Stability and bifurcation analysis on a three-species food chain system with two delays

Guo-hu, Cui; Yan, Xiang-Ping

doi:10.1016/j.cnsns.2010.12.042

Cited by 42 publications

(12 citation statements)

References 18 publications

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“…In contrast, Wang and Jiang have demonstrated that different values of delay can generate/eliminate chaos in a predator-prey model with dormancy in predators [Wang & Jiang, 2012]. Moreover, predatorprey models with multiple delays also explored the chaotic dynamics [Cui & Yan, 2011;Gakkhar & Singh, 2012;Guo et al, 2014;Nakaoka et al, 2006].…”

Section: Introductionmentioning

confidence: 99%

Cannibalistic Predator–Prey Model with Disease in Predator — A Delay Model

Biswas

Samanta

Chattopadhyay

2015

Int. J. Bifurcation Chaos

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In this paper, we propose and analyze a cannibalistic predator-prey model with a transmissible disease in the predator population. The disease can be transmitted through contacts with infected individuals as well as the cannibalism of an infected predator. We also consider incubation delay in disease transmission, where the incubation period represents the time in which the infectious agent develops in the host. Local stability analysis of the system around the biologically feasible equilibria is studied. Bifurcation analysis of the system around interior equilibrium is also studied. Applying the normal form theory and central manifold theorem, the direction of Hopf bifurcation, the stability and the period of bifurcating periodic solutions are derived. Under appropriate conditions, the permanence of the system with time delay is proved. Our results suggest that incubation delay destabilizes the system and can produce chaos. We also observe that cannibalism can control disease and population oscillations. Extensive numerical simulations are performed to support our analytical results.

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Section: Introductionmentioning

confidence: 99%

Cannibalistic Predator–Prey Model with Disease in Predator — A Delay Model

Biswas

Samanta

Chattopadhyay

2015

Int. J. Bifurcation Chaos

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“…There are also many papers on the bifurcation of predator-prey systems with two or multiple delays [12][13][14][15][16][17][18][19]. Cui and Yan [16] investigated the stability and existence of Hopf bifurcations of a three-species Lotka-Volterra food chain system by taking the sum of the two delays in the system as the bifurcation parameter. Further, they determined the direction and stability of the Hopf bifurcation by using the normal form method and center manifold argument and using a computation process similar to that in [10].…”

Section: Introductionmentioning

confidence: 99%

“…Dynamical systems with single delay have been investigated by many researchers [4][5][6][7][8][9][10][11]. There are also many papers on the bifurcation of predator-prey systems with two or multiple delays [12][13][14][15][16][17][18][19]. Cui and Yan [16] investigated the stability and existence of Hopf bifurcations of a three-species Lotka-Volterra food chain system by taking the sum of the two delays in the system as the bifurcation parameter.…”

Section: Introductionmentioning

confidence: 99%

Dynamical Analysis of a Food Chain System with Two Delays

Liu

Sun

2015

Qual. Theory Dyn. Syst.

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This paper is concerned with a three species food chain system with the Beddington-DeAngelis functional response and two delays. The local stability of the positive equilibrium and the existence of periodic solutions via Hopf bifurcation with respect to both delays at the positive equilibrium are established by analyzing the distribution of the roots of the associated characteristic equation. Further, the properties of the bifurcating periodic solutions such as the direction and the stability are determined by using the normal form method and center manifold argument. Numerical simulations are presented for supporting the analytical results.

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“…Therefore, it is important and meaningful to study dynamical behavior of the food-chain system. And the food-chain system is always a hot issue for the mathematical ecologists in the last decades, see [1][2][3][4][5][6][7][8][9][10][11][12] and so on. When modeling a predator-prey system, one of the most important factors should be involved is functional response.…”

Section: Introductionmentioning

confidence: 99%

Extinction and persistence of a nonautonomous stochastic food-chain system with impulsive perturbations

2016

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In this paper, a nonautonomous stochastic food-chain system with functional response and impulsive perturbations is studied. By using Itô's formula, exponential martingale inequality, differential inequality and other mathematical skills, some sufficient conditions for the extinction, nonpersistence in the mean, persistence in the mean, and stochastic permanence of the system are established. Furthermore, some asymptotic properties of the solutions are also investigated. Finally, a series of numerical examples are presented to support the theoretical results, and effects of different intensities of white noises perturbations and impulsive effects are discussed by the simulations.

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Stability and bifurcation analysis on a three-species food chain system with two delays

Cited by 42 publications

References 18 publications

Cannibalistic Predator–Prey Model with Disease in Predator — A Delay Model

Cannibalistic Predator–Prey Model with Disease in Predator — A Delay Model

Dynamical Analysis of a Food Chain System with Two Delays

Extinction and persistence of a nonautonomous stochastic food-chain system with impulsive perturbations

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