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 Lotkaâ€“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.

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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.