Question

Constant Yield Harvesting. In this problem, we assume that fish are caught at a constant rate h independent of the size of the fish population, that is, the harvesting rate H(y, t) = h. Then y satisfies dy/dt = r(1 − y/K )y − h = f (y). (ii) The assumption of a constant catch rate h may be reasonable when y is large but becomes less so when y is small.

(a) If h < rK/4, show that Eq. (ii) has two equilibrium points y1 and y2 with y1 < y2; determine these points.

(b) Show that y1 is unstable and y2 is asymptotically stable.

(c) From a plot of f (y) versus y, show that if the initial population y0 > y1, then y → y2 as t → ∞, but if y0 < y1, then y decreases as t increases. Note that y = 0 is not an equilibrium point, so if y0 < y1, then extinction will be reached in a finite time.

(d) If h > rK/4, show that y decreases to zero as t increases regardless of the value of y0. (e) If h = rK/4, show that there is a single equilibrium point y = K/2 and that this point is semistable. Thus the maximum sustainable yield is hm = rK/4, corresponding to the equilibrium value y=K/2. Observe that hm has the same value as Y m in Problem 1

(d). The fishery is considered to be overexploited if y is reduced to a level below K/2.

(e) If h = rK/4, show that there is a single equilibrium point y = K/2 and that this point is semistable. Thus the maximum sustainable yield is hm = rK/4, corresponding to the equilibrium value y=K/2. Observe that hm has the same value as Y m in Problem 1(d). The fishery is considered to be overexploited if y is reduced to a level below K/2

*Using Matlab

Answer 1