Macroscopically-observable probability currents in finite populations

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.