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

dP/dt = P(a - bP) - h, P(0) = P₀,

where a, b, h, and P₀ 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 P₀ > 4, 1 < P₀ , 4, and 0 < P₀ < 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 infi­nite time. If so, fi­nd that time.