Pattern formation in a diffusion-ODE model with hysteresis

TL;DR

This paper investigates how coupling diffusion with nonlinear intracellular processes and hysteresis can lead to pattern formation, especially sharp transitions, in biological systems modeled by reaction-diffusion and ODEs.

Contribution

It provides a systematic description of stationary solutions with transition layers in diffusion-ODE models exhibiting hysteresis, highlighting a novel pattern formation mechanism.

Findings

Stationary solutions include transition and boundary layers.

Discontinuous solutions arise with non-diffusing variables exhibiting hysteresis.

The model explains formation of sharp biological patterns.

Abstract

Coupling diffusion process of signaling molecules with nonlinear interactions of intracellular processes and cellular growth/transformation leads to a system of reaction-diffusion equations coupled with ordinary differential equations (diffusion-ODE models), which differ from the usual reaction-diffusion systems. One of the mechanisms of pattern formation in such systems is based on the existence of multiple steady states and hysteresis in the ODE subsystem. Diffusion tries to average different states and is the cause of spatio-temporal patterns. In this paper we provide a systematic description of stationary solutions of such systems, having the form of transition or boundary layers. The solutions are discontinuous in the case of non-diffusing variables whose quasi-stationary dynamics exhibit hysteresis. The considered model is motivated by biological applications and elucidates a possible mechanism of formation of patterns with sharp transitions.