Data was collected for 300 fish from the north atlantic the...

Data was collected for 300 fish from the North Atlantic. The length of the fish (in mm) is summarized in the GFDT below. Lengths (mm) Frequency 140 - 141 1 142 - 143 16 144 - 145 71 146 - 147 108 148 - 149 83 150 - 151 18 152 - 153 3 What is the lower class limit for the second class?lower class limit =

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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