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

Abstract

This paper concerns a general class of PDE–ODE reaction–diffusion systems, which exhibits a singular fast-reaction limit towards a reaction–diffusion equation coupled to a scalar hysteresis operator.

As applicational motivation, 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 natural models of consume rate functions such that the dynamics for the food stock 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 at the same time.

Finally, we present a new kind of hysteresis–diffusion-driven instability behaviour caused by the nonlinear coupling between a reaction–diffusion equation and a scalar generalised play operator. We discuss in detail how this coupling with a generalised play operator can lead to spatially inhomogeneous large-time behaviour or equilibration to a homogeneous state.