BOUNDEDNESS OF A CLASS OF SPATIALLY DISCRETE REACTION-DIFFUSION SYSTEMS.

Abstract

Although the spatially discrete reaction-diffusion equation is often used to describe biological processes, the effect of diffusion in this framework is not fully understood. In the spatially continuous case, the incorporation of diffusion can cause blow-up with respect to the norm, and criteria exist to determine whether the system is bounded for all time. However, no equivalent criteria exist for the discrete reaction-diffusion system. Due to the possible dynamical differences between these two system types and the advantage of using the spatially discrete representation to describe biological processes, it is worth examining the discrete system independently of the continuous system. Therefore, the focus of this paper is on determining sufficient conditions to guarantee that the discrete reaction-diffusion system is bounded for all time. We consider reaction-diffusion systems on a 1D domain with homogeneous Neumann boundary conditions and nonnegative initial data and solutions. We define a Lyapunov-like function and show that its existence guarantees that the discrete reaction-diffusion system is bounded. These results are considered in the context of four example systems for which Lyapunov-like functions can and cannot be found.