A reaction-diffusion equation for a cyclic system with three components

Abstract

The one-dimensional reaction-diffusion equations for the process

$$A + B \to 2A,B + C \to 2B,C + A \to 2C$$

((D))

are extended to include the counteracting reactions

$$A + 2B \to 3B,B + 2C \to 3C,C + 2A \to 3A$$

((R))

which have a reaction rate α relative to the direct process (D). This process can be seen as a three-component version of the reaction which is described by the Fisher-Kolmogorov equation. The fixed points of the equations are studied as a function of α. It is shown that the equations admit solutions which consist of a series of traveling fronts. Other solutions exist which are traveling periodic waves. A very remarkable fact is that for these waves exact expressions can be constructed.