Asymptotics of sign-changing patterns in hysteretic systems with diffusive thresholds

TL;DR

This paper analyzes a reaction-diffusion system with hysteresis that models phenotypic switching in populations, providing asymptotic formulas for pattern formation and its dynamics.

Contribution

It introduces asymptotic formulas for patterns in a reaction-diffusion system with hysteretic relay operators, advancing understanding of phenotype distribution dynamics.

Findings

Derived asymptotic formulas for pattern structures

Characterized the process of pattern formation

Linked phenotypic switching to environmental changes

Abstract

We consider a reaction-diffusion system including discontinuous hysteretic relay operators in reaction terms. This system is motivated by an epigenetic model that describes the evolution of a population of organisms which can switch their phenotype in response to changes of the state of the environment. The model exhibits formation of patterns in the space of distributions of the phenotypes over the range of admissible switching strategies. We propose asymptotic formulas for the pattern and the process of its formation.