Persistence of structured populations in random environments

Benaı̈m, Michel; Schreiber, Sebastian J.

doi:10.1016/j.tpb.2009.03.007

Cited by 73 publications

(86 citation statements)

References 52 publications

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“…Furthermore, we prove stronger convergence results and thus extend the work of Evans et al (2013). Analogous results for discrete-time versions of the model have been studied by Benaïm and Schreiber (2009) for discrete-space and by Hardin et al (1988a, b) for continuousspace.…”

Section: Remark 24 If the Dispersal Matrix (Dsupporting

confidence: 76%

“…Patch models of dispersion have been studied extensively in the deterministic setting (see for example Hastings 1983;Cantrell et al 2012). In the stochastic setting, there have been results for discrete time and space by Benaïm and Schreiber (2009), for continuous time and discrete space by Evans et al (2013) and for structured populations that evolve continuously both in time and space.…”

Section: Discussion and Generalizationsmentioning

confidence: 99%

“…The mathematical analysis for stochastic models with density-dependent feedbacks is less explored. In the setting of discrete-space discrete-time models there have been thorough studies by Benaïm and Schreiber (2009);Schreiber (2010); . Continuous-space discrete-time population models that disperse and experience uncorrelated, environmental stochasticity have been studied by Hardin et al (1988aHardin et al ( , b, 1990.…”

Section: Introductionmentioning

confidence: 99%

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Symmetries and pattern formation in hyperbolic versus parabolic models of self-organised aggregation

Buono

Eftimie

2014

J. Math. Biol.

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This work is devoted to studying the dynamics of a structured population that is subject to the combined effects of environmental stochasticity, competition for resources, spatio-temporal heterogeneity and dispersal. The population is spread throughout n patches whose population abundances are modeled as the solutions of a system of nonlinear stochastic differential equations living on [0, ∞) n . We prove that r , the stochastic growth rate of the total population in the absence of competition, determines the long-term behaviour of the population. The parameter r can be expressed as the Lyapunov exponent of an associated linearized system of stochastic differential equations. Detailed analysis shows that if r > 0, the population abundances converge polynomially fast to a unique invariant probability measure on (0, ∞) n , while when r < 0, the population abundances of the patches converge almost surely to 0 exponentially fast. 2013) and proves one of their conjectures. Compared to recent developments, our model incorporates very general density-dependent growth rates and competition terms. Furthermore, we prove that persistence is robust to small, possibly density dependent, perturbations of the growth rates, dispersal matrix and covariance matrix of the environmental noise. We also show that the stochastic growth rate depends continuously on the coefficients. Our work allows the environmental noise driving our system to be degenerate. This is relevant from a biological point of view since, for example, the environments of the different patches can be perfectly correlated. We show how one can adapt the nondegenerate results to the degenerate setting.As an example we fully analyze the two-patch case, n = 2, and show that the stochastic growth rate is a decreasing function of the dispersion rate. In particular, coupling two sink patches can never yield persistence, in contrast to the results from the nondegenerate setting treated by Evans et al. which show that sometimes coupling by dispersal can make the system persistent.

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Section: Remark 24 If the Dispersal Matrix (Dsupporting

confidence: 76%

Section: Discussion and Generalizationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Symmetries and pattern formation in hyperbolic versus parabolic models of self-organised aggregation

Buono

Eftimie

2014

J. Math. Biol.

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“…If the metapopulation growth rate defined here is positive for a linearization of a density dependent model about the extinction state, the work of Benaïm & Schreiber (2009) implies that the metapopulation persists in the sense of stochastic boundedness (Chesson 2000;Ellner 1984). Namely, with high probability, the metapopulation tends to stay bounded away from extinction.…”

Section: Discussionmentioning

confidence: 99%

Interactive effects of temporal correlations, spatial heterogeneity and dispersal on population persistence

Schreiber

2010

Proc. R. Soc. B.

