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# The size of an undisturbed fish population has been modeled by the formula $p_{n + 1} = \frac {bp_n}{a + p_n}$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 $\{ p_n \}$ 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 $\lim_{n \to \infty} p_n = 0$; in other words, the population dies out.(d) Now assume that $a < b$. Show that if $p_0 < b - a$, then $\{ p_n \}$ is increasing and $0 < p_n < b - a$. Show also that if $p_0 > b - a$, then $\{ p_n \}$ is decreasing and $p_n > b - a$. Deduce that if $a < b$, then $\lim_{n \to \infty} p_n = b - a$.

## a. $p=b-a$b. $1+\frac{p_{m}}{a}>1$c. $r=\frac{b}{a} \in(0,1)$d. it is convergent by the Monotonic Sequence Theorem. It then follows from part (a) that $\lim _{n \rightarrow \infty} p_{n}=b-a$

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