub t is the same as
X sub t+1