Problem 93E

93. 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 $p_{0}>0$.

(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 $p_{n+1}<(b / a) p_{n}$.

(c) Use part (b) to show that if $a>b$, then $\lim _{n \rightarrow \infty} p_{n}=0$; 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 $\lim _{n \rightarrow \infty} p_{n}=b-a$.

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