It is known that if the deer population falls below a certain level, m, then the deer will become extinct. It is also known that is the deer population goes above the maximum carrying capacity, M, the population will decrease to M.
(a) Discuss the reasonableness of the following model for the growth rate of the deer population as a function of time: dP/dt = kP(M-P)(P-m), where P is the population and k is a constant of proportionality.
(b) Explain how this growth rate model differs from the logistical model dP/dt = kp(M-P) Is it better or worse than the logistical model?
(c) Show that if P > M for all t, then the limit as P(t) approached infinity is M.
(d) Assuming m < P < M for all t, briefly explain the steps you would use to solve the differential equation.
(e) What are the equilibrium point of the model? Explain the dependence of the equilibrium level of P on the initial conditions. How many deer hunting permits should be issued?
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