Pattern formation in a cell migration model with aggregation and diffusion

TL;DR

This paper investigates pattern formation in a cell migration model combining aggregation and diffusion, revealing how interaction mechanisms lead to stable patterns and complex behaviors in both discrete and continuous settings.

Contribution

It introduces a discrete aggregation-diffusion model with Dirichlet boundary conditions and analyzes its convergence, steady states, and pattern formation mechanisms.

Findings

Discrete model converges to a positive steady state.

Aggregation induces rich asymptotic behaviors.

Patterns emerge even with minimal spatial points.

Abstract

In this paper, we study pattern formations in an aggregation and diffusion cell migration model with Dirichlet boundary condition. The formal continuum limit of the model is a nonlinear parabolic equation with a diffusivity which can become negative if the cell density is small and spatial oscillations and aggregation occur in the numerical simulations. In the classical diffusion migration model with positive diffusivity and non-birth term, species will vanish eventually with Dirichlet boundary. However, because of the aggregation mechanism under small cell density, the total species density is conservative in the discrete aggregation diffusion model. Also, the discrete system converges to a unique positive steady-state with the initial density lying in the diffusion domain. Furthermore, the aggregation mechanism in the model induces rich asymptotic dynamical behaviors or patterns even with 5 discrete space points which gives a theoretical explanation that the interaction between aggregation and diffusion induces patterns in biology. In the corresponding continuous backward forward parabolic equation, the existence of the solution, maximum principle, the asymptotic behavior of the solution is also investigated.