  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.
