Limit cycle and numerical similations for small and large delays in a predator–prey model with modified Leslie–Gower and Holling-type II schemes

Yafia, Radouane; Adnani, Fatiha El; Alaoui, Hamad Talibi

doi:10.1016/j.nonrwa.2006.12.017

Cited by 67 publications

(32 citation statements)

References 7 publications

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“…Using the same methods as the ones used in [13], for small delay, let − ≃ 1 − , together with (46), the following bifurcation results can be obtained. For large delay, by [13] we have the following results.…”

Section: Hopf Bifurcationmentioning

confidence: 99%

“…Remark. Because the proofs of Theorems 7 and 8 are the same as the proofs of Theorems 3.4 and 3.5 in [13], we omit them here.…”

Section: Hopf Bifurcationmentioning

confidence: 99%

“…The authors in [12,13] have studied system (1) without prey harvesting ( ), respectively, in [12], the global stability and persistence of the system are investigated. In [13], by using the Hopf bifurcation theorem and taking the delay as a parameter of bifurcation for small and large cases, the existence of the bifurcated limit cycle around a boundary equilibrium or an interior equilibrium is mainly considered.…”

Section: Introductionmentioning

confidence: 99%

“…In [13], by using the Hopf bifurcation theorem and taking the delay as a parameter of bifurcation for small and large cases, the existence of the bifurcated limit cycle around a boundary equilibrium or an interior equilibrium is mainly considered.…”

Section: Introductionmentioning

confidence: 99%

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Bogdanov-Takens and Triple Zero Bifurcations of a Delayed Modified Leslie-Gower Predator Prey System

Liu

Wang

2013

Abstract and Applied Analysis

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A delayed modified Leslie-Gower predator prey system with nonlinear harvesting is considered. The existence conditions that an equilibrium is Bogdanov-Takens (BT) or triple zero singularity of the system are given. By using the center manifold reduction, the normal form theory, and the formulae developed by Huang, 2008 andQiao et al., 2010, the normal forms and the versal unfoldings for this singularity are presented. The Hopf bifurcation of the system at another interior equilibrium is analyzed by taking delay (small or large) as bifurcation parameter.

show abstract

Section: Hopf Bifurcationmentioning

confidence: 99%

“…Remark. Because the proofs of Theorems 7 and 8 are the same as the proofs of Theorems 3.4 and 3.5 in [13], we omit them here.…”

Section: Hopf Bifurcationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Bogdanov-Takens and Triple Zero Bifurcations of a Delayed Modified Leslie-Gower Predator Prey System

Liu

Wang

2013

Abstract and Applied Analysis

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“…For more works on Leslie-Gower predator-prey model, one could refer to [1,4,7,8,19,21,26,29,30,[34][35][36][37] and the references cited therein.…”

Section: Introductionmentioning

confidence: 99%

Dynamic behaviors of a nonautonomous modified Leslie-Gower predator-prey model with Holling-type III schemes and a prey refuge

Chen¹,

Lin²,

Xie³

et al. 2017

J. Math. Computer Sci.

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A nonautonomous modified Leslie-Gower predator-prey model with Holling-type III schemes and a prey refuge is proposed and studied in this paper. Sufficient conditions which guarantee the permanence and global stability of the system are obtained, respectively. Our results indicate that the prey refuge has no influence on the persistent property of the system, while it has positive effect on the stability property of the system. Numeric simulations show the feasibility of the main results.

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Time delays and economic profit in fractional differential‐algebraic predator–prey model

Zhang,

Wang

2023

Math Methods in App Sciences

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In this paper, we formulate a fractional differential‐algebraic model with two discrete delays and double linear species harvesting. One of the delays denotes the time taken for digestion of the prey, and the other reflects the negative feedback of the predator's density. Taking into account of the economic profit, linear predator harvesting and prey harvesting have been incorporated into the proposed predator–prey system. From the biological viewpoint, we discuss the sufficient conditions for the existence of nontrivial positive equilibrium point. By jointly using the stability theory of fractional differential‐algebraic equations and the bifurcation theory, we study the delay‐induced instability and Hopf bifurcations. Finally, the correctness of the theoretical analysis is verified by numerical simulations, and the effects of delays, economic profit, and fractional exponents on the stability of the system are explored.

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Limit cycle and numerical similations for small and large delays in a predator–prey model with modified Leslie–Gower and Holling-type II schemes

Cited by 67 publications

References 7 publications

Bogdanov-Takens and Triple Zero Bifurcations of a Delayed Modified Leslie-Gower Predator Prey System

Bogdanov-Takens and Triple Zero Bifurcations of a Delayed Modified Leslie-Gower Predator Prey System

Dynamic behaviors of a nonautonomous modified Leslie-Gower predator-prey model with Holling-type III schemes and a prey refuge

Time delays and economic profit in fractional differential‐algebraic predator–prey model

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