Stability and Hopf bifurcation in a delayed predator–prey system with stage structure for prey

Hu, Haijun; Huang, Lihong

doi:10.1016/j.nonrwa.2009.10.001

Cited by 30 publications

(17 citation statements)

References 26 publications

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“…Remark 2 Since we consider a three-species-food-chain model, the dynamical behaviors are more complicated and the system has more equilibria than those in [4,10,12]. Although these conditions of (C2), (C3) and (C4) seem to be intricate, take Case I (please see the section of Numerical simulation (Sect.…”

Section: Existence Of Equilibriamentioning

confidence: 99%

“…Predator-prey type interaction is one of basic interspecies relations in the biology and ecology and it is also the basic block of the complicated food chain, food web and biochemical network structure [1][2][3][4]. Since the seminal work by Aiello and Freedman [5], species growth models with stage structure have drawn considerable attention (for more details as regards these studies, one can refer to [6,7]).…”

Section: Introductionmentioning

confidence: 99%

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Dynamical behaviors of a food-chain model with stage structure and time delays

et al. 2018

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Incorporating two delays (τ 1 represents the maturity of predator, τ 2 represents the maturity of top predator), we establish a novel delayed three-species food-chain model with stage structure in this paper. By analyzing the characteristic equations, constructing a suitable Lyapunov functional, using Lyapunov-LaSalle's principle, the comparison theorem and iterative technique, we investigate the existence of nonnegative equilibria and their stability. Some interesting findings show that the delays have great impacts on dynamical behaviors for the system: on one hand, if τ 1 ∈ (m 1 , m 2 ) and τ 2 ∈ (m 4 , +∞), then the boundary equilibrium E 2 (x 0 , y 0 1 , y 0 2 , 0, 0) is asymptotically stable (AS), i.e., the prey species and the predator species will coexist, the top-predator species will go extinct; on the other hand, if τ 1 ∈ (m 2 , +∞), then the axial equilibrium E 1 (k, 0, 0, 0, 0) is AS, i.e., all predators will go extinct. Numerical simulations are great well agreement with the theoretical results.

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Section: Existence Of Equilibriamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Dynamical behaviors of a food-chain model with stage structure and time delays

et al. 2018

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“…It is obvious that taking the effective factors of stage structure and time delay into account is more meaningful when we study the interaction of population. The models with stagestructure and time delay have been considered by many authors (see [2][3][4][5][6] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

“…[2] discussed a predator-prey model of Lotka-Volterra type with time delays and stage 1 E-mail addresses: weifengying@fzu.edu.cn (F. Wei), 245060584@qq.com (Q. Fu).…”

Section: Introductionmentioning

confidence: 99%

Hopf bifurcation and stability for predator–prey systems with Beddington–DeAngelis type functional response and stage structure for prey incorporating refuge

Wei

2016

Applied Mathematical Modelling

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Please cite this article as: F. Wei, Q. Fu, Hopf bifurcation and stability for predator-prey systems with BeddingtonDeAngelis type functional response and stage structure for prey incorporating refuge, Appl. Math. Modelling (2015), doi: http://dx.Abstract: A kind of stage-structured predator-prey model with Beddington-DeAngelis functional response incorporating a prey refuge is investigated in this paper. By analyzing the corresponding characteristic equations, the local stability of the equilibria is investigated. Moreover, Hopf bifurcations occur at the positive equilibrium as the delay τ crosses some critical values. Further, the influence of prey refuge on densities of predator species and prey species is investigated. Numerical simulations are carried out to illustrate our main results.

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“…But studies on dynamical system not only involve the persistence and stability, but also involve many other behaviors such as periodic phenomenon [5,26,27], global attractivity [11,28] and chaos [20]. In particular, the properties of periodic solutions are of great interest [3,6,7,8,10,16]. Based on this consideration, and motivated by the work of Wang [19] and Xu [22], we shall consider the bifurcation phenomenon and the properties of periodic solutions of the following predator-prey system with multiple delays:…”

Section: Introductionmentioning

confidence: 99%

Dynamics of a predator-prey system with stage structure and two delays

Liu¹,

Zhang²

2016

J. Nonlinear Sci. Appl.

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A Holling type III predator-prey system with stage structure for the predator and two delays is investigated. At first, we study the stability and the existence of periodic solutions via Hopf bifurcation with respect to both delays at the positive equilibrium by analyzing the distribution of the roots of the associated characteristic equation. Then, explicit formulas that can determine the direction of the Hopf bifurcation and the stability of the periodic solutions bifurcating from the Hopf bifurcation are established by using the normal form method and center manifold argument. Finally, some numerical simulations are carried out to support the main theoretical results.

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Stability and Hopf bifurcation in a delayed predator–prey system with stage structure for prey

Cited by 30 publications

References 26 publications

Dynamical behaviors of a food-chain model with stage structure and time delays

Dynamical behaviors of a food-chain model with stage structure and time delays

Hopf bifurcation and stability for predator–prey systems with Beddington–DeAngelis type functional response and stage structure for prey incorporating refuge

Dynamics of a predator-prey system with stage structure and two delays

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