Timescales of population rarity and commonness in random environments

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)(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<1λ<1, the population process is generally transient with escape towards extinction. When λ⩾1λ⩾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>1λ>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 λλ is close to one, the population will persistently experience wide fluctuations between states of rarity and commonness. When λ=1λ=1 and density dependence is weak at low density, rarity follows a universal power law with exponent -32. 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 -32 power law generally approximates the law of rarity of ‘weakly invading’ species with λλ values close to one. Some preliminary results for the dynamics of multispecific systems are presented.