Stable patterns with jump discontinuity in systems with Turing instability and hysteresis

TL;DR

This paper explores a novel pattern formation mechanism in reaction-diffusion-ODE systems where diffusion-driven instability and hysteresis lead to stable patterns with jump discontinuities, extending classical Turing pattern theory.

Contribution

It introduces conditions for stability of discontinuous patterns and demonstrates a new pattern formation mechanism involving DDI and hysteresis in reaction-diffusion-ODE models.

Findings

Discontinuous patterns can be stable under certain conditions.

Hysteresis effects enable far-from-equilibrium pattern formation.

The model explains de novo pattern formation involving DDI and hysteresis.

Abstract

Classical models of pattern formation are based on diffusion-driven instability (DDI) of constant stationary solutions of reaction-diffusion equations, which leads to emergence of stable, regular Turing patterns formed around that equilibrium. In this paper we show that coupling reaction-diffusion equations with ordinary differential equations (ODE) may lead to a novel pattern formation phenomenon in which DDI causes destabilization of both constant solutions and Turing patterns. Bistability and hysteresis effects in the null sets of model nonlinearities yield formation of far from the equilibrium patterns with jump discontinuity. We derive conditions for stability of stationary solutions with jump discontinuity in a suitable topology which allows disclosing the discontinuity points and leads to the definition of ({\epsilon}0 , A)-stability. Additionally, we provide conditions on stability of patterns in a quasi-stationary model reduction. The analysis is illustrated on the example of three-component model of receptor-ligand binding. The proposed model provides an example of a mechanism of de novo formation of far from the equilibrium patterns in reaction-diffusion-ODE models involving co-existence of DDI and hysteresis.