Sharks and Sardines with Fishing We apply the classical predator-prey model of Volterra: d...

Sharks and Sardines with Fishing We apply the classical predator-prey model of Volterra: 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 model becomes

where the constant f ≥ 0 denotes the “fishing” effort.

(a) Find the equilibrium point of the system under fishing, and skctch 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?