Timescales of population rarity and commonness in random environments

Abstract

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 ⩾ 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 - 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 - 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.