Boundedness of a class of discretized reaction-diffusion systems

TL;DR

This paper investigates conditions under which discretized reaction-diffusion systems with Neumann boundary conditions remain bounded over time, highlighting differences from continuous systems and introducing Lyapunov-like functions for analysis.

Contribution

It provides the first sufficient conditions for the boundedness of discretized reaction-diffusion systems with Neumann boundaries using Lyapunov-like functions.

Findings

Lyapunov-like functions guarantee boundedness of discretized systems.

Boundedness depends on the existence of specific Lyapunov-like functions.

Application to example systems illustrates when boundedness criteria are met or not.

Abstract

Although the spatially continuous version of the reaction-diffusion equation has been well studied, in some instances a spatially-discretized representation provides a more realistic approximation of biological processes. Indeed, mathematically the discretized and continuous systems can lead to different predictions of biological dynamics. It is well known in the continuous case that the incorporation of diffusion can cause diffusion-driven blow-up with respect to the $L^{\infty}$ norm. However, this does not imply diffusion-driven blow-up will occur in the discretized version of the system. For example, in a continuous reaction-diffusion system with Dirichlet boundary conditions and nonnegative solutions, diffusion-driven blow up occurs even when the total species concentration is non-increasing. For systems that instead have homogeneous Neumann boundary conditions, it is currently unknown whether this deviation between the continuous and discretized system can occur. Therefore, it is worth examining the discretized system independently of the continuous system. Since no criteria exist for the boundedness of the discretized system, the focus of this paper is to determine sufficient conditions to guarantee the system with diffusion remains bounded for all time. We consider reaction-diffusion systems on a 1D domain with homogeneous Neumann boundary conditions and non-negative initial data and solutions. We define a Lyapunov-like function and show that its existence guarantees that the discretized reaction-diffusion system is bounded. These results are considered in the context of three example systems for which Lyapunov-like functions can and cannot be found.