Revisiting Cross-Diffusion for Overcrowding Dispersal in Interacting Species System

This work introduces and study a novel class of reaction-diffusion systems to model the evolution of two interacting species that disperse to avoid overcrowding. Our approach employs the principle of proximal minimization energy, implemented through a minimum flow proximal algorithm. This framework offers a potential generalization of existing systems commonly used in the theory of segregation for interacting species dynamics. Beyond existence and uniqueness, this approach allows us to capture the dynamic interplay between diffusion rates and concentration gradients of each species, and has the potential to significantly advance our understanding of how cross-diffusion shapes the spatial distribution, coexistence, and pattern formation of multiple species within a system.