Bifurcations of a singular prey–predator economic model with time delay and stage structure

Xue, Zhang; Zhang, Qingling; Liu, Chao; Xiang, Zhongyi

doi:10.1016/j.chaos.2009.03.051

Cited by 27 publications

(13 citation statements)

References 17 publications

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“…[1][2][3][4][5][6] The complex dynamics in the model are triggered by many natural factors, viz, delay, seasonal effect and diurnal effect. [7][8][9][10] The delay due to maturation time of prey, gestation time of predator, and hunting can induce periodic solutions as well as chaos in prey-predator model. [11][12][13][14][15] Hence, the exploration of prey-predator model with several natural factors is getting more popularity because of interesting biological phenomenon.…”

Section: Introductionmentioning

confidence: 99%

Complex dynamics of a discrete fractional‐order Leslie‐Gower predator‐prey model

Singh

Elsadany

Elsonbaty

2019

Math Methods in App Sciences

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A proposed discretized form of fractional‐order prey‐predator model is investigated. A sufficient condition for the solution of the discrete system to exist and to be unique is determined. Jury stability test is applied for studying stability of equilibrium points of the discretized system. Then, the effects of varying fractional order and other parameters of the systems on its dynamics are examined. The system undergoes Neimark‐Sacker and flip bifurcation under certain conditions. We observe that the model exhibits chaotic dynamics following stable states as the memory parameter α decreases and step size h increases. Theoretical results illustrate the rich dynamics and complexity of the model. Numerical simulation validates theoretical results and demonstrates the presence of rich dynamical behaviors include S‐asymptotically bounded periodic orbits, quasi‐periodicity, and chaos. The system exhibits a wide range of dynamical behaviors for fractional‐order α key parameter.

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Section: Introductionmentioning

confidence: 99%

Complex dynamics of a discrete fractional‐order Leslie‐Gower predator‐prey model

Singh

Elsadany

Elsonbaty

2019

Math Methods in App Sciences

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“…On the other hand, it should be noted that almost all the existing bioeconomic models (see [20,14,3,15]) only investigate the simplest case of a logistic prey growth function. Compared with these work, the introduction of Allee effect makes the work studied in this paper novel.…”

Section: Discussionmentioning

confidence: 99%

“…Substituting (3.12)-(3.15) into (3.9) and noticing that [8,20,19]. Then the model (2.1) with given values of parameters takes the following form:…”

Section: By (33) We Getmentioning

confidence: 99%

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Hopf analysis of a differential-algebraic predator–prey model with Allee effect and time delay

Zhang

2015

Int. J. Biomath.

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A differential-algebraic prey-predator model with time delay and Allee effect on the growth of the prey population is investigated. Using differential-algebraic system theory, we transform the prey-predator model into its normal form and study its dynamics in terms of local analysis and Hopf bifurcation. By analyzing the associated characteristic equation, it is observed that the model undergoes a Hopf bifurcation at some critical value of time delay. In particular, we study the direction of Hopf bifurcation and the stability of bifurcated periodic solutions, and an explicit algorithm is given by applying the normal form theory and the center manifold reduction for functional differential equations. Finally, numerical simulations supporting the theoretical analysis are also included.

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“…In [17], the authors investigated dynamic behavior of a differential-algebraic model system which considered a prey-predator system with stage structure for prey and harvest effort on predator. In [18], Zhang et al studied a singular prey-predator economic model with time delay and stage structure. Hu and Huang [19] considered a predator-prey system of Lotka-Volterra type with delays and stage structure for prey, and the local stability of the equilibrium points was investigated and Hopf bifurcations occurring at the positive equilibrium point under some conditions were demonstrated.…”

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confidence: 99%

Stability and Hopf bifurcation in a reaction–diffusion predator–prey system with interval biological parameters and stage structure

Zhao

Wang

2014

Nonlinear Dyn

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This paper deals with a delayed reactiondiffusion predator-prey system combined with stage structure for prey and interval biological parameters. By taking the sum of delays as the bifurcation parameter, the local stability of the equilibrium points is investigated and the condition of Hopf bifurcation is obtained. In succession, using the normal form theory and the center manifold reduction for partial functional differential equations, we derive the explicit formulas determining stability, direction and other properties of bifurcating periodic solutions. Some numerical simulations are also included to illustrate our theoretical results.

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Bifurcations of a singular prey–predator economic model with time delay and stage structure

Cited by 27 publications

References 17 publications

Complex dynamics of a discrete fractional‐order Leslie‐Gower predator‐prey model

Complex dynamics of a discrete fractional‐order Leslie‐Gower predator‐prey model

Hopf analysis of a differential-algebraic predator–prey model with Allee effect and time delay

Stability and Hopf bifurcation in a reaction–diffusion predator–prey system with interval biological parameters and stage structure

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