Stability and Hopf bifurcation analysis in a prey–predator system with stage-structure for prey and time delay

Abstract In this paper, we consider a Beddington-DeAngelis predator-prey system with stage structure for predator and time delay incorporating prey refuge. By analyzing the characteristic equations, we study the local stability of the equilibrium of the system. Using the delay as a bifurcation parameter, the model undergoes a Hopf bifurcation at the coexistence equilibrium when the delay crosses some critical values. After that, by constructing a suitable Lyapunov functional, sufficient conditions are derived for the global stability of the system. Finally, the influence of prey refuge on densities of prey species and predator species is discussed.

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Stability and Hopf Bifurcation Analysis on a Bazykin Model with Delay

Abstract and Applied Analysis ◽

10.1155/2014/539684 ◽

2014 ◽

Vol 2014 ◽

pp. 1-7 ◽

Cited By ~ 2

Author(s):

Jianming Zhang ◽

Lijun Zhang ◽

Chaudry Masood Khalique

Keyword(s):

Hopf Bifurcation ◽

Normal Form ◽

Periodic Solutions ◽

Bifurcation Analysis ◽

Center Manifold ◽

Positive Equilibrium ◽

Hopf Bifurcations ◽

Finite Delay ◽

The Stability ◽

Predator System

The dynamics of a prey-predator system with a finite delay is investigated. We show that a sequence of Hopf bifurcations occurs at the positive equilibrium as the delay increases. By using the theory of normal form and center manifold, explicit expressions for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived.

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Hopf Bifurcation Analysis of a Three-Stage-Structured Prey-Predator System with Multi-Delays

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.756-759.2857 ◽

2013 ◽

Vol 756-759 ◽

pp. 2857-2862

Author(s):

Shun Yi Li ◽

Wen Wu Liu

Keyword(s):

Hopf Bifurcation ◽

Bifurcation Analysis ◽

Local Stability ◽

Positive Equilibrium ◽

Characteristic Equations ◽

Numerical Examples ◽

Predator Model ◽

Predator System

A three-stage-structured prey-predator model with multi-delays is considered. The characteristic equations and local stability of the equilibrium are analyzed, and the conditions for the positive equilibrium occurring Hopf bifurcation are obtained by applying the theorem of Hopf bifurcation. Finally, numerical examples and brief conclusion are given.

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BIFURCATION ANALYSIS OF A DETRITUS-BASED ECOSYSTEM WITH TIME DELAY

Journal of Biological System ◽

10.1142/s0218339000000183 ◽

2000 ◽

Vol 08 (03) ◽

pp. 255-261 ◽

Cited By ~ 11

Author(s):

DEBASIS MUKHERJEE ◽

SANTANU RAY ◽

DILIP KUMAR SINHA

Keyword(s):

Hopf Bifurcation ◽

Time Delay ◽

Bifurcation Analysis ◽

Local Stability ◽

Mangrove Ecosystem ◽

Stability Criteria ◽

Delay Effect

This article concentrates on the study of delay effect of a mangrove ecosystem of detritus, detritivores and predator of detritivores. Local stability criteria are derived in the absence of delays. Conditions are found out for which the system undergoes a Hopf bifurcation. Further conditions are derived for which there can be no change in stability.

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Chaos and Hopf Bifurcation Analysis of the Delayed Local Lengyel-Epstein System

Discrete Dynamics in Nature and Society ◽

10.1155/2014/139375 ◽

2014 ◽

Vol 2014 ◽

pp. 1-7

Author(s):

Qingsong Liu ◽

Yiping Lin ◽

Jingnan Cao ◽

Jinde Cao

Keyword(s):

Hopf Bifurcation ◽

Time Delay ◽

Numerical Simulations ◽

Bifurcation Analysis ◽

Center Manifold ◽

Local Reaction ◽

Reaction Diffusion ◽

Critical Value ◽

Hopf Bifurcations ◽

The Stability

The local reaction-diffusion Lengyel-Epstein system with delay is investigated. By choosingτas bifurcating parameter, we show that Hopf bifurcations occur when time delay crosses a critical value. Moreover, we derive the equation describing the flow on the center manifold; then we give the formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Finally, numerical simulations are performed to support the analytical results and the chaotic behaviors are observed.

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