Bang–Bang Control of a Prey–Predator Model with a Stable Food Stock and Hysteresis

A nonlinear partial differential control system is considered. This system arises, for instance, when modeling the evolution of populations in the vegetation–prey–predator framework. Our system accounts for the situation when the dependence of the vegetation density on the densities of prey and predators exhibits hysteretic character. At the same time, we do not allow for the diffusion of vegetation which is a reasonable assumption in many biological models of practical interest. Under a minimal set of requirements on the functions defining the hysteresis region, we first prove the existence of a solution to the corresponding (uncontrolled) system. Then, the existence of solutions for the control system is established and the bang–bang principle for it is obtained. The latter asserts the proximity of extremal solutions to a given solution of the control system.