Controlling a population—The fish and game department in a certain state is planning to issue hunting permits to control the deer population (one deer per permit). It is known that if the deer population falls below a certain level in, the deer will become extinct. It is also known that if the deer population rises above the carrying capacity M, the population will decrease back to M through disease and malnutrition.

Require:
a. Discuss the reasonableness of the following model for the growth rate of the deer population as a function of time:

Dp/dt= rP(M-P) (P-m)

where P is the population of the deer and r is a positive constant of proportionality. Include a phase line.
b. Explain how this model differs from the logistic model dP/dt= r P(M — P). Is it better or worse than the logistic model?
c. Show that if P >M for all t, then limt→[infinity] P(t)= M.
d. What happens if P e. Discuss the solutions to the differential equation. What are the equilibrium points of the model? Explain the dependence of the steady-state value of P on the initial values of P. About how many permits should be issued?