Deterministic Competition

Consider a simplified system, one that can be described by N_t objects at time t. For example, one could consider the number of grasshoppers on the plains of Africa, or on some small plot of land. Let there be competition between the growth processes and the decay processes so that the number of objects at time t + 1 is N_t + 1 = N_t exp[r (1 – N_t)], an exponential growth relationship. This equation is deterministic, for N_t determines N_t + 1 unambiguously. One can think of r as a measure of the ratio between growth and decay. Numerous mechanical, hydrodynamic, chemical, and electrical systems can be approximately modeled by this relationship.

How does the number of objects behave with elapsed time? If N_t = 1, then N remains 1 forever. In the general case, we can determine N_t as t→∞ to find out whether N approaches the equilibrium value 1. For instance, let r = 1 and begin with N₀ = 0.5, and calculate with a calculator or personal computer. Now try different values for r. What behavior do you predict?

Answer

The time evolution here depends on the value of r. One finds that N_t = 1 is a stable equilibrium only when r lies between 0 and 2. If r = 2.3 with N₀ = 0.5, then successive N_t will oscillate between about 1.59 and about 0.40 as a stable 2 cycle. For r > 3.102, no cycle is stable, all cycles are possible, etc.

In the chaotic regimes, the equation results are deterministic, but the time evolution is indistinguishable from that governed by probability laws. One really needs to see the calculations proceed to appreciate the amazing behavior of this simple-looking equation.