A piecewise deterministic model for a prey-predator community

Costa, Manon

doi:10.1214/16-aap1182

Cited by 18 publications

(12 citation statements)

References 51 publications

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“…This would make our system a piecewise deterministic Markov process (PDMP). These processes have recently been studied in a biological context and have offered new insight regarding the competitive exclusion principle (see [BL16]) and the long term behavior of predator-prey communities (see [Cos16]).…”

Section: Discussionmentioning

confidence: 99%

Stochastic Lotka–Volterra food chains

Hening

Nguyen

2017

J. Math. Biol.

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Abstract. We study the persistence and extinction of species in a simple food chain that is modelled by a Lotka-Volterra system with environmental stochasticity. There exist sharp results for deterministic Lotka-Volterra systems in the literature but few for their stochastic counterparts. The food chain we analyze consists of one prey and n − 1 predators. The jth predator eats the j − 1th species and is eaten by the j + 1th predator; this way each species only interacts with at most two other species -the ones that are immediately above or below it in the trophic chain. We show that one can classify, based on an explicit quantity depending on the interaction coefficients of the system, which species go extinct and which converge to their unique invariant probability measure. Our work can be seen as a natural extension of the deterministic results of Gard and Hallam '79 to a stochastic setting.As one consequence we show that environmental stochasticity makes species more likely to go extinct. However, if the environmental fluctuations are small, persistence in the deterministic setting is preserved in the stochastic system. Our analysis also shows that the addition of a new apex predator makes, as expected, the different species more prone to extinction.Another novelty of our analysis is the fact that we can describe the behavior the system when the noise is degenerate. This is relevant because of the possibility of strong correlations between the effects of the environment on the different species.

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Section: Discussionmentioning

confidence: 99%

Stochastic Lotka–Volterra food chains

Hening

Nguyen

2017

J. Math. Biol.

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“…The process X = (X, Ξ) belongs to the class of Piecewise Deterministic Markov Processes (PDMP), a term coined by Davis in one of the first general study on this kind of processes (see [22]). In the last decades, PDMPs generated by switching ODEs have been extensively studied by numerous authors in the context of population dynamics and epidemiology, see e.g , Takueshi et al [45], Du, Dang, and Feng [24], Benaïm and Lobry [13], Costa [20], Hening and Strickler [30] for population dynamics or Gray et al [28] and Li, Liu and Cui [37] for epidemiology.…”

Section: Introductionmentioning

confidence: 99%

Large population asymptotics for a multitype stochastic SIS epidemic model in randomly switched environment

Prodhomme,

Strickler

2021

Preprint

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We consider an epidemic SIS model described by a multitype birth-and-death process in a randomly switched environment. That is, the infection and cure rates of the process depend on the state of a finite Markov jump process (the environment), whose transitions also depend on the number of infectives. The total size of the population is constant and equal to some K ∈ N * , and the number of infectives vanishes almost surely in finite time. We prove that, as K → ∞, the process composed of the proportions of infectives of each type X K and the state of the environment Ξ K , converges to a piecewise deterministic Markov process (PDMP) given by a system of randomly switched ODEs. The long term behaviour of this PDMP has been previously investigated by Benaïm and Strickler, and depends only on the sign of the top Lyapunov exponent Λ of the linearised PDMP at 0: if Λ < 0, the proportion of infectives in each group converges to zero, while if Λ > 0, the disease becomes endemic. In this paper, we show that the large population asymptotics of X K also strongly depend on the sign of Λ: if negative, then from fixed initial proportions of infectives the disease disappears in a time of order at most log(K), while if positive, the typical extinction time grows at least as a power of K. We prove that in the situation where the origin is accessible for the linearised PDMP, the mean extinction time of X K is logarithmically equivalent to K p * , where p * > 0 is fully characterised. We also investigate the quasi-stationary distribution µ K of (X K , Ξ K ) and show that, when Λ < 0, weak limit points of (µ K ) K>0 are supported by the extinction set, while when Λ > 0, limit points belong to the (non empty) set of stationary distributions of the limiting PDMP which do not give mass to the extinction set.

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“…This enables to reduce the infinite-dimensional model to a two-dimensional system of equations. The averaging phenomenon in finite dimension is classical [KKP14,BKPR06,Cos16,MT12]. The reduction of the infinite setting to a finite one describing the number of preys and predators may be less.…”

Section: Introductionmentioning

confidence: 99%

From the distributions of times of interactions to preys and predators dynamical systems

Bansaye¹,

Cloez²

2021

Preprint

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We consider a stochastic individual based model where each predator searches during a random time and then manipulates its prey or rests. The time distributions may be non-exponential. An age structure allows to describe these interactions and get a Markovian setting. The process is characterized by a measure-valued stochastic differential equation. We prove averaging results in this infinite dimensional setting and get the convergence of the slow-fast macroscopic prey predator process to a two dimensional dynamical system. We recover classical functional responses. We also get new forms arising in particular when births and deaths of predators are affected by the lack of food.

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A piecewise deterministic model for a prey-predator community

Cited by 18 publications

References 51 publications

Stochastic Lotka–Volterra food chains

Stochastic Lotka–Volterra food chains

Large population asymptotics for a multitype stochastic SIS epidemic model in randomly switched environment

From the distributions of times of interactions to preys and predators dynamical systems

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