Q. 52 - Calculus

Question

Leila, in her capacity as a population biologist in Idaho, is trying to figure out how many salmon a local hatchery should release annually in order to revitalize the fishery. She knows that if $p_k$ salmon spawn in Redfish Lake in a given year, then only $0.2p_k$ fish will return to the lake from the offspring of that run, because of all the dams on the rivers between the sea and the lake. Thus, if she adds the spawn from h fish, from a hatchery, then the number of fish that return from that run k will be $p_{k+1} = 0.2( p_k+h).$.

(a) Show that the sustained number of fish returning approaches $p_{\infty }=h\sum _{k+1}^{\infty }0.2^k$ as k→∞.

(b) Evaluate $p_{\infty }$.

(c) How should Leila choose h, the number of hatchery fish to raise in order to hold the number of fish returning in each run at some constant P?

Step-by-Step Solution

Answer

Ans: 

Step 1: Identify the problem

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Step 2: Approach

Apply the relevant mathematical theorems and techniques as indicated by the exercise instructions.