Spatiotemporal stability of periodic travelling waves in a heteroclinic-cycle model

Abstract

We study a rock–paper–scissors model for competing populations that exhibits travelling waves in one spatial dimension and spiral waves in two spatial dimensions. A characteristic feature of the model is the presence of a robust heteroclinic cycle that involves three saddle equilibria. The model also has travelling fronts that are heteroclinic connections between two equilibria in a moving frame of reference, but these fronts are unstable. However, we find that large-wavelength travelling waves can be stable in spite of being made up of three of these unstable travelling fronts. In this paper, we focus on determining the essential spectrum (and hence, stability) of large-wavelength travelling waves in a cyclic competition model with one spatial dimension. We compute the curve of transition from stability to instability with the continuation scheme developed by Rademacher et al (2007 Physica D 229 166–83). We build on this scheme and develop a method for computing what we call belts of instability, which are indicators of the growth rate of unstable travelling waves. Our results from the stability analysis are verified by direct simulation for travelling waves as well as associated spiral waves. We also show how the computed growth rates accurately quantify the instabilities of the travelling waves.

Travelling waves and heteroclinic networks in models of spatially-extended cyclic competition

“…e (0)e λ + e T + D 6 y (1) e (0)e λ − e T (16) where 14) and ( 16). ) Therefore, the following relationship exists between these four constants, through the expression for tan θ(1) e ,…”

“…We are thus able to consider only the composition Ψ 13 ψ 1 , as fixed points of this composition will also be fixed points of the full return map Ψ 1 . We consider (16) without any dependence on time or the superscripts corresponding to the equilibria. Equation ( 16) becomes the following five equations…”

“…These are a resonance bifurcation, in which an algebraic condition on the eigenvalues is satisfied; a bifurcation of Belyakov-Devaney type, in which the imaginary part of a complex-conjugate pair of eigenvalues vanishes; or a bifurcation of orbit flip type. More recently, with Hasan and Osinga, the stability of these waves has been determined [16].…”