The aim of this paper is to show with a functional Beddington-DeAngelis response, that cross-diffusion plays an important role in the phenomenon of toxin-productionphytoplankton (TPP) in the dynamics of zooplankton and phytoplankton system. The
demonstration tools are mainly based on the theory of fixed point indices and analytical techniques. Two-dimensional numerical simulations have established the formation
of spatial patterns and a release threshold of the toxin, above which we talk about the
phytoplankton proliferation

Toxin effect; reaction-diffusion system; index theory; pattern formation