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106

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It is an ecological truism that population persistence depends on a population's growth rate when rare. To understand the interplay between temporal correlations, spatial heterogeneity and dispersal on persistence, an analytic approximation for this growth rate is derived for partially mixing populations. Partial mixing has two effects on population growth. In the absence of temporal correlations in relative fitness, greater movement to patches with, on average, higher relative fitness increases population growth rates. In the absence of spatial heterogeneity in the average relative fitnesses, lower dispersal rates enhance population growth when temporal autocorrelations of relative fitness within a patch exceed temporal cross-correlations in relative fitness between patches. This approximation implies that metapopulations whose expected fitness in every patch is less than 1 can persist if there are positive temporal autocorrelations in relative fitness, sufficiently weak spatial correlations and the population disperses at intermediate rates. It also implies that movement into lower quality habitats increases the population growth rate whenever the net temporal variation in per capita growth rates is sufficiently larger than the difference in the means of these per capita growth rates. Moreover, temporal autocorrelations, whether they be negative or positive, can enhance population growth for optimal dispersal strategies.

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“…It may be possible to strengthen the type of convergence in Theorem 4.8, similar to the results that Benaïm and Schreiber [21] proved for a discrete time model. That is, if condition (4.30) is satisfied, then x(t) → 0 almost surely.…”

Section: A Model With a Dynamic Environmentsupporting

confidence: 58%

Spatially structured metapopulation models within static and dynamic environments

Smith¹

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Traditionally in population modelling, the mixing of individuals has been assumed to be homogeneous; that is, every individual can come into contact with every other individual. Within the last 40 years, however, a number of population models have been proposed that do not assume homogeneous mixing but rather assume populations are divided into disjoint habitable patches that are separated by uninhabitable space. Populations with this structure are known as metapopulations.When metapopulation modelling was first proposed, the habitable patches would be classified as either colonised or extinct and the dynamics of colonisation and extinction would be the only dynamics accounted for in the model. For example, the logistic model adapted to a metapopulation wouldwhere x(t) is the proportion of occupied patches at time t, λ is the rate an unoccupied patch becomes colonised when all patches are unoccupied and µ is the rate an occupied patch becomes extinct. Adding more detail to metapopulation models, the model examined in this thesis records the number of individuals in each location, thereby accounting for an individual's dynamics, such as the rates of births, deaths and migrations, within the model.The model is a continuous time Markov process that will be used to account for the demographic stochasticity within populations such as births, deaths and migration events. Results by Kurtz, and extended by Pollett, which can be applied to a family of Markov processes, termed asymptotically density dependent, will be used to determine an approximating system of ii differential equations. These differential equations are then analysed to determine conditions for persistence and extinction. Furthermore, an Allee effect, where the initial conditions of the population determine whether it persists or goes extinct, is confirmed to exist in a two patch system that has a large difference between the migration rate for the two patches.The model is extended in two ways. The first extension accounts for a deterministically changing environment. This is done by allowing the parameters of the system to depend on time. A new functional limit law is derived which can be applied to time inhomogeneous, asymptotically density dependent Markov processes. This functional limit law is used to derive a nonautonomous system of differential equations. This system is then analysed to provide conditions for persistence and extinction of the metapopulation.The second extension to the model accounts for a stochastically changing environment. Again, the parameters of the system are allowed to vary in time. However, the parameters vary stochastically according to an underlying Markov process, designed to model a stochastically changing environment. A functional limit law provides a way to approximate the process with a random dynamical system. The random dynamical system is then analysed to determine a sufficient condition for extinction. While not a complete description of the long term behaviour, such an approach facilitates more research ...

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Persistence of structured populations in random environments

Cited by 73 publications

References 52 publications

Symmetries and pattern formation in hyperbolic versus parabolic models of self-organised aggregation

Symmetries and pattern formation in hyperbolic versus parabolic models of self-organised aggregation

Interactive effects of temporal correlations, spatial heterogeneity and dispersal on population persistence

Spatially structured metapopulation models within static and dynamic environments

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