now let's turn to exploring the dynamics at the logistic model
so you might remember that
n of sub (t+1) is the population that time t + 1
and that was equal to the birthrate – the death rate times the population at time t minus the numbers individuals who died
times the population that time
T minus the
numbers individuals who died due to overcrowding which is the population
a time T-square divided by maximum population or the carrying capacity
I'm going to start by reading this
in the simplest format. First I'm going to let
R represents the birth rate minus the death rate
and I'm going to let K equal
the maximum population. now I can write
this top equation using these new symbols
now I'm going to do
a little bit algebra so if you don't like algebra
then just try and follow along and if you don't understand it doesn't really
matter it's only the endpoint
that you need to know. So I'm gonna do is divide
both sides of this equation by K
the carrying capacity. And now I'm gonna define
one more new symbol and that's going to be
X sub t it equals to
n sub t over K. So now I can rewrite
this equation using my new symbol
X
This equation represents the fraction
the current population is of the carrying capacity
at a given time and that's equal to
R times the fraction that the previous time step
minus that same fraction square and this
is known as the logistic
map. And this turns out to be
the most famous equation in the field of chaos theory.
Let's rewrite the logistic map here for clarity
Really simple huh
as it happens though it's more interesting than it appears
many people have studied this equation in depth since Verhulst proposed
Two prominent examples have people who have studied it
our Lord Robert May a theoretical biologist who had a very influential
paper about sequestering in the 1970s
and Michelle Feigenbaum a theoretical physicist
who worked extensively on this equation in the nineteen eighties
as probably the person most commonly associated with it in the scientific
community
note that X
is the population at time
sometime divided by the carrying capacity
for the maximum population so X always is a real number between 0&1
this is why the question is called a map that is it takes
in this on this side a current value of x between 0&1
and maps it into a new value of x which is also between 0&1
so let's look at an example let's let
are equal 2 at our initial
population over carrying capacity X 0
equal 0.2 that is our population is 20 percent
at at the carrying capacity. Now i can
iterate this map so get out a calculator let's calculate
x1. That's gonna be equal to 2.
that's our value times 0.2
0.2^2 and
quite my calculator that's equal to 0.32
so we've gone from twenty percent are carrying capacity to 32 percent
now what happens the next year. At the next year we have
2 times well now we gotta take this value for
our previous generation 0.32 minus 0.32^2
and that's equal 0.4352
(writing)
okay let's keep going but a little faster now
(writing)
and forever after will get .5 is our answer.
This means that if your growth rate that is the
birthrate months the death rate or R is equal to 2
and you start out at twenty percent of the carrying capacity
under this model the population what always end up at fifty percent of the
carrying capacity.
There are two things I should note here
first I am using the term model
here to refer to a mathematical equation that is
the logistic map. This is the model.
It's called the model because its simplified representation
of real phenomenon of population growth, also refer to the computer programs we
write or use
in NetLogo as models since they're also simplified representations of real
phenomena.
the word model is very general term in science for any simplified
representation of nature
whether it be an equation, computer program, a drawing or what have you.
The second thing to note
is that this value of .5 is called
an attractor.
It's an attractor for the system because the system
is in some sense attracted to it. It turns out that even if we had started
with
different initial population say
x0 equals 0.8 the system would still end up with a value
0.5 after some number of steps. When the system and sub
a single value like 0.5 this value is called
a fixed point.
since the value with the point stays fixed
Thus for this system 0.5
is called a fixed-point attractor and by this
system i mean again this equation with R
equals to 2. Often the terms model and system
are used to synonymously I hope this doesn't get too confusing.
In any case we'll see some other kinds of attractor in the next subunit.
Finally let's look at a different way of visualizing the dynamics at the logistic
map
that is how it changes as you iterate it. I'm going to draw
a plot logistic map equation for R equals 2
now I am plotting down here is X sub t
and I am plotting over here
X sub t+1. OK, so
not a great drawing but this is roughly what it looks like it's a parabola
and it goes between 0&1
here and here for E equals 2
right that goes between
0 and 0.5 just the maximum. Okay let's put 0.5 here on the
x-axis and then we can follow
the steps we took before in calculating the
values taken. Okay so our first value if you remember
x1 was point 0.32 thats x1
we find this on the .32 to its about right here
and the y value for that on the parabola
was point 0.4352, that was x2
okay so this was the point
(X1,X2)
then we take our value for X2
and we find it down here on that
x-axis here because we're gonna calculate the next value
of the function so
0.4352 you down here is
around here and that corresponds up to
this point on the problem that's X3
.49160192
okay this point here is
(X2,X3)
Okay and we take our X3
which was .49160192
We find that on the x-axis
to make up here to find X4 which was .4 999999...
et cetera okay I keep doing that
and finally we get to you exactly .5
and .5 and once these are both .5
the system doesn't go anywhere
you know it just stays at this point. So you can think of hopping
from one point to another along
this parabola as an example of the dynamics of
the system with this r-value and
are starting point and
that is called a trajectory. Now it's time for our next quiz.
You need a calculator for this quiz. Said R
equals 2.5 and X0 equals to that 0.2.
then you see equation for the logistic map filling in
2.5 for R and starting out with X0 of 0.2
To calculate X1 X2 X3 and so on until you reach a fixed-point.
What is that fixed-point? and recall
the fixed-point is the value of X
such that X sub t is the same as X sub t+1