Deterministic extinction by mixing in cyclically competing species

TL;DR

This paper analyzes a cyclically competing species model with global mixing, demonstrating that deterministic extinction occurs at sufficiently high mixing rates for any number of species, supported by analytical and numerical evidence.

Contribution

It provides the first analytical proof that finite mixing rates induce deterministic extinction in cyclically competing species models.

Findings

Extinction occurs deterministically at high mixing rates for all species numbers N≥3.

Numerical results show trajectories toward extinction and bifurcations as mixing rate varies.

Abstract

We consider a cyclically competing species model on a ring with global mixing at finite rate, which corresponds to the well-known Lotka-Volterra equation in the limit of infinite mixing rate. Within a perturbation analysis of the model from the infinite mixing rate, we provide analytical evidence that extinction occurs deterministically at sufficiently large but finite values of the mixing rate for any species number $N\ge3$. Further, by focusing on the cases of rather small species numbers, we discuss numerical results concerning the trajectories toward such deterministic extinction, including global bifurcations caused by changing the mixing rate.