Extinction in Lotka-Volterra model

TL;DR

This paper studies how stochastic fluctuations in the Lotka-Volterra predator-prey model lead to eventual extinction, with extinction times scaling as a power-law of population sizes, contrasting with exponential scaling in stable models.

Contribution

It demonstrates how population discreteness causes extinction in marginally stable oscillatory systems like Lotka-Volterra, revealing power-law scaling of extinction times.

Findings

Extinction time scales as a power-law of population size.

Fluctuation effects destroy mean-field stability.

Contrasts with exponential scaling in stable models.

Abstract

Competitive birth-death processes often exhibit an oscillatory behavior. We investigate a particular case where the oscillation cycles are marginally stable on the mean-field level. An iconic example of such a system is the Lotka-Volterra model of predator-prey competition. Fluctuation effects due to discreteness of the populations destroy the mean-field stability and eventually drive the system toward extinction of one or both species. We show that the corresponding extinction time scales as a certain power-law of the population sizes. This behavior should be contrasted with the extinction of models stable in the mean-field approximation. In the latter case the extinction time scales exponentially with size.