Biological Waves: Multi-Species Reaction Diffusion Models

Abstract

In the last chapter we saw that if we allowed spatial dispersal in the single reactant or species, travelling wave front solutions were possible. Such solutions effected a smooth transition between two steady states of the space independent system. For example, in the case of the Fisher equation (11.6), wavefront solutions joined the steady state u = 0 to the one at u = 1 as shown in the evolution to a propagating wave in Fig. 11.2. In Section 11.5, where we considered a model for the spatial spread of the spruce budworm, we saw how such travelling wave solutions could be found to join any two steady states of the spatially independent dynamics. In this and the next three chapters, we shall be considering systems where several species or reactants are involved, concentrating on reaction diffusion mechanisms, of the type derived in Section 9.2 (Equation (9.18)) namely

$$ \frac{{\partial u}}{{\partial t}} = f(u) + D{\nabla ^2}u $$

((12.1))

where u is the vector of reactants, f the nonlinear reaction kinetics and D the matrix of diffusivities, taken here to be constant.