A fitness-driven cross-diffusion system from polulation dynamics as a gradient flow

TL;DR

This paper models dispersal of multiple interacting populations using a gradient flow framework in the space of Radon measures, proving existence and long-term convergence to equilibrium in a complex cross-diffusion PDE system.

Contribution

It extends previous single-population models to multiple populations, establishing a gradient flow structure and proving exponential convergence to equilibrium.

Findings

Existence of global non-negative weak solutions.

Solutions converge exponentially to an ideal free distribution.

Model is identified as a gradient flow in the space of Radon measures.

Abstract

We consider a fitness-driven model of dispersal of $N$ interacting populations, which was previously studied merely in the case $N=1$. Based on some optimal transport distance recently introduced, we identify the model as a gradient flow in the metric space of Radon measures. We prove existence of global non-negative weak solutions to the corresponding system of parabolic PDEs, which involves degenerate cross-diffusion. Under some additional hypotheses and using a new multicomponent Poincar\'e-Beckner functional inequality, we show that the solutions converge exponentially to an ideal free distribution in the long time regime.