Boundedness of a class of discretized reaction-diffusion systems

Research in systems biology has led to the development of network-analysis tools that assume well-mixed pools of reactants. If space is broken up into discrete compartments, these same tools can be applied to study spatially heterogeneous systems. For instance, diffusion can be modeled as an additional reaction across compartment boundaries. It is well known that in spatially continuous systems, the incorporation of diffusion can lead to the blow-up of solutions. As shown here, diffusion-driven blow-up can also occur in spatially discretized systems. This phenomenon is biologically unrealistic and implies the system cannot be studied under steady-state assumptions. Therefore, the focus of this paper is to determine sufficient conditions for the discretized system with diffusion to remain bounded for all time. We consider reaction-diffusion systems on a 1D domain with 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. For some systems, it is possible to quickly determine whether or not a Lyapunov-like function exists. We discuss these types of systems and present examples of a bounded and unbounded system. In the future, we would like to extend these results to determine conditions for the system to remain bounded in the continuum limit.