Quasi-stationary distributions for randomly perturbed dynamical systems

Faure, Mathieu; Schreiber, Sebastian J.

doi:10.1214/13-aap923

Cited by 39 publications

(84 citation statements)

References 74 publications

(99 reference statements)

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“…(6) converge on the dynamics of the mean-field model (eqn (3) ;Kurtz 1981;Faure & Schreiber 2014). A second set of equations with species subscripts reversed describes dynamics of species 2.…”

Section: Effect On Coexistence Of Variation Between Discrete Individualsmentioning

confidence: 99%

“…S describes landscape size and consequently

\frac{G_{1}}{S}

is the density of germinants in the landscape. In the limit of landscapes of infinite size there are essentially an infinite number of invading individuals (though at density approaching zero) and the dynamics of the stochastic model (6) converge on the dynamics of the mean‐field model (eqn ; Kurtz ; Faure & Schreiber ). A second set of equations with species subscripts reversed describes dynamics of species 2.…”

Section: Introductionmentioning

confidence: 99%

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How variation between individuals affects species coexistence

2016

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Although the effects of variation between individuals within species are traditionally ignored in studies of species coexistence, the magnitude of intraspecific variation in nature is forcing ecologists to reconsider. Compelling intuitive arguments suggest that individual variation may provide a previously unrecognised route to diversity maintenance by blurring species-level competitive differences or substituting for species-level niche differences. These arguments, which are motivating a large body of empirical work, have rarely been evaluated with quantitative theory. Here we incorporate intraspecific variation into a common model of competition and identify three pathways by which this variation affects coexistence: (1) changes in competitive dynamics because of nonlinear averaging, (2) changes in species' mean interaction strengths because of variation in underlying traits (also via nonlinear averaging) and (3) effects on stochastic demography. As a consequence of the first two mechanisms, we find that intraspecific variation in competitive ability increases the dominance of superior competitors, and intraspecific niche variation reduces species-level niche differentiation, both of which make coexistence more difficult. In addition, individual variation can exacerbate the effects of demographic stochasticity, and this further destabilises coexistence. Our work provides a theoretical foundation for emerging empirical interests in the effects of intraspecific variation on species diversity.

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Section: Effect On Coexistence Of Variation Between Discrete Individualsmentioning

confidence: 99%

“…S describes landscape size and consequently

\frac{G_{1}}{S}

Section: Introductionmentioning

confidence: 99%

How variation between individuals affects species coexistence

2016

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“…(16). Since it has derivatives of infinite order, presumably an infinite number of boundary conditions need to be specified.…”

Section: Mesoscopic Formulationmentioning

confidence: 99%

“…In the mathematical literature, on the other hand, one can view the desired behavior (i.e., trajectories that remain in the interval for the duration of a simulation) as metastable behavior-see for example Ref. [16], and references therein. The interpretation in this case is that, if the system is allowed to evolve for a sufficiently long period of time, all of the probability distribution will 'leak' out of the interval, representing the true, stationary behavior of the system.…”

Section: Introductionmentioning

confidence: 99%

Suppressing escape events in maps of the unit interval with demographic noise

et al. 2016

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We explore the properties of discrete-time stochastic processes with a bounded state space, whose deterministic limit is given by a map of the unit interval. We find that, in the mesoscopic description of the system, the large jumps between successive iterates of the process can result in probability leaking out of the unit interval, despite the fact that the noise is multiplicative and vanishes at the boundaries. By including higher-order terms in the mesoscopic expansion, we are able to capture the non-Gaussian nature of the noise distribution near the boundaries, but this does not preclude the possibility of a trajectory leaving the interval. We propose a number of prescriptions for treating these escape events, and we compare the results with those obtained for the metastable behavior of the microscopic model, where escape events are not possible. We find that, rather than truncating the noise distribution, censoring this distribution to prevent escape events leads to results which are more consistent with the microscopic model. The addition of higher moments to the noise distribution does not increase the accuracy of the final results, and it can be replaced by the simpler Gaussian noise.

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“…However, our approach can be adapted to work in systems that deviate from several of the other characteristics. For instance, characteristic 1 is not a limitation of the quasi-potential framework; Kifer ( 1990 ) describes how analogous concepts can be applied to discrete-time Markov chains (Kifer 1990 , Faure andSchreiber 2014 ). Variable transformations (see Appendix S1: Section S6 ) can be used to compute quasi-potentials for systems that deviate from characteristic 3 (e.g.…”

Section: Limitations and Generalizationsmentioning

confidence: 99%