Hopf bifurcations in a predator–prey system with a discrete delay and a distributed delay

Zhang, Cun-Hua; Yan, Xiang-Ping; Guo-hu, Cui

doi:10.1016/j.nonrwa.2010.05.001

Cited by 50 publications

(27 citation statements)

References 12 publications

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“…When G(s) takes the "weak" generic kernel function and the "strong" generic kernel function G(s) = (2) and Hopf bifurcations of nonconstant periodic solutions have been investigated respectively by using the normal form theory and the center manifold reduction for FDEs [14,15]. See [5,16] for details.…”

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

confidence: 99%

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

confidence: 99%

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

Wang¹,

Liu²

2012

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“…Yang and Li [10] studied the permanence of species for a delayed discrete ratio-dependent predator-prey model with monotonic functional response. Study of discrete dynamical behavior of systems is usually focussed on boundedness and persistence, existence and uniqueness of equilibria, periodicity, and there local and global stability (see for example, [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]), but there are few articles that discuss the dynamical behavior of discrete-time predator-prey models for exploring the possibility of bifurcation and chaos phenomena [26][27][28][29][30].…”

Section: Introductionmentioning

confidence: 99%

Stability and Neimark–Sacker bifurcation of a ratio‐dependence predator–prey model

Khan

2017

Math Methods in App Sciences

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In this paper, stability and bifurcation of a two‐dimensional ratio‐dependence predator–prey model has been studied in the close first quadrant double-struckR+2. It is proved that the model undergoes a period‐doubling bifurcation in a small neighborhood of a boundary equilibrium and moreover, Neimark–Sacker bifurcation occurs at a unique positive equilibrium. We study the Neimark–Sacker bifurcation at unique positive equilibrium by choosing b as a bifurcation parameter. Some numerical simulations are presented to illustrate theocratical results. Copyright © 2017 John Wiley & Sons, Ltd.

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“…It is very important to understand these periodic activities of neural network [1]. The stability and periodic phenomenon of neuron models and predator-prey systems have also been widely studied [8][9][10][11][12][13][14][15][16].…”

Section: Introductionmentioning

confidence: 99%

Stability and Hopf bifurcation of a neural network model with distributed delays and strong kernel

2016

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In this paper, the dynamical behaviors of a two-neuron network model with distributed delays and strong kernel are investigated. Considering the mean delay as a bifurcation parameter, explicit algorithms for determining the conditions of Hopf bifurcation are derived. A family of periodic solutions bifurcate from an equilibrium when the bifurcation parameter exceeds a critical value. The direction and stability of the bifurcating periodic solutions are determined in detail by using the theory of normal form and center manifold.Numerical simulations are performed to illustrate the effectiveness of the results found.

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Hopf bifurcations in a predator–prey system with a discrete delay and a distributed delay

Cited by 50 publications

References 12 publications

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

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

Stability and Neimark–Sacker bifurcation of a ratio‐dependence predator–prey model

Stability and Hopf bifurcation of a neural network model with distributed delays and strong kernel

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