Coexistence in the Face of Uncertainty

Abstract

Over the past century, nonlinear difference and differential equations have been used to understand conditions for coexistence of interacting populations. However, these models fail to account for random fluctuations due to demographic and environmental stochasticity which are experienced by all populations. I review some recent mathematical results about persistence and coexistence for models accounting for each of these forms of stochasticity. Demographic stochasticity stems from populations and communities consisting of a finite number of interacting individuals, and often are represented by Markovian models with a countable number of states. For closed populations in a bounded world, extinction occurs in finite time but may be preceded by long-term transients. Quasi-stationary distributions (QSDs) of these Makov models characterize this meta-stable behavior. For sufficiently large "habitat sizes", QSDs are shown to concentrate on the positive attractors of deterministic models. Moreover, the probability extinction decreases exponentially with habitat size. Alternatively, environmental stochasticity stems from fluctuations in environmental conditions which influence survival, growth, and reproduction. Stochastic difference equations can be used to model the effects of environmental stochasticity on population and community dynamics. For these models, stochastic persistence corresponds to empirical measures placing arbitrarily little weight on arbitrarily low population densities. Sufficient and necessary conditions for stochastic persistence are reviewed. These conditions involve weighted combinations of Lyapunov exponents corresponding to "average" per-capita growth rates of rare species. The results are illustrated with how climatic variability influenced the dynamics of Bay checkerspot butterflies, the persistence of coupled sink populations, coexistence of competitors through the storage effect, and stochastic rock-paper-scissor communities. Open problems and conjectures are presented.

From noise to knowledge: how randomness generates novel phenomena and reveals information

“…; Coulson ; Lande et al . ; Schreiber ; Ovaskainen & Meerson ). Stochasticity can also do the opposite, allowing many species to coexist in scenarios where the deterministic skeleton would predict all but one of them to be doomed to extinction (Tuljapurkar & Orzack ; Chesson & Warner ; Chesson ; Chesson & Ellner ; Melbourne et al .…”

“…; Hart et al . ; Schreiber ). Two important trends in this literature have been establishing existing results in more general and precise language and theorems (e.g.…”

“…) or the coexistence of multiple species (Caswell ; Tuljapurkar & Orzack ; Chesson & Warner ; Chesson ; Chesson & Ellner ; Melbourne et al . ; Schreiber ); correlated (Roughgarden ; Ripa & Lundberg ; Petchey et al . ; Schreiber ; Lee et al .…”

Public goods games in populations with fluctuating size

“…We now characterize the extinction time for our model, inspired by techniques used in classical branching processes [see [47][48][49]. Let x " px C {K, x D {Kq be normalized quantities of cooperators and defectors, where K ą 0 parametrizes the baseline reproductive capacity, f N .…”

Biodiversity maintenance may be lower under partial niche differentiation than under neutrality

“…However, stochastic formulations, which model the influence of variability in finite populations whose deterministic details are unknown or unrelated to species differences, are more relevant for natural systems. In such settings, coexistence is defined in a probabilistic sense (Schreiber ). Furthermore, many if not most communities in nature are subject to propagule pressure from regional pools.…”