Fluctuation-induced dissipation in evolutionary dynamics

TL;DR

This paper demonstrates that stochastic fluctuations in evolutionary dynamics induce dissipation towards extinction due to positive curvature of trajectories, revealing a fundamental non-equilibrium aspect of ecosystems.

Contribution

It uncovers a universal mechanism where fluctuations cause dissipation in evolutionary systems, extending beyond traditional replicator models.

Findings

Fluctuations generate dissipation towards extinction.

Trajectory curvature determines the direction of dissipation.

Agreement between theoretical predictions and Lotka-Volterra simulations.

Abstract

Biodiversity and extinction are central issues in evolution. Dynamical balance among different species in ecosystems is often described by deterministic replicator equations with moderate success. However, fluctuations are inevitable, either caused by external environment or intrinsic random competitions in finite populations, and the evolutionary dynamics is stochastic in nature. Here we show that, after appropriate coarse-graining, random fluctuations generate dissipation towards extinction because the evolution trajectories in the phase space of all competing species possess positive curvature. As a demonstrating example, we compare the fluctuation-induced dissipative dynamics in Lotka-Volterra model with numerical simulations and find impressive agreement. Our finding is closely related to the fluctuation-dissipation theorem in statistical mechanics but the marked difference is the non-equilibrium essence of the generic evolutionary dynamics. As the evolving ecosystems are far from equilibrium, the relation between fluctuations and dissipations is often complicated and dependent on microscopic details. It is thus remarkable that the generic positivity of the trajectory curvature warrants dissipation arisen from the seemingly harmless fluctuations. The unexpected dissipative dynamics is beyond the reach of conventional replicator equations and plays a crucial role in investigating the biodiversity in ecosystems.