About reaction–diffusion systems involving the Holling-type II and the Beddington–DeAngelis functional responses for predator–prey models

We consider in this paper a microscopic model (that is, a system of three reaction–diffusion equations) incorporating the dynamics of handling and searching predators, and show that its solutions converge when a small parameter tends to 0 towards the solutions of a reaction–cross diffusion system of predator–prey type involving a Holling-type II or Beddington–DeAngelis functional response. We also provide a study of the Turing instability domain of the obtained equations and (in the case of the Beddington–DeAngelis functional response) compare it to the same instability domain when the cross diffusion is replaced by a standard diffusion.