Macroscopically-observable probability currents in finite populations

TL;DR

This paper explores how probability currents in finite populations can create observable cycles in species abundances, especially when the deterministic dynamics form a neutrally-stable circle, using Fokker-Planck equations and escape rate theory.

Contribution

It identifies conditions under which probability currents cause observable population cycles and analyzes the effectiveness of simplified models in capturing these phenomena.

Findings

Probability currents can produce observable cycles in population models.

Circular neutrally-stable manifolds enable such probability currents.

Estimates of current magnitudes are derived using Kramers' escape rate theory.

Abstract

In finite-size population models, one can derive Fokker-Planck equations to describe the fluctuations of the species numbers about the deterministic behaviour. In the steady state of populations comprising two or more species, it is permissible for a probability current to flow. In such a case, the system does not relax to equilibrium but instead reaches a non-equilibrium steady state. In a two-species model, these currents form cycles (e.g., ellipses) in probability space. We investigate the conditions under which such currents are solely responsible for macroscopically-observable cycles in population abundances. We find that this can be achieved when the deterministic limit yields a circular neutrally-stable manifold. We further discuss the efficacy of one-dimensional approximation to the diffusion on the manifold, and obtain estimates for the macroscopically-observable current around this manifold by appealing to Kramers' escape rate theory.