scholarly journals Stability and Hopf Bifurcation Analysis for a Stage-Structured Predator-Prey Model with Discrete and Distributed Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Ruiqing Shi ◽  
Junmei Qi ◽  
Sanyi Tang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Three Dimensional ◽  
Distributed Delays ◽  
Positive Equilibrium ◽  
Dimensional System ◽  
Normal Form Theory ◽  
Predator Prey Model ◽  
Prey Model ◽  
Form Theory

We propose a three-dimensional stage-structured predatory-prey model with discrete and distributed delays. By use of a new variable, the original three-dimensional system transforms into an equivalent four-dimensional system. Firstly, we study the existence and local stability of positive equilibrium of the new system. And, by choosing the time delayτas a bifurcation parameter, we show that Hopf bifurcation may occur as the time delayτpasses through some critical values. Secondly, by use of normal form theory and central manifold argument, we establish the direction and stability of Hopf bifurcation. At last, some simple discussion is presented.

scholarly journals Stability and Bifurcation Analysis for a Predator-Prey Model with Discrete and Distributed Delay

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Ruiqing Shi ◽  
Junmei Qi ◽  
Sanyi Tang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Three Dimensional ◽  
Distributed Delay ◽  
Dimensional System ◽  
Two Dimensional ◽  
Normal Form Theory ◽  
Predator Prey Model ◽  
Prey Model ◽  
Form Theory

We propose a two-dimensional predatory-prey model with discrete and distributed delay. By the use of a new variable, the original two-dimensional system transforms into an equivalent three-dimensional system. Firstly, we study the existence and local stability of equilibria of the new system. And, by choosing the time delayτas a bifurcation parameter, we show that Hopf bifurcation can occur as the time delayτpasses through some critical values. Secondly, by the use of normal form theory and central manifold argument, we establish the direction and stability of Hopf bifurcation. At last, an example with numerical simulations is provided to verify the theoretical results. In addition, some simple discussion is also presented.

scholarly journals Bifurcation Analysis for a Predator-Prey Model with Time Delay and Delay-Dependent Parameters

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Changjin Xu
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Form Theory ◽  
The Stability ◽  
Manifold Theory ◽  
Delay Dependent

A class of stage-structured predator-prey model with time delay and delay-dependent parameters is considered. Its linear stability is investigated and Hopf bifurcation is demonstrated. Using normal form theory and center manifold theory, some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained. Finally, numerical simulations are performed to verify the analytical results.

scholarly journals Dynamical Analysis in a Delayed Predator-Prey Model with Two Delays

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Changjin Xu ◽  
Peiluan Li
Keyword(s):  
Hopf Bifurcation ◽  
Dynamical Analysis ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Form Theory ◽  
The Stability ◽  
Manifold Theory

A class of Beddington-DeAngelis functional response predator-prey model is considered. The conditions for the local stability and the existence of Hopf bifurcation at the positive equilibrium of the system are derived. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using the normal form theory and center manifold theory. Some numerical simulations for justifying the theoretical analysis are also provided. Finally, main conclusions are given.

Hopf Bifurcation Analysis and Stability for a Ratio-Dependent Predator–Prey Diffusive System with Time Delay

2020 ◽  
Vol 30 (03) ◽  
pp. 2050037
Author(s):  
Longyue Li ◽  
Yingying Mei ◽  
Jianzhi Cao
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Periodic Solutions ◽  
Positive Equilibrium ◽  
Dimensional System ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Predator Prey ◽  
Ratio Dependent ◽  
Low Dimensional

In this paper, we are focused on a new ratio-dependent predator–prey system that introduced the diffusive and time delay effect simultaneously. By analyzing the characteristic equations and the distribution of eigenvalues, we examine the stability and boundary of positive equilibrium states, and the existence of spatially homogeneous and spatially inhomogeneous bifurcating periodic solutions, respectively. Further, we prove that when [Formula: see text], the system has Hopf bifurcation at the positive equilibrium state. By using the center manifold reduction, we simplify the system so that we can convert an infinite-dimensional system into a low-dimensional finite-dimensional system. By using the normal form theory, we obtain explicit expressions for the direction, stability and period of Hopf bifurcation periodic solutions. Finally, we have illustrated the main results in this thesis by numerical examples, our work may provide some useful measures to save time or cost and to control the ecosystem.

