Timescales of population rarity and commonness in random environments

R. Ferriere (1) , Alice Guionnet (2) , I. Kurkova (3)

1 FESE - Fonctionnement et évolution des systèmes écologiques
2 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
3 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

This paper investigates the influence of environmental noise on the characteristic timescale of the dynamics of density-dependent populations. General results are obtained on the statistics of time spent in rarity (i.e.\ below a small threshold on population density) and time spent in commonness (i.e. above a large threshold). The nonlinear stochastic models under consideration form a class of Markov chains on the state space $]0, \infty[$ which are transient if the intrinsic growth rate is negative and recurrent if it is positive or null. In the recurrent case, we obtain a necessary and sufficient condition for positive recurrence and precise estimates for the distribution of times of rarity and commonness. In the null recurrent, critical case that applies to ecologically neutral species, the distribution of rarity time is a universal power law with exponent $-3/2$. This has implications for our understanding of the long-term dynamics of some natural populations, and provides a rigorous basis for the statistical description of on-off intermittency known in physical sciences.

Mots clés: Population dynamics; Stochastic non-linear difference equations; Environmental stochasticity; Rarity; Ecological timescales; L Power law; On-off intermittency; Markov chains; Martingales

Domaines: Probabilités [math.PR]

R. Ferriere, Alice Guionnet, I. Kurkova. Timescales of population rarity and commonness in random environments. 2004. ⟨hal-00002998⟩