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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.If J * 11 < 0, we have J * 11− J * ∆22 + + J * 22 − J * ∆11 + − J * 12 − J * ∆21 − , so that condition (90) does not hold and, as in the case of linear diffusion, no Turing instability can appear. On the opposite, if J * 11 > 0, we have J * 11 + J * ∆22 + + J * 22 − J * ∆11 + − J * 12 − J * ∆21 − .Then, for any k = 0 (so that λ k > 0, we can select D 2 large enough and D 3 ∼ D 2 so that det J − (J * 11 J * ∆22 − J * 12 J * ∆21 )λ k < 0. Then, when D 1 > 0 is small enough, det M < 0 and the Turing instability appears.We then compare the Turing instability regions in the cases 2 and 3, that is when the characteristic matrices are

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