Discrete Population Models – An introduction
In many species there is no overlap in generations: The previous generation is dead
before the next matures. In this case, we can think about the populations at discrete
intervals in time, such as one year.
Let’s call the population at time t , N t then the population at time t  1 is Nt 1 . These
values are related by a simple function like this:
Nt 1  f ( Nt )
For example the function could be
f ( Nt )  2* Nt
which means
Nt 1  2* Nt
So if we start at time 0 with 2, then we get 4,8,16,32,…
Leonardo of Pisa (13th century) suggested this model of a growing rabbit population:
At the start of the breeding season there is a pair of immature rabbits. After one
reproductive season they produce two pairs of male and female rabbits, and these
parents stop reproducing. So if we call the number of pairs of male and female rabbits
N t , then we have
Nt 1  Nt  Nt 1
I guess you can see why this equation is as it is. So if we start with 1 pair, then we get
1,1,2,3,5,8,13. This is called the Fibonacci sequence because in the 18th century
Leonardo of Pisa was nicknamed Fibonacci.
In general the function f ( Nt ) has she shape sketched below. This is not hard to
understand. At low population levels of N t , the growth in population is quite high, but
at high levels of N t the growth is low because of crowding and self-regulation. The
larger the N t in this region, the lower the growth rate. So there is some peak in the
growth function. This has interesting consequences as we shall soon see.
Nt 1
Nt
There’s an interesting way to find how the population changes over time – and it does
not involve any arithmetic calculation. It’s a graphical technique called
“cobwebbing”. To understand how this works, think again what we are actually doing
in using the above equations. We start with N1 and put it into the function and we get
N 2 as output, and what do we do? Stick this back into the function as input and we
get N3 out and so on. Recursion! How do we show this output-back-in-as-input on the
graph? Well we draw a line at 45 degrees. This line connects points which have equal
values on the horizontal and vertical axes of the graph. We start with N1 on the
bottom axis of the graph and draw a vertical line up to the function, so getting the
output of the function N 2 . Now to return this output as the next input, we draw a
horizontal line from this point until it reaches the 45-degree line, so the output value is
transferred to the input. This is sketched below:
Nt 1
Nt
FIRST INVESTIGATIONS
1. Try cobwebbing the following functions. Choose at least 2 different starting values
and draw the cobweb as accurately as you can. Confirm that the final “equilibrium
state” does not depend on the starting value.
2. From the samples given, try and deduce some conclusions about how the shape of
the function affects the final population.
Nt 1
Nt 1
Nt
Nt
Nt 1
Nt 1
Nt
Nt
Nt 1
Nt 1
Nt
4. Look up the “Allee effect”. It corresponds to one of these curves.
FURTHER INVESTIGATIONS
Nt
A very famous function is the “logistic”. This has the shape of an upside-down
parabola and has the mathematical form
Nt 1  4 Nt (1  Nt )
Here  is a “parameter”, a number which changes how high the curve is. This must
be greater or equal to zero and less than 1.
To investigate this, we have provided an Applet where you can set the parameter
value and observe the behaviour of the system. Here’s some values of the parameter
which you ought to try:
0.5, 0.75, 0.8, 0.866, 0.9
Of course you will also try other values, but there are some keys. For each you should
graph out the time series by hand. Press the “show-data” button to show the data.
RESEARCH
Try
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Fibonacci’s rabbits
Cross and Cotton model for tumor cell growth
Fishery Management Model.
Allee Effect
May’s equation