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.

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 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 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.

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.

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.

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 Stability and Bifurcation Analysis on an Ecoepidemiological Model with Stage Structure and Time Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Lingshu Wang ◽  
Guanghui Feng
Keyword(s):  
Time Delay ◽  
Lyapunov Functions ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Positive Equilibrium ◽  
Prey Refuge ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Iteration Technique

An ecoepidemiological predator-prey model with stage structure for the predator and time delay due to the gestation of the predator is investigated. The effects of a prey refuge with disease in the prey population are concerned. By analyzing the corresponding characteristic equations, the local stability of each of the feasible equilibria of the model is discussed. Further, it is proved that the model undergoes a Hopf bifurcation at the positive equilibrium. By means of appropriate Lyapunov functions and LaSalle’s invariance principle, sufficient conditions are obtained for the global stability of the semitrivial boundary equilibria. By using an iteration technique, sufficient conditions are derived for the global attractiveness of the positive equilibrium.

scholarly journals A Predator-Prey Model with Functional Response and Stage Structure for Prey

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Xiao-Ke Sun ◽  
Hai-Feng Huo ◽  
Xiao-Bing Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Functional Response ◽  
Stage Structure ◽  
Transcendental Equation ◽  
Positive Equilibrium ◽  
Type Ii ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Type Ii Functional Response

A predator-prey system with Holling type II functional response and stage structure for prey is presented. The local and global stability are studied by analyzing the associated characteristic transcendental equation and using comparison theorem. The existence of a Hopf bifurcation at the positive equilibrium is also studied. Some numerical simulations are also given to illustrate our results.

scholarly journals On the Nature of Bifurcation in a Ratio-Dependent Predator-Prey Model with Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-17
Author(s):  
Changjin Xu ◽  
Yusen Wu
Keyword(s):  
Hopf Bifurcation ◽  
Local Stability ◽  
Time Delays ◽  
Dynamical Behavior ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Two Delays ◽  
Ratio Dependent

A ratio-dependent predator-prey model with two delays is investigated. The conditions which ensure the local stability and the existence of Hopf bifurcation at the positive equilibrium of the system are obtained. It shows that the two different time delays have different effects on the dynamical behavior of the system. An example together with its numerical simulations shows the feasibility of the main results. Finally, main conclusions are included.

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.

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