Timescales of population rarity and commonness in random environments

Regis Ferriere, Alice Guionnet, Irina Kurkova

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


This is a mathematical study of the interactions between non-linear feedback (density dependence) and uncorrelated random noise in the dynamics of unstructured populations. The stochastic non-linear dynamics are generally complex, even when the deterministic skeleton possesses a stable equilibrium. There are three critical factors of the stochastic non-linear dynamics; whether the intrinsic population growth rate ( λ ) is smaller than, equal to, or greater than 1; the pattern of density dependence at very low and very high densities; and whether the noise distribution has exponential moments or not. If λ < 1, the population process is generally transient with escape towards extinction. When λ {greater than or slanted equal to} 1, our quantitative analysis of stochastic non-linear dynamics focuses on characterizing the time spent by the population at very low density (rarity), or at high abundance (commonness), or in extreme states (rarity or commonness). When λ > 1 and density dependence is strong at high density, the population process is recurrent: any range of density is reached (almost surely) in finite time. The law of time to escape from extremes has a heavy, polynomial tail that we compute precisely, which contrasts with the thin tail of the laws of rarity and commonness. Thus, even when λ is close to one, the population will persistently experience wide fluctuations between states of rarity and commonness. When λ = 1 and density dependence is weak at low density, rarity follows a universal power law with exponent - frac(3, 2). We provide some mathematical support for the numerical conjecture [Ferriere, R., Cazelles, B., 1999. Universal power laws govern intermittent rarity in communities of interacting species. Ecology 80, 1505-1521.] that the - frac(3, 2) power law generally approximates the law of rarity of 'weakly invading' species with λ values close to one. Some preliminary results for the dynamics of multispecific systems are presented.

Original languageEnglish (US)
Pages (from-to)351-366
Number of pages16
JournalTheoretical Population Biology
Issue number4
StatePublished - Jun 2006


  • Ecological timescales
  • Environmental stochasticity
  • Markov chains
  • Martingales
  • On-off intermittency
  • Population dynamics
  • Power law
  • Rarity
  • Stochastic non-linear difference equations

ASJC Scopus subject areas

  • Ecology, Evolution, Behavior and Systematics


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