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

Abstract

Dynamical systems containing heteroclinic cycles and networks can be invoked as models of intransitive competition between three or more species. When populations are assumed to be well-mixed, a system of ordinary differential equations (ODEs) describes the interaction model. Spatially extending these equations with diffusion terms creates a system of partial differential equations which captures both the spatial distribution and mobility of species. In one spatial dimension, travelling wave solutions can be observed, which correspond to periodic orbits in ODEs that describe the system in a steady-state travelling frame of reference. These new ODEs also contain a heteroclinic structure. For three species in cyclic competition, the topology of the heteroclinic cycle in the well-mixed model is preserved in the steady-state travelling frame of reference. We demonstrate that with four species, the heteroclinic cycle which exists in the well-mixed system becomes a heteroclinic network in the travelling frame of reference, with additional heteroclinic orbits connecting equilibria not connected in the original cycle. We find new types of travelling waves which are created in symmetry-breaking bifurcations and destroyed in an orbit flip bifurcation with a cycle between only two species. These new cycles explain the existence of ‘defensive alliances’ observed in previous numerical experiments. We further describe the structure of the heteroclinic network for any number of species, and we conjecture how these results may generalise to systems of any arbitrary number of species in cyclic competition.

Heteroclinic networks for brain dynamics

“…Possible effects of diffusive coupling may go far beyond simple synchronization of the entire grid. When its effect is studied for cyclic competition between four species in one dimension and the system is described in a steady-state traveling frame of reference, a bifurcation analysis, in particular as a function of the speed of the wave, becomes possible and reveals additional HNs with additional heteroclinic orbits due to the diffusive coupling (Dijkema and Postlethwaite, 2022). The traveling wave solutions amount to periodic orbits in ODEs in this new reference frame.…”

“…The traveling wave solutions amount to periodic orbits in ODEs in this new reference frame. Within this approach, it seems possible to reproduce the formation of alliances between players in cyclic competition games (Durney et al, 2011;Roman et al, 2012;Dobramysl et al, 2018;Dijkema and Postlethwaite, 2022) 1 What is specific for a coupling of HUs on a spatial grid is coupling via diffusion between repetitive units; an alternative interpretation of this set is a system of specifically coupled HUs in a large phase space (assigned to a single site).…”