In this lab, we examine a simple fish harvesting problem adapted from [68]. Suppose at some point, designated as t = 0, a fish population is introduced into a fishery of some kind (for example, an artificial tank or a netted area in a body of water). Let x(t) be the population level (scaled) at time t, where x(0) = x0 > 0 is the initial concentration, as determined by the introduction. Suppose that, when introduced, the fish are very small and that the average mass of the fish at time t = 0 is essentially 0. Further, the average mass of the fish as a function of time is given by

fmass(t) = kt

t+ 1 ,

where k is the maximum mass of this species. We will assume the time interval [0, T ], over which we are to consider harvesting, is small enough that no reproduction will occur. Specifically, the population will have no natural growth. Let u(t) be the harvest rate at time t andm be the natural death rate of the fish. We wish to maximize the total mass harvested over the interval taking into account the cost of harvesting. So, the problem can be stated

max u

A kt

t+ 1 x(t)u(t)− u(t)2 dt

subject to x′(t) = −(m+ u(t))x(t), x(0) = x0, 0 ≤ u(t) ≤M.