Fixation in the stochastic Lotka-Volterra model with small fitness trade-offs

TL;DR

This paper derives an explicit formula for fixation probability in a stochastic two-species Lotka-Volterra model, revealing how small fitness trade-offs significantly influence stochastic outcomes despite minimal effects on deterministic dynamics.

Contribution

It introduces a novel approximation method based on a fast timescale to solve the backward Kolmogorov equation explicitly for fixation probabilities.

Findings

Small fitness trade-offs can greatly affect fixation outcomes in stochastic models.

The derived formula provides insights into the role of trade-offs in evolutionary dynamics.

Deterministic dynamics may not predict stochastic fixation probabilities accurately.

Abstract

We study the probability of fixation in a stochastic two-species competition model. By identifying a naturally occurring fast timescale, we derive an approximation to the associated backward Kolmogorov equation that allows us to obtain an explicit closed form solution for the probability of fixation of either species. We use our result to study fitness tradeoff strategies and show that, despite some tradeoffs having nearly negligible effects on the corresponding deterministic dynamics, they can have large implications for the outcome of the stochastic system.