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.

The Stability and Hopf Bifurcation for a Predator-Prey Model with Delays

2014 ◽  
Vol 926-930 ◽  
pp. 3314-3317
Author(s):  
Hong Bing Chen
Keyword(s):  
Periodic Solution ◽  
Hopf Bifurcation ◽  
Normal Form ◽  
Center Manifold ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Form Theory ◽  
The Stability

In this paper, a predator–prey model with discrete and distributed delays is investigated. the direction of Hopf bifurcation as well as stability of periodic solution are studied. The method which we used is the normal form theory and center manifold. At last, an example showed the feasibility of results.

scholarly journals Stability and Hopf bifurcation analysis for a Lotka-Volterra predator-prey model with two delays

2011 ◽  
Vol 21 (1) ◽  
pp. 97-107 ◽  
Author(s):  
Changjin Xu ◽  
Maoxin Liao ◽  
Xiaofei He
Keyword(s):  
Hopf Bifurcation ◽  
Bifurcation Analysis ◽  
Transcendental Equation ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Two Delays ◽  
Manifold Theory

Stability and Hopf bifurcation analysis for a Lotka-Volterra predator-prey model with two delays In this paper, a two-species Lotka-Volterra predator-prey model with two delays is considered. By analyzing the associated characteristic transcendental equation, the linear stability of the positive equilibrium is investigated and Hopf bifurcation is demonstrated. Some explicit formulae for determining the stability and direction of Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using normal form theory and center manifold theory. Some numerical simulations for supporting the theoretical results are also included.

BIFURCATION ANALYSIS FOR A PREDATOR-PREY SYSTEM WITH PREY DISPERSAL AND TIME DELAY

2008 ◽  
Vol 01 (02) ◽  
pp. 209-224 ◽  
Author(s):  
QINTAO GAN ◽  
RUI XU ◽  
PINGHUA YANG
Keyword(s):  
Time Delay ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Theoretical Predictions ◽  
Form Theory ◽  
The Stability ◽  
Manifold Reduction

In this paper, a predator-prey model with prey dispersal and time delay is investigated. By analyzing the corresponding characteristic equation of a positive equilibrium, the local stability of the positive equilibrium and the existence of Hopf bifurcation are discussed. By using the normal form theory and center manifold reduction, explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. Numerical simulations are given to illustrate the theoretical predictions.

Hopy Bifurcation and Stability Analysis in a Predator-Prey Model with Distributed Delays

2014 ◽  
Vol 1049-1050 ◽  
pp. 1400-1402 ◽  
Author(s):  
Hong Bing Chen
Keyword(s):  
Hopf Bifurcation ◽  
Stage Structure ◽  
Distributed Delays ◽  
Stable Equilibrium Point ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Form Theory ◽  
Necessary And Sufficient ◽  
Bifurcation And Stability

In this paper, A mathematical model of two species with stage structure and distributed delays is investigated, the necessary and sufficient of the stable equilibrium point are studied. Further, by analyze the associated characteristic equation, it is founded that Hopf bifurcation occurs when τ crosses some critical value. The direction of Hopf bifurcation as well as stability of periodic solution are studied. Using the normal form theory and center manifold method.

scholarly journals Dynamical analysis of a giving up smoking model with time delay

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Zizhen Zhang ◽  
Ruibin Wei ◽  
Wanjun Xia
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Local Stability ◽  
Center Manifold ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Dynamical Analysis ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory

AbstractIn this paper, we are concerned with a delayed smoking model in which the population is divided into five classes. Sufficient conditions guaranteeing the local stability and existence of Hopf bifurcation for the model are established by taking the time delay as a bifurcation parameter and employing the Routh–Hurwitz criteria. Furthermore, direction and stability of the Hopf bifurcation are investigated by applying the center manifold theorem and normal form theory. Finally, computer simulations are implemented to support the analytic results and to analyze the effects of some parameters on the dynamical behavior of the model.

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