Macroscopically observable probability currents in finite populations

D. I. Russell*, R. A. Blythe

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


In finite size population models, one can derive Fokker-Planck equations to describe 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 nonequilibrium 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.

Original languageEnglish
Article number06008
Number of pages26
Journal Journal of Statistical Mechanics: Theory and Experiment
Publication statusPublished - Jun 2013


  • driven diffusive systems (theory)
  • population dynamics (theory)
  • stationary states
  • diffusion

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