Analysis of a splitting-differentiation population model leading to cross-diffusion

TL;DR

This paper extends a population splitting model to derive cross-diffusion PDEs, proves solution existence in 1D, and demonstrates improved numerical stability over existing methods.

Contribution

It introduces a new analytical approach to cross-diffusion models from population splitting, proving existence and stability results.

Findings

Proved existence of solutions in 1D for a specific cross-diffusion model.

Demonstrated the regularity of solutions is best represented in bounded variation spaces.

Numerical experiments show the new scheme outperforms existing models in stability.

Abstract

Starting from the dynamical system model capturing the splitting-differentiation process of populations, we extend this notion to show how the speciation mechanism from a single species leads to the consideration of several well known evolution cross-diffusion partial differential equations. Among the different alternatives for the diffusion terms, we study the model introduced by Busenberg and Travis, for which we prove the existence of solutions in the one-dimensional spatial case. Using a direct parabolic regularization technique, we show that the problem is well posed in the space of bounded variation functions, and demonstrate with a simple example that this is the best regularity expected for solutions. We numerically compare our approach to other alternative regularizations previously introduced in the literature, for the particular case of the contact inhibition problem. Simulation experiments indicate that the numerical scheme arising from the approximation introduced in this article outperforms those of the existent models from the stability point of view.