Dynamics of nested, self-similar winnerless competition in time and space

Abstract

We construct n levels of nested, self-similar winnerless competition dynamics of which we explicitly work out the first three levels in the framework of generalized Lotka-Volterra equations. We choose microscopic rules such that the competition in the form of rock-paper-scissors is played between metapopulations, populations, and individuals at the same time. The trajectory of individual activities moves through a hierarchically structured heteroclinic network in a desired way. The hierarchy in structure is able to induce a separation of timescales that translates into nested spirals if the heteroclinic networks are coupled via diffusion on a spatial grid. For sufficiently strong diffusion the dynamics of interacting heteroclinic networks gets synchronized between the sites, which amounts to a large dimensional reduction of phase space. Possible applications lie in ecology and in brain dynamics. Our model reproduces in particular chunking dynamics with slow oscillations modulating fast oscillations modulating faster ones as observed in brain dynamics.