The effects of unequal diffusion coefficients on periodic travelling waves in oscillatory reaction–diffusion systems

Many oscillatory biological systems show periodic travelling waves. These are often modelled using coupled reaction–diffusion equations. However, the effects of different movement rates (diffusion coefficients) of the interacting components on the predictions of these equations are largely unknown. Here we investigate the ways in which varying the diffusion coefficients in such equations alters the wave speed, time period, wavelength, amplitude and stability of periodic wave solutions. We focus on two sets of kinetics that are commonly used in ecological applications: lambda–omega equations, which are the normal form of an oscillatory coupled reaction–diffusion system close to a supercritical Hopf bifurcation, and a standard predator–prey model. Our results show that changing the ratio of the diffusion coefficients can significantly alter the shape of the one-parameter family of periodic travelling wave solutions. The position of the boundary between stable and unstable waves also depends on the ratio of the diffusion coefficients: in all cases, stability changes through an Eckhaus (‘sideband’) instability. These effects are always symmetrical in the two diffusion coefficients for the lambda–omega equations, but are asymmetric in the predator–prey equations, especially when the limit cycle of the kinetics is of large amplitude. In particular, there are two separate regions of stable waves in the travelling wave family for some parameter values in the predator–prey scenario. Our results also show the existence of a one-parameter family of travelling waves, but not necessarily a Hopf bifurcation, for all values of the diffusion coefficients. Simulations of the full partial differential equations reveals that varying the ratio of the diffusion coefficients can significantly change the properties of periodic travelling waves that arise from particular wave generation mechanisms, and our analysis of the travelling wave families assists in the understanding of these effects.