Asymptotic properties of a continuous-space discrete-time population model in a random environment

Hardin, Douglas P.; Takáč, Peter; Webb, Glenn

doi:10.1007/bf00276367

Cited by 45 publications

(62 citation statements)

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“…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. They show that the leading Lyapunov exponent r of the linearization of the system around the extinction state almost determines the persistence and extinction of the population.…”

Section: Introductionmentioning

confidence: 99%

“…In the current paper we explore the question of persistence and extinction when the population dynamics is given by a system of stochastic differential equations. In our setting, even though our methods and techniques are very different from those used by Hardin et al (1988a) and Mierczyński and Shen (2004), we still make use of the system linearized around the extinction state. The Lyapunov exponent of this linearized system plays a key role throughout our arguments.…”

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: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Buono

Eftimie

2014

J. Math. Biol.

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“…Intuitively, if the stochastic growth rate is positive, the population tends to increase when rare and, consequently, is more likely to persist. Under suitable conditions, Chesson (1982), Ellner (1984), and Hardin et al (1988a) placed this heuristic on a mathematically rigorous foundation for populations in serially uncorrelated environments. One of our main goals is to extend these results to correlated random environments.…”

Section: Introductionmentioning

confidence: 99%

“…Alternatively, if γ < 0, the population is driven to extinction. To contend with the nonlinearities in structured population models, Hardin et al (1988a) extended the work of Ellner (1984) to structured populations in serially uncorrelated environments. Under suitable assumptions about the matrices A (e.g.…”

Section: Introductionmentioning

confidence: 99%

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Persistence of structured populations in random environments

Benaı̈m

Schreiber

2009

Theoretical Population Biology

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Coexistence in the Face of Uncertainty

Schreiber

2017

Fields Institute Communications

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Over the past century, nonlinear difference and differential equations have been used to understand conditions for coexistence of interacting populations. 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", QSDs are shown to concentrate on the positive attractors of deterministic models. Moreover, the probability extinction decreases exponentially with habitat size. Alternatively, environmental stochasticity stems from fluctuations in environmental conditions which influence survival, growth, and reproduction. Stochastic difference equations can be used to model the effects of environmental stochasticity on population and community dynamics. For these models, stochastic persistence corresponds to empirical measures placing arbitrarily little weight on arbitrarily low population densities. Sufficient and necessary conditions for stochastic persistence are reviewed. These conditions involve weighted combinations of Lyapunov exponents corresponding to "average" per-capita growth rates of rare species. The results are illustrated with how climatic variability influenced the dynamics of Bay checkerspot butterflies, the persistence of coupled sink populations, coexistence of competitors through the storage effect, and stochastic rock-paper-scissor communities. Open problems and conjectures are presented.

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Asymptotic properties of a continuous-space discrete-time population model in a random environment

Cited by 45 publications

References 7 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

Persistence of structured populations in random environments

Coexistence in the Face of Uncertainty

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