Pattern formation in parabolic equations containing hysteresis with diffusive thresholds

TL;DR

This paper studies a reaction-diffusion system with hysteresis-driven discontinuous reactions, modeling bacterial phenotype switching, and proves pattern formation in the phenotype space driven by memory effects.

Contribution

It introduces a mathematical framework for pattern formation in systems with hysteresis and demonstrates the emergence of patterns due to memory effects in a biological context.

Findings

Pattern formation in phenotype space is proven mathematically.

Hysteresis induces stable pattern formation.

The model applies to bacterial phenotype evolution.

Abstract

We consider a reaction-diffusion system with discontinuous reaction terms modeled by non-ideal relays. The system is motivated by an epigenetic population model of evolution of two-phenotype bacteria which switch phenotype in response to variations of environment. We prove the formation of patterns in the phenotype space. The mechanism responsible for pattern formation is based on memory (hysteresis) of the non-ideal relays.