Consider the survival of a population of whales and assume that if the number of whales falls below a minimum survival level m, then the species will become extinct. In addition, assume that the population is limited by the carrying capacity M of the environment. The following model is proposed to describe this population: w(t + 1) = w(t) + k(M - w(t)) (w(t) - m),
where k > 0.
a. Does this model make sense in terms of the description above? Explain.
b. Find the equilibria of the model and determine their stability.
c. Perform a graphical equilibrium and stability analysis assuming that M = 5000, m = 100, k = 0.0001. (Use Matlab or a graphing calculator to graph the right-hand side of the model.) Compare your results to the general case under b).
d. The model has two serious shortcomings. What are they? (Consider what happens if w(0) < m, or when w(0) is much larger than M.)