This is a very interesting process - one in which simple deterministic dynamics can produce a wide range of behaviors, including oscillations and apparently random trajectories. These kinds of results fall under the general rubric of deterministic chaos (see Connections).

A bit about bifurcations

The results of the previous section suggest that when we encounter a differential or difference equation, we should consider not only the solution, but how the solution depends upon the parameters of the equation. This subject is generally called bifurcation theory (because, as we will see, solutions "branch" as parameters vary). In this section, we will consider the two simplest bifurcations and some of their implications. As we discuss the material, do not try to apply biological interpretations to the equations; I have picked them to make illustrating the main points as simple as possible. At the end of this section, I will do one biological example and in Connections point you towards the literature for other ones.

We begin with the differential equation

dt for the variable x(t) depending upon the single parameter a. When we first encounter a differential equation, we may ask "What is the solution of this equation?''. The trouble is, the vast majority of differential equations do not have explicit solutions. Given that restriction, a good first question is "What are the steady states, that is for what values of x is dx/dt equal to 0?''. This is always a good question, and can often be answered. For the dynamics in Eq. (2.31), the steady states are given by xs = ±y/a. We thus conclude that if a < 0 there are no steady states (more precisely, there are no real steady states) and that if a > 0 there are one (when a = 0) or two steady states. We will call these steady solutions branches; there are thus two branches, one of which is positive and one of which is negative. Along these branches, dx/dt = 0. What about elsewhere in the plane? Between the branches, a is greater than x2, so we conclude that dx/dt < 0 and that x(t) will decrease, thus moving towards the lower branch. Anywhere else in the plane a is less than x2, so that dx/dt > 0 and x(t) will increase; I have summarized this analysis in Figure 2.11.

Before going on with the analysis, a few stylistic comments. First, note that I have put x on the ordinate and a on the abscissa. Thus, one might say "x is on they axis, how confusing.'' However, the labeling of axes is a convention, not a rule, and one just needs to be careful when conducting the analysis (more of this to come with the next bifurcation). Second, I have used x(t) and x interchangeably; this is done for convenience (and for avoiding writing things in a more cumbersome manner). Once again, this is not a problem if one is careful in understanding and presentation.

Returning to the figure, imagine that a is fixed, but x may vary, and that we are at some point along the positive branch. Then dx/dt = 0 and

Figure 2.11. The steady states of the differential equation dx/dt = x2 — a, showing the positive and negative branches.

Positive branch

Figure 2.11. The steady states of the differential equation dx/dt = x2 — a, showing the positive and negative branches.

Positive branch egative branch i-1-1-1-1

0 0

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