  This paper addresses a nonlinear partial differential control system arising
in population dynamics. The system consist of three diffusion equations
describing the evolutions of three biological species: prey, predator, and food
for the prey or vegetation. The equation for the food density incorporates a
hysteresis operator of generalized stop type accounting for underlying
hysteresis effects occurring in the dynamical process. We study the problem of
minimization of a given integral cost functional over solutions of the above
system. The set-valued mapping defining the control constraint is
state-dependent and its values are nonconvex as is the cost integrand as a
function of the control variable. Some relaxation-type results for the
minimization problem are obtained and the existence of a nearly optimal
solution is established.
