Hopf bifurcation in a delayed Lokta–Volterra predator–prey system

Yan, Xiang-Ping; Zhang, Cun-Hua

doi:10.1016/j.nonrwa.2006.09.007

Cited by 80 publications

(47 citation statements)

References 14 publications

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“…Yan and Zhang [9] studied the effect of delay on the dynamics of system (3) when τ = η. Furthermore, for the study of system (3) with delayed in-X.…”

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

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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“…Yan and Zhang [9] studied the effect of delay on the dynamics of system (3) when τ = η. Furthermore, for the study of system (3) with delayed in-X.…”

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

confidence: 99%

“…It is well known that the stability of the zero equilibrium (0, 0) of system (8) is determined by the real parts of the roots of Equation (9). If all roots of Equation (9) locate the left-half complex plane, then the zero equilibrium (0, 0) of system (8) is asymptotically stable.…”

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%

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

Wang¹,

Liu²

2012

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“…Since any time delay does not change the equilibrium of the system and, according to Yan and Zhang (2008), we know that the delayed prey predator model (5) …”

Section: Stability Of the Positive Equilibrium And Local Hopf Bifurcamentioning

confidence: 99%

“…According to the view point of Kuang (1993), Yan and Zhang (2008) considered the stability and Hopf bifurcation of the following delayed system:…”

mentioning

confidence: 99%

Stability and Hopf bifurcation analysis for a Lotka-Volterra predator-prey model with two delays

Xu¹,

Liao²,

He³

2011

International Journal of Applied Mathematics and Computer Science

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In this paper, a two-species Lotka-Volterra predator-prey model with two delays is considered. By analyzing the associated characteristic transcendental equation, the linear stability of the positive equilibrium is investigated and Hopf bifurcation is demonstrated. Some explicit formulae for determining the stability and direction of Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using normal form theory and center manifold theory. Some numerical simulations for supporting the theoretical results are also included.

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“…With the deepening of research, entropy theory has been widely used in the research methods of system complexity, and many valuable results have been obtained. Yan et al [12] cited the entropy change in a delayed Lokta-Volterra predator-prey system. Meanwhile, Chen et al [13] followed the former results and used entropy control to a financial system with time-delayed feedbacks.…”

Section: Introductionmentioning

confidence: 99%

Research Entropy Complexity about the Nonlinear Dynamic Delay Game Model

Zhan

Ren

2017

Entropy

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Based on the research of domestic and foreign scholars, this paper has improved and established a double oligopoly market model of renewable energy, and analyzed the complex dynamic characteristics of a system based on entropy theory and chaos theory, such as equilibrium point, stability, Hopf bifurcation conditions, etc. This paper also studied and simulated the effects of the natural growth rate of energy and the single delay decision on the renewable energy system by minimizing the entropy of the system and reducing the system instability to a minimum, so that the degree of disorder within the system was reduced. The results show that with the increase of the natural growth rate of energy, the stability of the system is not affected, but the market demand of the oligopoly 1 is gradually reducing and the market demand of the oligopoly 2 is gradually increasing. At the same time, a single oligopoly making the time delay decision will affect the stability of the two oligopolies. With the increase of delay, the time required to reach the stable state will grow, and the system will eventually enter the Hopf bifurcation, thus the system will have its entropy increased and fall into an unstable state. Therefore, in the actual market of renewable energy, oligopolies should pay attention to the natural growth rate of energy and time delay, ensuring the stability of the game process and the orderliness of the system.

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Hopf bifurcation in a delayed Lokta–Volterra predator–prey system

Cited by 80 publications

References 14 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 Hopf bifurcation analysis for a Lotka-Volterra predator-prey model with two delays

Research Entropy Complexity about the Nonlinear Dynamic Delay Game Model

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