Existence and stability of periodic travelling waves: who will prevail in a Rock-Paper-Scissors game?

Cris Hasan, University of Auckland. Part of the LAND seminar series.

We study a Rock-Paper-Scissors model that describes the spatiotemporal evolution of three competing populations, or strategies, in evolutionary game theory and biology. The dynamics of the model is determined by a set of partial differential equations (PDEs) that features travelling waves (TWs) in one spatial dimension and spiral waves in two spatial dimensions. We focus on periodic (in both space and time) TWs, a phenomenon that is also inherently found in other reaction-diffusion models. The existence of periodic TWs can be established via the transformation of the PDE model into a system of ordinary differential equations (ODEs) under the assumption that the wave speed is constant. We explore the bifurcation diagram of the ODE system and investigate the existence of TWs as different parameters are varied. Determining the stability of periodic TWs is more challenging and requires a study of the essential spectrum of the linear operator of the periodic TWs. We compute this spectrum and the curve of instability with the continuation method developed in [Rademacher et al., Physica D, 2007]. We also develop a method for computing what we call belts of instability, which are indicators of the temporal expansion rates of unstable TWs. We finally show how these results compare with simulations of the PDE model.