We apply the classical predator prey model of Volterrs dx/dt = ax - bxy, dy/dt = -cx + dxy, where x denotes the population of sardines (prey) and y the population of sharks (predators). We now subtract a term from each equation that accounts for the depletion of both species due to external fishing. If we fish each species at the same rate, then the Volterra
dx/dt = ax - bxy -fx.
dy/dt = -cy + dxy - fy
Where the constant f ≥ 0 denotes the "fishing" effort
(a) Find the equilibrium point of the system under fishing, and sketch a phase portrait for f = 0.5.
(b) Describe how the position of this fishing equilibrium has moved relative to the equilibrium point with no fishing (i.e., f = 0).
(c) When is it best to fish for sardines? For sharks? Just use common sense.
(d) Explain how this model describes the often unwanted consequences of spraying insecticide when a natural predator (good guys) controls an insect population (bad guys), but the insecticide kills both the natural predator and the insects?