Stability and bifurcation analysis for a delayed Lotka–Volterra predator–prey system

Yan, Xiang-Ping; Chu, Yan-Dong

doi:10.1016/j.cam.2005.09.001

Cited by 71 publications

(49 citation statements)

References 13 publications

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“…Recently, researchers are using more than one delay to study the effect of past history of the system populations [14][15][16][17][18], as in reality time delays occur in almost every biological situation [19] and are assumed to be one of the reasons of regular fluctuations in population density [20]. Delay is frequently introduced in a biologically realistic predator-prey model.…”

Section: International Journal Of Stochastic Analysismentioning

confidence: 99%

Influence of Gestation Delay and Predator’s Interference in Predator-Prey Interaction under Stochastic Environment

Jana

2014

International Journal of Stochastic Analysis

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Previous experimental and theoretical studies suggest that predator's interference in predator-prey relationship provides better descriptions of predator's feeding over a range of predator-prey abundances. Also biological delays and environmental stochasticity play an important role to describe the system and its values. In this present study, I consider a Gaussian white-noise induced stochastic predator-prey model with the Beddington-DeAngelis functional response and gestation delay. Stochastic stability is measured by second order moment terms by calculating the nonequilibrium fluctuation of the nondelayed system and Fourier transform technique depicts the fluctuation of stochastic stability by introducing time lag. Different dynamical behaviors for both situations have been illustrated numerically also. The biological implications of my analytical and numerical findings are discussed critically.

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Section: International Journal Of Stochastic Analysismentioning

confidence: 99%

Influence of Gestation Delay and Predator’s Interference in Predator-Prey Interaction under Stochastic Environment

Jana

2014

International Journal of Stochastic Analysis

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“…He [17] and Lu andWang [18] investigated the stability of the positive equilibrium of the system, and they found that the positive equilibrium is globally asymptotically stable for any values of delays τ and η when the coefficients of the system satisfy the condition 11 22 a a  12 21 > 0 when η > 0, and consider η or the sum of two delays τ and η as the bifurcation parameter, one can see [6][7][8] for details. Yan and Zhang [9] studied the effect of delay on the dynamics of system (3) when τ = η.…”

Section: T X T R a X T Y T Y T R A X A Y Tmentioning

confidence: 99%

“…and the positive equilibrium of system (6) is transformed into the zero equilibrium (0, 0) of system (8). It is easy to see that the characteristic equation of the linearized system of system (8) at the zero equilibrium (0, 0) is…”

Section: T MX T Nx T a X T A X T X T A X T X T X T Dx T Ex T B X T mentioning

confidence: 99%

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Hopf Bifurcations in a Predator-Prey System of Population Allelopathy with Discrete Delay

Wang¹,

Liu²

2012

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A delayed Lotka-Volterra two-species predator-prey system of population allelopathy with discrete delay is considered. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the positive equilibrium is investigated and Hopf bifurcations are demonstrated. Furthermore, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations (FDEs). Finally, some numerical simulations are carried out for illustrating the theoretical results.

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“…They studied the effect of diffusion and obtained the stability of the positive equilibrium and the location of Hopf bifurcation points. Yan and Chu [22] analyzed the stability for a delayed Lotka-Volterra predator-prey system and found conditions for oscillatory solutions to occur. They also examined the stability of the oscillatory solutions.…”

Section: Introductionmentioning

confidence: 99%

The diffusive Lotka–Volterra predator–prey system with delay

Noufaey

Marchant

Edwards

2015

Mathematical Biosciences

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Semi-analytical solutions for the diffusive Lotka-Volterra predator-prey system with delay are considered in one and two-dimensional domains. The Galerkin method is applied, which approximates the spatial structure of both the predator and prey populations. This approach is used to obtain a lower-order, ordinary differential delay equation model for the system of governing delay partial differential equations. Steady-state and transient solutions and the region of parameter space, in which Hopf bifurcations occur, are all found. In some cases simple linear expressions are found as approximations, to describe steady-state solutions and the Hopf parameter regions. An asymptotic analysis for the periodic solution near the Hopf bifurcation point is performed for the one-dimensional domain. An excellent agreement is shown in comparisons between semianalytical and numerical solutions of the governing equations. The diffusive Lotka-Volterra predator-prey system with delay Abstract. Semi-analytical solutions for the diffusive Lotka-Volterra predator-prey system with delay are considered in one and two-dimensional domains. The Galerkin method is applied, which approximates the spatial structure of both the predator and prey populations. This approach is used to obtain a lower-order, ordinary differential delay equation model for the system of governing delay partial differential equations. Steady-state and transient solutions and the region of parameter space, in which Hopf bifurcations occur, are all found. In some cases simple linear expressions are found as approximations, to describe steady-state solutions and the Hopf parameter regions. An asymptotic analysis for the periodic solution near the Hopf bifurcation point is performed for the one-dimensional domain. An excellent agreement is shown in comparisons between semi-analytical and numerical solutions of the governing equations.

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Stability and bifurcation analysis for a delayed Lotka–Volterra predator–prey system

Cited by 71 publications

References 13 publications

Influence of Gestation Delay and Predator’s Interference in Predator-Prey Interaction under Stochastic Environment

Influence of Gestation Delay and Predator’s Interference in Predator-Prey Interaction under Stochastic Environment

Hopf Bifurcations in a Predator-Prey System of Population Allelopathy with Discrete Delay

The diffusive Lotka–Volterra predator–prey system with delay

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