Applications - A Guide to MATLAB for Beginners and Experienced Users - page 159

0.00012900700782
0.00007617734805
0.00004498196225
0.00002656139889

In the first case, the population is growing rapidly; in the second, it is
decaying rapidly. In fact, it is clear from the model that, for any n ,the
quotient P n + 1 / P n = (1 + r ), and therefore it follows that P n = P 0 (1 + r ) n

, n ≥ 0.
This accounts for the expression exponential growth and decay . The model
predicts a population growthwithout bound (for growing populations) and is
therefore not realistic. Our next model allows for a check on the population
caused by limited space, limited food supply, competitors and predators.

LogisticGrowth
The previous model assumes that the relative change in population is
constant, that is,

( P n + 1 − P n ) / P n = r .

Now let's build in a term that holds down the growth, namely

( P n + 1 − P n ) / P n = r − uP n .

We shall simplify matters by assuming that u = 1 + r , so that our recursion
relation becomes

P n + 1 = uP n (1 − P n ) ,

where u is a positive constant. In this model, the population P is constrained
to lie between 0 and 1, and should be interpreted as a percentage of a
maximum possible population in the environment in question. So let us set
up the function we will use in the iterative procedure:

clear f; f = inline('u*x*(1 - x)', 'x', 'u');

Now let's compute a few examples; and use plot to display the results.

u = 0.5; Xinit = 0.5; X = itseq(f, Xinit, 20, u); plot(X)