Fast reaction limit for a Leslie–Gower model including preys, meso-predators and top-predators

We consider a system of three reaction–diffusion equations modeling the interaction between a prey species and two predators species including functional responses of Holly type-II and Leslie–Gower type. We propose a reaction–diffusion model with five equations with simpler functional responses which, in the fast reaction limit, allows to recover the zero-th order terms of the initially considered system. The diffusive part of the initial equations is however modified and cross diffusion terms pop up. We first study the equilibria of this new system and show that no Turing instability appears. We then rigorously prove a partial result of convergence for the fast reaction limit (in 1D and 2D).