Periodic traveling waves generated by invasion in cyclic predator-prey systems: The effect of unequal dispersal

Periodic traveling waves (wavetrains) have been an invaluable tool in the understanding of spatiotemporal oscillations observed in ecological data sets. Various mechanisms are known to trigger this behavior, but here we focus on invasion, resulting in a predator--prey-type interaction. Previous work has focused on the normal form reduction of PDE models to the well-understood $\lambda$-$\omega$ equations near a Hopf bifurcation, though this is valid only when assuming an equal rate of dispersion for both predators and prey---an unrealistic assumption for many ecosystems. By relaxing this constraint, we obtain the complex Ginzburg--Landau normal form equation, which has a one-parameter family of periodic traveling wave solutions, parametrized by the amplitude. We derive a formula for the wave amplitude selected by invasion before investigating the stability of the solutions. This gives us a complete description of small-amplitude periodic traveling waves in the governing model ecosystem.