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 is $p_n$ 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 $p_0>0$.

(a) Show that if $\left\{p_n\right\}$ is convergent, then the only possible values for its limit are 0 and $b-a$.

(b) Show that $p_{n+1}<\left(b/a\right)p_n$.

(c) Use part (b) to show that if $a>b$, then ${{\displaystyle \lim }}_{n\to \infty }{p}_n=0$; in other words, the population dies out.

(d) Now assume that $a<b$ Show that if $p_0<b-a$, then $\left\{p_n\right\}$ is increasing and $0<p_n<b-a$. Show also that if $p_0>b-a$, then $\left\{p_n\right\}$ is decreasing and $p_n>b-a$. Deduce that if $a<b$ then ${{\displaystyle \lim }}_{n\to \infty }{p}_n=b-a$.