Stability Analysis in a Delayed Predator-Prey Model

2012 ◽  
Vol 472-475 ◽  
pp. 2940-2943
Author(s):  
Zhi Chao Jiang ◽  
Hui Chen
Keyword(s):  
Stability Analysis ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Local Stability ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Boundary Equilibrium ◽  
System With Time Delay

A stage-structured predator-prey system with time delay is considered. By analyzing the characteristic equations, the local stability of a positive equilibrium and a boundary equilibrium is discussed, respectively. Furthermore, it is proved that the system undergoes a Hopf bifurcation at the positive equilibrium when . The estimation of the length of delay to preserve stability has also been calculated.

scholarly journals Hopf bifurcation analysis in a stage-structure predator–prey model with two time delay

2022 ◽  
Vol 355 ◽  
pp. 03048
Author(s):  
Bochen Han ◽  
Shengming Yang ◽  
Guangping Zeng
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Local Stability ◽  
Time Delays ◽  
Stage Structure ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Predator Model

In this paper, we consider a predator-prey system with two time delays, which describes a prey–predator model with parental care for predators. The local stability of the positive equilibrium is analysed. By choosing the two time delays as the bifurcation parameter, the existence of Hopf bifurcation is studied. Numerical simulations show the positive equilibrium loses its stability via the Hopf bifurcation when the time delay increases beyond a threshold.

scholarly journals Stability and Hopf Bifurcation of a Predator-Prey Model with Distributed Delays and Competition Term

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Lv-Zhou Zheng
Keyword(s):  
Hopf Bifurcation ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Distributed Delays ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Positive Equilibrium Point ◽  
The Stability

A class of predator-prey system with distributed delays and competition term is considered. By considering the time delay as bifurcation parameter, we analyze the stability and the Hopf bifurcation of the predator-prey system. According to the theorem of Hopf bifurcation, some sufficient conditions are obtained for the local stability of the positive equilibrium point.

scholarly journals The Effect of Time Delay on Dynamical Behavior in an Ecoepidemiological Model

2012 ◽  
Vol 2012 ◽  
pp. 1-18
Author(s):  
Changjin Xu ◽  
Yusen Wu
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Local Stability ◽  
Time Delays ◽  
Dynamical Behavior ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model

A delayed predator-prey model with disease in the prey is investigated. The conditions for the local stability and the existence of Hopf bifurcation at the positive equilibrium of the system are derived. The effect of the two different time delays on the dynamical behavior has been given. Numerical simulations are performed to illustrate the theoretical analysis. Finally, the main conclusions are drawn.

scholarly journals Stability and Hopf Bifurcation in a Delayed Predator-Prey System with Herd Behavior

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Chaoqun Xu ◽  
Sanling Yuan
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Positive Equilibrium ◽  
Herd Behavior ◽  
Bifurcation Parameter ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Self Defense ◽  
Prey System ◽  
The Stability

A special predator-prey system is investigated in which the prey population exhibits herd behavior in order to provide a self-defense against predators, while the predator is intermediate and its population shows individualistic behavior. Considering the fact that there always exists a time delay in the conversion of the biomass of prey to that of predator in this system, we obtain a delayed predator-prey model with square root functional response and quadratic mortality. For this model, we mainly investigate the stability of positive equilibrium and the existence of Hopf bifurcation by choosing the time delay as a bifurcation parameter.

scholarly journals Hopf bifurcation and stability in a Beddington-DeAngelis predator-prey model with stage structure for predator and time delay incorporating prey refuge

2019 ◽  
Vol 17 (1) ◽  
pp. 141-159 ◽  
Author(s):  
Zaowang Xiao ◽  
Zhong Li ◽  
Zhenliang Zhu ◽  
Fengde Chen
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Prey Refuge ◽  
Predator Species ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Bifurcation And Stability

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.

