Persistence, coexistence and spatial spread in a fluctuating environment

Sebastian Schreiber (June 26, 2012)

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All populations experience stochastic
uctuations in abiotic factors such as temperature, nutrient avail-
ability, precipitation. This environmental stochasticity in conjunction with biotic interactions can facilitate
or disrupt persistence. One approach to examining the interplay between these deterministic and stochastic
forces is the construction and analysis of stochastic di erence equations and stochastic di erential equations.
Many theoretical biologists are interested in whether the models are stochastically bounded and persis-
tent. Stochastic boundedness asserts that asymptotically the population process tends to remain in compact
sets. In contrast, stochastic persistence requires that the population process tends to be epelled" by some
"extinction set". Here, I will review recent results on both of these proprieties are reviewed for models
of multi-species interactions and spatially-structured populations. Basic results about random products of
matrices, Lyapunov exponents, stationary distributions, and small-noise approximations will be discussed.
Applications include bet-hedging, coexistence via the storage e ect, and evolutionary games in stochastic