Constant Yield Harvesting. In this problem, we assume that fishare caught at a constant rate h independent of the size of the fishpopulation, that is, the harvesting rate H(y, t) = h. Then ysatisfies dy/dt = r(1 ? y/K )y ? h = f (y). (ii) The assumption ofa constant catch rate h may be reasonable when y is large butbecomes less so when y is small.

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

(b) Show that y1 is unstable and y2 is asymptoticallystable.

(c) From a plot of f (y) versus y, show that if the initialpopulation y0 > y1, then y ? y2 as t ? ?, but if y0 < y1,then y decreases as t increases. Note that y = 0 is not anequilibrium point, so if y0 < y1, then extinction will bereached in a finite time.

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

(d). The fishery is considered to be overexploited if y isreduced 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 maximumsustainable yield is hm = rK/4, corresponding to the equilibriumvalue y=K/2. Observe that hm has the same value as Y m in Problem1(d). The fishery is considered to be overexploited if y is reducedto a level below K/2

*Using Matlab