A hierarchical heteroclinic network: Controlling the time evolution along its paths

TL;DR

This paper investigates a hierarchical heteroclinic network model in winnerless competition, demonstrating how tuning parameters and noise influence the transition between different species coexistence states and time scales.

Contribution

It introduces a hierarchical heteroclinic network framework with tunable parameters to control time scales and state transitions in winnerless competition models.

Findings

Tuning predation rates creates long time scales on the higher level.

A single bifurcation parameter can switch the system between different heteroclinic cycles.

Increasing noise strength can induce transitions between species coexistence states.

Abstract

We consider a heteroclinic network in the framework of winnerless competition of species. It consists of two levels of heteroclinic cycles. On the lower level, the heteroclinic cycle connects three saddles, each representing the survival of a single species; on the higher level, the cycle connects three such heteroclinic cycles, in which nine species are involved. We show how to tune the predation rates in order to generate the long time scales on the higher level from the shorter time scales on the lower level. Moreover, when we tune a single bifurcation parameter, first the motion along the lower and next along the higher-level heteroclinic cycles are replaced by a heteroclinic cycle between 3-species coexistence-fixed points and by a 9-species coexistence-fixed point, respectively. We also observe a similar impact of additive noise. Beyond its usual role of preventing the slowing-down of heteroclinic trajectories at small noise level, its increasing strength can replace the lower-level heteroclinic cycle by 3-species coexistence fixed-points, connected by an effective limit cycle, and for even stronger noise the trajectories converge to the 9-species coexistence-fixed point. The model has applications to systems in which slow oscillations modulate fast oscillations with sudden transitions between the temporary winners.