Control of biological models with hysteresis

This article presents a control problem in population dynamics when diffusive and hysteretic effects in the evolution of population are taken into account. Namely, we consider a diffusive prey–predator model, in a bounded domain, with a parameter corresponding to the quantity of food for the preys. In the model, the evolution of the food density is described by a hysteresis operator of the generalized play type whose inputs are the densities of preys/predators. Roughly speaking, the rate of change of the food density depends not only on the present state of preys/predators, but also on their immediately preceding density history. Our main concern is to extend the previous research of this model by controlling the food supply and to study the well-posedness of the corresponding dynamical control problem.