On hysteresis-reaction-diffusion systems: Singular fast-reaction limit derivation and nonlinear hysteresis feedback

TL;DR

This paper analyzes reaction-diffusion PDE-ODE systems with a fast-reaction limit leading to hysteresis effects, providing models applicable to biological systems and revealing potential for spatially inhomogeneous instabilities.

Contribution

It derives a general framework for PDE-ODE systems with singular fast-reaction limits resulting in hysteresis operators, applicable to biological and chemical models.

Findings

Fast-reaction limit leads to generalized play hysteresis operator

Models demonstrate convergence of food stock dynamics to hysteresis behavior

Example shows potential for spatially inhomogeneous large-time behavior

Abstract

This paper concerns a general class of PDE-ODE reaction-diffusion systems, which features a singular fast-reaction limit towards a reaction-diffusion equation coupled to a scalar hysteresis operator. As prototypical application, we present a PDE model for the growth of a population according to a given food supply coupled to an ODE for the turnover of a food stock. Under realistic conditions the stock turnover is much faster than the population growth yielding an intrinsic scaling parameter. We present two models of consume rate functions such that the resulting food stock dynamics converges to a generalised play operator in the associated fast-reaction-limit. We emphasise that the structural assumptions on the considered PDE-ODE models are quite general and that analogue systems might describe e.g. cell-biological buffer mechanisms, where proteins are stored and used in parallel. Finally, we discuss an explicit example showing that nonlinear coupling with a scalar generalised play operator can lead to spatially inhomogeneous large-time behaviour in a kind of hysteresis-diffusion driven instability.