Persistence for stochastic difference equations: A mini-review

Understanding under what conditions populations, whether they be plants, animals, or viral particles, persist is an issue of theoretical and practical importance in population biology. Both biotic interactions and environmental fluctuations are key factors that can facilitate or disrupt persistence. One approach to examining the interplay between these deterministic and stochastic forces is the construction and analysis of stochastic difference equations $X_{t+1}=F(X_t,\xi_{t+1})$ where $X_t \in \R^k$ represents the state of the populations and $\xi_1,\xi_2,...$ is a sequence of random variables representing environmental stochasticity. In the analysis of these stochastic models, many theoretical population biologists are interested in whether the models are bounded and persistent. Here, boundedness asserts that asymptotically $X_t$ tends to remain in compact sets. In contrast, persistence requires that $X_t$ tends to be "repelled" by some "extinction set" $S_0\subset \R^k$. Here, results on both of these proprieties are reviewed for single species, multiple species, and structured population models. The results are illustrated with applications to stochastic versions of the Hassell and Ricker single species models, Ricker, Beverton-Holt, lottery models of competition, and lottery models of rock-paper-scissor games. A variety of conjectures and suggestions for future research are presented.Comment: Accepted for publication in the Journal of Difference Equations and Application

Rodent population dynamics affect seedling recruitment in alpine habitats

Questions: How do rodents with cyclic population dynamics affect seedling recruitment in alpine habitats? Does disturbance fromrodents have larger implications on seedling recruitment in some plant communities than in others? Location: Snowbeds and sheltered heaths in the low-alpine zone in areas of Børgefjell and Dovrefjell, Norway. Methods: We recorded seedling emergence, rodent activity and cover of mosses, lichens, litter and bare ground in 270 plots in snowbeds and sheltered heaths in a rodent population peak year and in the following low-density year. Results: Seedling recruitment was positively correlated with disturbances from lemmings and voles in both years. More seedlings emerged in the low-density year than in the year of the population peak. Snowbeds had higher seedling recruitment than the sheltered heaths, but both habitats were equally affected by disturbances fromrodents. Conclusions: Rodent activity created gaps and increased seedling emergence in these alpine plant communities, particularly in the year after the rodent peak, both in snowbeds and sheltered heath habitats. Our study therefore suggests that regeneration patterns in alpine vegetation are tightly linked to the population cycles of lemmings and voles,which peak in density at 3- to 5-yr intervals.publishedVersionPaid Open Acces

Acquiring skill Module 1 - skill

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