The stability and Hopf bifurcation for a predator–prey system with time delay

Çelik, Canan

doi:10.1016/j.chaos.2007.10.045

Cited by 63 publications

(30 citation statements)

References 28 publications

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“…For the analysis of delayed system, we refer to [7,16,17]. Equilibrium points of the system without maturation delay and of that with maturation delay are the same, as time delay does not change the equilibria of the system.…”

Section: System With Maturation Delaymentioning

confidence: 99%

A dynamical approach to the legal and illegal logging of forestry population and conservation using taxation

Chaudhary

Pathak

2017

Adv Differ Equ

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Over-harvesting of forestry resource, primarily trees, through illegal logging has been exceptionally regular for decades. In the context of this worldwide issue, a mathematical model considering the joined impact of legal and illegal logging of trees from forestry biomass using delay-driven ordinary differential equations is proposed. For the set of equations, we have taken immature, mature forestry biomass, and industrial densities as three state variables. Additionally, the effect of time-lag for the conversion of immature forestry biomass to mature forestry biomass is considered. System boundedness, feasible equilibrium analysis and the stability of all the feasible equilibria is examined using the differential equation theory. From the detailed analysis of the system, it is observed that with the delay in time, the system bifurcates as it reaches the critical threshold. While without the delay, the system is asymptotically stable. Biologic and bio-economic results of the system are also interpreted for the optimal equilibrium solutions. The optimal path is obtained by constructing the Hamiltonian, which is further solved using Pontryagin"s principle associated with the control problem. Further, numerical simulation is also provided in support of analytical results. Moreover, the normalized forward sensitivity index is used to analyze the parameter sensitivity.MSC: 34D; 34H; 90A; 92B

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Section: System With Maturation Delaymentioning

confidence: 99%

A dynamical approach to the legal and illegal logging of forestry population and conservation using taxation

Chaudhary

Pathak

2017

Adv Differ Equ

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“…The predator-prey system is important in dynamical population models and has been discussed by many authors [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

Analysis of a Predator-Prey Model with Switching and Stage-Structure for Predator

Suebcharoen

2017

International Journal of Differential Equations

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This paper studies the behavior of a predator-prey model with switching and stage-structure for predator. Bounded positive solution, equilibria, and stabilities are determined for the system of delay differential equation. By choosing the delay as a bifurcation parameter, it is shown that the positive equilibrium can be destabilized through a Hopf bifurcation. Some numerical simulations are also given to illustrate our results.

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“…In system (1.1), there is no delay. In [1], Celik incorporated the delay into the system, then system (1.1) becomes dN (t) dt = r 1 N (t) − εP (t)N (t), dP (t) dt = P (t)(r 2 − θ P (t−τ ) N (t) ), (1.2) where τ ≥ 0 denotes the delay time for the predator density. In system (1.2) predator density is logistic with time delay and the carrying capacity proportional to prey density.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation analysis for a ratio-dependent predator-prey system with multiple delays

Lv¹,

Zhang²,

Tang³

2016

J. Nonlinear Sci. Appl.

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In this paper, we consider a ratio-dependent predator-prey system with multiple delays where the dynamics are logistic with the carrying capacity proportional to prey population. By choosing the sum τ of two delays as the bifurcation parameter, the stability of the positive equilibrium and the existence of Hopf bifurcation are investigated. 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. Finally, some numerical simulations are carried out for illustrating the theoretical results.

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The stability and Hopf bifurcation for a predator–prey system with time delay

Cited by 63 publications

References 28 publications

A dynamical approach to the legal and illegal logging of forestry population and conservation using taxation

A dynamical approach to the legal and illegal logging of forestry population and conservation using taxation

Analysis of a Predator-Prey Model with Switching and Stage-Structure for Predator

Bifurcation analysis for a ratio-dependent predator-prey system with multiple delays

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