Coexistence in the face of uncertainty
Sebastian J. Schreiber

TL;DR
This paper reviews recent mathematical results on species coexistence under demographic and environmental stochasticity, highlighting how stochastic models predict persistence, extinction probabilities, and the influence of habitat size and environmental fluctuations.
Contribution
It synthesizes new theoretical insights into stochastic persistence and coexistence, extending classical models to include demographic and environmental randomness.
Findings
QSDs concentrate on deterministic attractors for large habitats
Extinction probability decreases exponentially with habitat size
Conditions for stochastic persistence involve Lyapunov exponents
Abstract
Over the past century, nonlinear difference and differential equations have been used to understand conditions for species coexistence. 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",…
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