Mean Evolutionary Dynamics for Stochastically Switching Environments

TL;DR

This paper investigates how populations evolve in environments that change suddenly or periodically, analyzing the impact on evolutionary outcomes across various models and establishing relationships between environmental switching and long-term behavior.

Contribution

It introduces a simple relationship linking environmental switching probability, fitness gains, and long-term evolutionary outcomes across multiple models.

Findings

Derived a relationship between switching probability and fixation probabilities.

Analyzed stability when switching between different dynamic regimes.

Provided Lyapunov stability results for dynamic switching scenarios.

Abstract

Populations of replicating entities frequently experience sudden or cyclical changes in environment. We explore the implications of this phenomenon via a environmental switching parameter in several common evolutionary dynamics models including the replicator dynamic for linear symmetric and asymmetric landscapes, the Moran process, and incentive dynamics. We give a simple relationship between the probability of environmental switching, the relative fitness gain, and the effect on long term behavior in terms of fixation probabilities and long term outcomes for deterministic dynamics. We also discuss cases where the dynamic changes, for instance a population evolving under a replicator dynamic switching to a best-reply dynamic and vice-versa, giving Lyapunov stability results.