Stochastic Gilpin–Ayala competition model with infinite delay

Vasilova, Maja; Jovanović, Miljana

doi:10.1016/j.amc.2010.11.043

Cited by 40 publications

(27 citation statements)

References 16 publications

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“…It is useful to point out that some special cases of (2.1) have been studied extensively in literature, for example, [11, 12, 15-17, 37, 38] considered model (2.1) with τ ij = 0 and γ i (u) = 0, [23,39,42] investigated model (2.1) with γ i (u) = 0. Model (2.1) with τ ij = 0 was analyzed in [3].…”

Section: Resultsmentioning

confidence: 99%

Permanence of a stochastic delay competition model with Levy jumps

Li¹,

Deng²,

Wang³

2017

J. Nonlinear Sci. Appl.

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Permanence is one of the most important topics in biomathematics. The question of permanence of stochastic multi-species models is challenging because the current approaches can not be used. In this paper, an asymptotic approach is used, and sufficient criteria for permanence of a general n-species stochastic delay Lotka-Volterra competition model with Lévy jumps are established. It is also shown that these criteria are sharp in some cases. The results reveal that the stochastic noises play a key role in the permanence. This approach can be also applied to investigate the permanence of other stochastic population models with/without time delay and/or Lévy noises.

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Section: Resultsmentioning

confidence: 99%

Permanence of a stochastic delay competition model with Levy jumps

Li¹,

Deng²,

Wang³

2017

J. Nonlinear Sci. Appl.

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“…Cushing [5] studies the LotkaVolterra equations for two competing species under the assumption that the coefficients are periodic functions of a common period. Linear assumptions in Lotka-Volterra models for the interspecific interference are relaxed in GilpinAyala models, recently analyzed by Jovanović and Vasilova [11,19]. In the case of some models of two species competing in a randomly varying environment, Ellner [6] obtains sufficient conditions for convergence to the corresponding stationary distribution.…”

Section: Introductionmentioning

confidence: 99%

“…In a more general setting, we may cite the work by Cushing [5], Ellner [6], Gopalsamy [9], Jovanović and Vasilova [11,19], Li and Smith [12,13], Qi-Min et al [15], and Zhang and Han [20], among others, who study a variety of models under stochastic and deterministic perspectives, such as age-dependent mortality and fertility functions (Gopalsamy [9]), age-structured models (Qi-Min et al [15], Zhang and Han [20]), and four species that coexist in competition for three essential resources (Li and Smith [12]). Cushing [5] studies the LotkaVolterra equations for two competing species under the assumption that the coefficients are periodic functions of a common period.…”

Section: Introductionmentioning

confidence: 99%

Lifetime and reproduction of a marked individual in a two-species competition process

Gómez‐Corral

López‐García

2015

Applied Mathematics and Computation

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The interest is in a stochastic model for the competition of two species, which was first introduced by Reuter [16] and Iglehart [10], and then analyzed by . The model is related to the two-species autonomous competitive model (Zeeman [21]), where individuals compete either directly or indirectly for a limited food supply and, consequently, births and death rates depend on the population sizes of one or both of the species. The aim is to complement the treatment of the model we started in [7,8] by focusing here on probabilistic descriptors that are inherently linked to an individual: its residual lifetime and the number of direct descendants. We present an approximating model based on the maximum size distribution, and we discuss on various models defined in terms of the underlying killing and reproductive strategies. Numerical examples are presented to show the effects of the killing and reproductive strategies on the behavior of an individual, and how the impact of these strategies on the descriptors vanishes in highly competitive ecosystems.

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“…For various forms about the Gilpin-Ayala system readers can see [6,11,15] and references therein for details. Gilpin-Ayala competition models in random environments have recently also been studied by many authors, for example, [13,14,22,23]. A stochastic Gilpin-Ayala predator-prey model with time delay and a special case of it have been studied by Vasilova [21]: …”

Section: Introductionmentioning

confidence: 99%

Asymptotic behavior of non-autonomous stochastic Gilpin-Ayala predator-prey model with jumps

Zhang¹

2017

J. Nonlinear Sci. Appl.

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In this paper, a non-autonomous stochastic Gilpin-Ayala predator-prey model with jumps is studied. Firstly, we show that this model has a unique global positive solution under certain conditions. Then, we discuss the sufficient conditions for stochastically ultimate boundedness and obtain the asymptotic behavior of the solution. Finally, sufficient criteria for extinction of all prey and predator species, stochastic weak persistence in the mean of prey species are established.

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Stochastic Gilpin–Ayala competition model with infinite delay

Cited by 40 publications

References 16 publications

Permanence of a stochastic delay competition model with Levy jumps

Permanence of a stochastic delay competition model with Levy jumps

Lifetime and reproduction of a marked individual in a two-species competition process

Asymptotic behavior of non-autonomous stochastic Gilpin-Ayala predator-prey model with jumps

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