Solved: Let A have the properties described in Exercise

Chapter 5, Problem 7E

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Problem 7E

Let A have the properties described in Exercise 1.

a. Is the origin an attractor, a repeller, or a saddle point of the dynamical system  xk + 1 = Axk?

b. Find the directions of greatest attraction and/or repulsion for this dynamical system.

c. Make a graphical description of the system, showing the directions of greatest attraction or repulsion. Include a rough sketch of several typical trajectories (without computing specific points).

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.