Reaction-Diffusion Equations and Biochemical Pattern Formation

Abstract

Many biochemical reactions can be modeled using reaction-diffusion equations, some of which form patterns. Reaction-diffusion equations are partial differential equations that describe how the concentrations of two species change over time when subjected to diffusion and chemical reaction. Diffusion is represented in all cases by a first derivative with respect to time and a second derivative with respect to space. Chemical interactions are represented differently depending on the model, though they are typically described by nonlinear combinations of the dependent variables or concentrations. The two models explored here are the activator-substrate and the activator-inhibitor reaction-diffusion equations. We perform a stability analysis of each system and investigate solutions numerically under sinusoidal and random conditions with various parameters. The most interesting results come from the activator-inhibitor model under random conditions.