Zero-one survival behavior of cyclically competing species

TL;DR

This paper investigates the survival probabilities of species in cyclic competition, revealing a zero-one law where the weakest species survives with probability approaching one in large populations, supported by simulations and analysis.

Contribution

It uncovers a simple zero-one survival law in cyclic competition models with asymmetric interactions, providing analytical and simulation evidence.

Findings

Weakest species survives with probability approaching one in large populations.

Other species are guaranteed to go extinct under asymmetric interactions.

The zero-one law holds in models with neutrally stable coexistence.

Abstract

Coexistence of competing species is, due to unavoidable fluctuations, always transient. In this Letter, we investigate the ultimate survival probabilities characterizing different species in cyclic competition. We show that they often obey a surprisingly simple, though non-trivial behavior. Within a model where coexistence is neutrally stable, we demonstrate a robust zero-one law: When the interactions between the three species are (generically) asymmetric, the `weakest' species survives at a probability that tends to one for large population sizes, while the other two are guaranteed to extinct. We rationalize our findings from stochastic simulations by an analytic approach.