Chapter 11.1, Problem 93E

The size of an undisturbed fish population has been modeled by the formula

$p_{n+1}=\frac{b{p}_n}{a+p_n}$

where p_n is the fish population after n years and a and b are positive constants that depend on the species and its environment. Suppose that the population in year 0 is p₀ > 0.

1. (a) Show that if {p_n} is convergent, then the only possible values for its limit are 0 and b − a.
2. (b) Show that p_n+1 < (b/a)p_n.
3. (c) Use part (b) to show that if a > b, then lim_n→∞ p_n = 0; in other words, the population dies out.
4. (d) Now assume that a < b. Show that if p₀ < b − a, then {p_n} is increasing and 0 < p_n < b − a. Show also that if p₀ > b − a, then {p_n} is decreasing and p_n > b − a. Deduce that if a < b, then lim_n→∞ p_n = b − a.