Cyclic competition models
We now consider an example of a reactiondiffusion system based on the following reaction kinetics:
\[\begin{aligned}\pd{u}{t}&=D_u\nabla^2 u+u(1uavbw),\\ \pd{v}{t}&=D_v\nabla^2 v+v(1buvaw),\\ \pd{w}{t}&=D_w\nabla^2 w+w(1aubvw).\end{aligned}\]These are an example of a generalised Lotka–Volterra system. If we set $a < 1 < b$, then each population outcompetes another, and hence their relative fitness forms a cycle. This kind of model is also known as a spatial rockpaperscissors game.
To make things more interesting, we will allow the species to diffuse at different rates.

Load the interactive simulation

This begins with an initially structured population that eventually devolves into a complex spatiotemporal motion, with spiral waves a dominant feature.

This system exhibits a wide range of behaviours. One of the most interesting is that if you allow spiral waves to form, and then set all diffusion coefficients to be equal (e.g. by changing $D_u$ to $0.5$), the system will still admit spiral waves despite not having a Turinglike instability.