Timescales of population rarity and commonness in random environments

Abstract

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 and time spent in commonness. 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. These non- trivial results should be of interest to biologists involved in the conservation of threatened populations, and to epidemiologists facing the need to better understanding the dynamics of pest or disease outbreaks.