Two simple criterion to prove the existence of patterns in reaction-diffusion models of two components

The aim of this work is to study the effect of diffusion on the stability of the equilibria in a general two-components reaction-diffusion system with Neumann boundary conditions in the space of continuous functions. As by product, we establish sufficient conditions on the diffusive coefficients and other parameters for such a reaction-diffusion model to exhibit patterns and we analyze their stability. We apply the results obtained in this paper to explore under which parameters values a Turing bifurcation can occur, given rise to non uniform stationary solutions (patterns) for a reaction-diffusion predator-prey model with variable mortality and Hollyn’s type II functional response.