STABILITY AND HOPF BIFURCATION FOR AN ECO-EPIDEMIOLOGY MODEL WITH HOLLING-III FUNCTIONAL RESPONSE AND DELAYS

2008 ◽  
Vol 01 (03) ◽  
pp. 377-389 ◽  
Author(s):  
WEI ZOU ◽  
JIEHUA XIE ◽  
ZUOLIANG XIONG
Keyword(s):  
Hopf Bifurcation ◽  
Functional Differential Equations ◽  
Positive Equilibrium ◽  
Asymptotically Stable ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Retarded Functional Differential Equations ◽  
Boundary Equilibrium ◽  
Functional Differential ◽  
Holling Iii Functional Response

In this paper, a system of retarded functional differential equations is proposed as a predator-prey model with disease in the prey. The invariance of non-negativity, nature of boundary equilibrium and global stability are analyzed. It also shows that positive equilibrium is locally asymptotically stable when time delay τ = τ1 + τ2 is suitable small, while a loss of stability by a Hopf bifurcation can occur around the positive equilibrium as the delays increase.

scholarly journals Hopf Bifurcation of a Predator-Prey System with Delays and Stage Structure for the Prey

2012 ◽  
Vol 2012 ◽  
pp. 1-28 ◽  
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Prey System ◽  
The Stability

This paper is concerned with a Holling type III predator-prey system with stage structure for the prey population and two time delays. The main result is given in terms of local stability and bifurcation. By choosing the time delay as a bifurcation parameter, sufficient conditions for the local stability of the positive equilibrium and the existence of periodic solutions via Hopf bifurcation with respect to both delays are obtained. In particular, explicit formulas that can determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are established by using the normal form method and center manifold theorem. Finally, numerical simulations supporting the theoretical analysis are also included.

Rank One Strange Attractors in Periodically Kicked Predator–Prey System with Time-Delay

2016 ◽  
Vol 26 (07) ◽  
pp. 1640114 ◽  
Author(s):  
Wenjie Yang ◽  
Yiping Lin ◽  
Yunxian Dai ◽  
Huitao Zhao
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Strange Attractors ◽  
Periodic Force ◽  
Delayed Systems ◽  
Predator Prey ◽  
Prey System ◽  
Rank One ◽  
Delayed System ◽  
System With Time Delay

This paper is devoted to the study of the problem of rank one strange attractor in a periodically kicked predator–prey system with time-delay. Our discussion is based on the theory of rank one maps formulated by Wang and Young. Firstly, we develop the rank one chaotic theory to delayed systems. It is shown that strange attractors occur when the delayed system undergoes a Hopf bifurcation and encounters an external periodic force. Then we use the theory to the periodically kicked predator–prey system with delay, deriving the conditions for Hopf bifurcation and rank one chaos along with the results of numerical simulations.

scholarly journals Time delay induced Hopf bifurcation in a diffusive predator–prey model with prey toxicity

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ruizhi Yang ◽  
Yuxin Ma ◽  
Chiyu Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Functional Differential Equations ◽  
Positive Equilibrium ◽  
Center Manifold Reduction ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Functional Differential ◽  
The Stability

AbstractIn this paper, we consider a diffusive predator–prey model with a time delay and prey toxicity. The effect of time delay on the stability of the positive equilibrium is studied by analyzing the eigenvalue spectrum. Delay-induced Hopf bifurcation is also investigated. By utilizing the normal form method and center manifold reduction for partial functional differential equations, the formulas for determining the property of Hopf bifurcation are given.

On the Existence of Periodic Solutions of a Three-Patch Diffusion Predator-Prey System

2009 ◽  
Vol 64 (7-8) ◽  
pp. 405-410
Author(s):  
Mohammed Ismail ◽  
Atta A. K. Abu Hany ◽  
Aysha Agha
Keyword(s):  
Mathematical Model ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Periodic Orbit ◽  
Time Delays ◽  
Positive Equilibrium ◽  
Bifurcation Parameter ◽  
Predator Prey ◽  
Prey System ◽  
Existence Of Periodic Solutions

AbstractWe establish a mathematical model for the three-patch diffusion predator-prey system with time delays. The theory of Hopf bifurcation is implemented, choosing the time delay parameter as a bifurcation parameter. We present the condition for the existence of a periodic orbit of the Hopf-type from the positive equilibrium.

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