dr/dt = kr (1 − r/r∗) − αfr

where k > 0 is a constant representing the rabbit breeding rate, r∗ > 0 is the (constant) maximum sustainable rabbit population size in the absence of predation, f > 0 is the population of foxes, and α > 0 is the (constant) rate of predation of rabbits by foxes.

Suppose that the fox population, f, is constant. Solve the differential equation above, and determine

(a) the size of the rabbit population as t → ∞;

(b) the maximum predation rate α for which the rabbit population does not die out as t → ∞;

(c) the value of α which maximises αfr (the total number of rabbits caught) as t → ∞, and the corresponding rabbit population.