Solved: (a) If a constant number h of fish are harvested from a fishery per unit time

Chapter 2, Problem 5

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(a) If a constant number h of fish are harvested from a fishery per unit time, then a model for the population P(t) of the fishery at time t is given by

\(\frac{d P}{d t}=P(a-b P)-h, \quad P(0)=P_{0}\),

where a, b, h, and \(P_{0}\) are positive constants. Suppose \(a=5\), \(b=1\), and \(h=4\).  Since the DE is autonomous, use the phase portrait concept of Section 2.1 to sketch representative solution curves corresponding to the cases \(P_{0}>4\), \(1<P_{0}<4\), and \(0<P_{0}<1\).  Determine the long-term behavior of the population in each case.

(b) Solve the IVP in part (a). Verify the results of your phase portrait in part (a) by using a graphing utility to plot the graph of P(t) with an initial condition taken from each of the three intervals given.

(c) Use the information in parts (a) and (b) to determine whether the fishery population becomes extinct in finite time. If so, find that time.

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