Dynamics of a linearly-perturbed May-Leonard competition model

TL;DR

This paper investigates how small linear mutations in the May-Leonard competition model alter its dynamics, revealing new bifurcation behaviors and stable states while disrupting some original heteroclinic connections.

Contribution

It introduces a linear mutation perturbation to the May-Leonard model and analyzes its impact on the system's dynamics, uncovering new bifurcations and stable equilibria.

Findings

Retains some classical dynamics at small mutation rates

Identifies fold bifurcations with emerging and coalescing equilibria

Discovers new stable fixed points absent in the original model

Abstract

The May--Leonard model was introduced to examine the behavior of three competing populations where rich dynamics, such as limit cycles and nonperiodic cyclic solutions, arise. In this work, we perturb the system by adding the capability of global mutations, allowing one species to evolve to the other two in a linear manner. We find that for small mutation rates the perturbed system not only retains some of the dynamics seen in the classical model, such as the three-species equal-population equilibrium bifurcating to a limit cycle, but also exhibits new behavior. For instance, we capture curves of fold bifurcations where pairs of equilibria emerge and then coalesce. As a result, we uncover parameter regimes with new types of stable fixed points that are distinct from the single- and dual-population equilibria characteristic of the original model. On the contrary, the linearly-perturbed system fails to maintain heteroclinic connections that exist in the original system. In short, a linear perturbation proves to be significant enough to substantially influence the dynamics, even with small mutation rates.