Complex dynamics in a prey predator system with multiple delays

Gakkhar, Sunita; Singh, Anuraj

doi:10.1016/j.cnsns.2011.05.047

Cited by 95 publications

(55 citation statements)

References 28 publications

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“…This form has the advantage that it simplifies the analytical work and also it is the form present in many population dynamical models involving delays [6], [7], [9], [10]. The DDE (2) may or may not have equilibrium points (or steady states) and these will depend on the values of µ.…”

Section: A One Equation With One Delaymentioning

confidence: 99%

“…Published papers have shown that the incorporation of discrete time delays can highly impact the dynamics of the system, since they can switch the stability of a steady state point, and can also cause the system to go through a Hopf bifurcation near that steady state point (Culshaw [6], Gakkhar [7], Bellen [3]). In this paper we consider a system of n delay differential equations (DDE's) with one parameter µ as the bifurcation parameter and also with one or more discrete time delays, τ , which can also behave as bifurcation parameters.…”

Section: Introductionmentioning

confidence: 99%

“…DDE's have been extensively studied by many researchers including pioneers Bellman [4], Driver [5], and in more recent years by Culshaw [6], Gakkhar [7], Bellen [3], and a superb monograph on the subject by Gopalsamy [8]. While most of these research papers focus on issue of the stability changes caused by the delay(s), the main motivation of this paper is to study how a local bifurcation parameter of the system may affect the changes in stability caused by the delay (s).…”

Section: Introductionmentioning

confidence: 99%

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Effects of Discrete Time Delays and Parameters Variation on Dynamical Systems

Diakite

2015

BIOMATH

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Abstract-Delay Differential Equations (DDE's) have received considerable attention in recent years. While most of these articles focused on the effects of the time delays on the stability of the equilibrium points and on the bifurcation that they may raised, very few papers address the key roles that system parameters play on if and how the discrete delays induce stability changes of the equilibria and produce bifurcations near such equilibria. In this article we focus on that question in a general setting, that is, if one has a system of DDE's with one or multiple discrete time delays, what are the results of changing the system parameters values on the effects of the discrete time delays on the dynamic of the system. We present general results for one equation with one and two delays and study a specific example of one equation with one delay. We then establish the procedure for n equations with multiple delays and do a specific example for two equations with two delays. We compute the steady states and analyze their stability as both chosen bifurcation parameters, the discrete time delay τ and a local equation parameter µ, cross critical values. Our analysis shows that while changes in both parameters can destabilize the steady state, the discrete time delay can only cause stability switches of the steady state for certain values of µ, while the effects of the local equation parameter on the steady state do not necessarily depend on the value of τ . While µ may cause the system to go through different type of bifurcations, the discrete time delay can only introduce a Hopf bifurcation for certain values of µ.

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Section: A One Equation With One Delaymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Effects of Discrete Time Delays and Parameters Variation on Dynamical Systems

Diakite

2015

BIOMATH

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“…Now the similar methods in [20,23,24] are applied to investigate the distribution of roots of (3.2). When…”

Section: ð2:1þmentioning

confidence: 99%

Dynamics for Phytoplankton-Zooplankton System with Time Delays

Jiang

Takeuchi

2017

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Abstract. In this paper, a system consisting of two harmful phytoplanktons and one zooplankton with two time delays is investigated. Firstly, the global existence, nonnegativity and boundedness of the solutions of the system are discussed. Secondly, using time delays as bifurcating parameters, the existence of local Hopf bifurcations at the positive equilibrium of the system is investigated in details. The phenomenon of stability switches is confirmed under some certain conditions. It is shown by numerical simulations under some suitable parameters that the system can exhibit complicated dynamic properties, and undergoes changes from stable periodic solution or equilibrium to chaos or from chaos to stable periodic solution or equilibrium. At last, some conclusions are given. The direction of the Hopf bifurcations and the stability of the bifurcation periodic solutions are given in Appendix by using the center manifold and normal form theory.

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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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Complex dynamics in a prey predator system with multiple delays

Cited by 95 publications

References 28 publications

Effects of Discrete Time Delays and Parameters Variation on Dynamical Systems

Effects of Discrete Time Delays and Parameters Variation on Dynamical Systems

Dynamics for Phytoplankton-Zooplankton System with Time Delays

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

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