Problem 3E

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 .2 in equation (3). (Give a formula for xk.) Does the owl population grow or decline? What about the wood rat population?

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.

Solution 3E

Step 1

The evolution of the dynamical system is,

Let and the parameter .

Write the dynamical system equation in the matrix form.

That is, where for .