In an open-access fishery, fishermen are free to come and go as they please. The fishing effort E is determined by the opportunity to make a profit. Let c be the cost of operation, and p the price the fishermen get for their catch H. Profit is given by
P = pH - cE.
where
H=qEN
is the harvest rate, which is proportional to the amount of fish N there are in the fishery and the effort E expended to catch them. When there is profit to be made, the fishermen would increase their effort in the hope of making even more profit. Thus
(d/dt)E = aP(t).(1)
where a is a proportionality constant.
The equation governing the fish population is
(d/dt)N = rN(1-N/K)-H. (2)
Determine the equilibria (N*, E*) and their stability of the coupled system (1) and (2).(Note that the only two unknows are N(t) and E(t).) Assume c/(pqk) < 1.