In Exercises 3–6, assume that any initial

Chapter 5, Problem 4E

(choose chapter or problem)

Problem 4E

In Exercises 3–6, assume that any initial vector x0 has an eigenvector decomposition such that the coefficient c1 in equation (1) of this section is positive.

Determine the evolution of the dynamical system in Example 1 when the predation parameter p is .125. (Give a formula for xk.) As time passes, what happens to the sizes of the owl and wood rat populations? The system tends toward what is sometimes called an unstable equilibrium. What do you think might happen to the system if some aspect of the model (such as birth rates or the predation rate) were to change slightly?

Reference Example 1:

Denote the owl and wood rat populations at time k by

where k is the time in months, Ok is the number of owls in the region studied, and Rk is the number of rats (measured in thousands). Suppose

where p is a positive parameter to be specified. The (.5) Ok in the first equation says that with no wood rats for food, only half of the owls will survive each month, while the (1.1) Rk in the second equation says that with no owls as predators, the rat population will grow by 10% per month. If rats are plentiful, the (.4) Rk will tend to make the owl population rise, while the negative term –p Ok measures the deaths of rats due to predation by owls. (In fact, 1000p is the average number of rats eaten by one owl in one month.) Determine the evolution of this system when the predation parameter p is .104.