Bifurcation Analysis of the Logistic Model

We recall the classic work of R. May. The simple logistic difference model is given by xn+ 1 — ( 1 X%

where r is the so-called driving parameter. In biology, this number Xn represents the population of the species.

The equation is used in the following manner. Start with a fixed value of the driving parameter, r, and an initial value of x0. One then runs the equation recursively, obtaining xi, x2,..., x„. For low values of r, xn (as n goes to infinity) eventually converges to a single number.

How can we get the bifurcation diagram? Certainly, via computers. It is well known that there are some international mathematical software, such as Matlab, Mathematica, Maple, etc. In this article, all of our computations and plottings finished in Maple. The bifurcation diagram of the logistic model [7] is shown in Figure 8.

It is when the driving parameter, r, slowly turns up that interesting things happen. When r = 3.0, xn no longer converges - it oscillates between two values. This characteristic change in behavior is called a bifurcation. Turn up the driving parameter even further and xn oscillates between not two, but four values. As one continues to increase the driving parameter, xn goes through bifurcations of period 8, then 16, ..., then chaos. When the value of the driving parameter r« 3.569945 672, xn neither converges nor oscillates - the values become completely random. For values of r> 3.569 945 672, the behavior is largely chaotic. However, there is a particular value of r where the sequence again oscillates with a period of three.

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