Cyclic competition models

Biology | Chaos | Parabolic | Waves

We now consider an example of a reaction-diffusion system based on the following reaction kinetics:

$$\begin{array}{rl}\frac{\partial u}{\partial t}& =D_u{\mathrm{\nabla }}^{2}u+u(1-u-av-bw),\\ \frac{\partial v}{\partial t}& =D_v{\mathrm{\nabla }}^{2}v+v(1-bu-v-aw),\\ \frac{\partial w}{\partial t}& =D_w{\mathrm{\nabla }}^{2}w+w(1-au-bv-w).\end{array}$$

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 rock-paper-scissors 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 Turing-like instability.

• Wave-induced spatiotemporal chaos can even occur without a Turing-like instability; see this wave simulation for an example.

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