Cross-Diffusion Theory for Overcrowding Dispersal in Interacting Species System

Abstract

This work introduces a new class of cross-diffusion systems for studying overcrowding dispersal of two species. The approach, based on proximal minimization energy through a minimum flow process, offers a potential generalization of existing segregation models. Unlike prior methods using PDEs or $W_2$-Wasserstein flows, it establishes a well-posed PDE framework for capturing the interplay between diffusion and concentration gradients. This framework has the potential to significantly improve our understanding of how cross-diffusion shapes spatial patterns, coexistence, and overall distribution of multiple species. Notably, for homogeneous cases, the approach definitely leads to a well-defined PDE grounded in a new general $H^{-1}$-theory specifically developed for overcrowding dispersal. This theory provides a robust foundation for further analysis.