The stability and Hopf bifurcation for a predator-prey system with discrete and distributed delays

2010 ◽  
Author(s):  
Zhiping Shu ◽  
Zuoliang Xiong ◽  
Zhifang He
Keyword(s):  
Hopf Bifurcation ◽  
Distributed Delays ◽  
Predator Prey ◽  
Prey System ◽  
The Stability

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 Hopf bifurcation in a diffusive predator–prey model with Smith growth rate and herd behavior

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Heping Jiang ◽  
Huiping Fang ◽  
Yongfeng Wu
Keyword(s):  
Hopf Bifurcation ◽  
Neumann Boundary ◽  
Herd Behavior ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Prey System ◽  
Prey Model ◽  
Homogeneous Neumann Boundary Condition ◽  
The Stability

Abstract This paper mainly aims to consider the dynamical behaviors of a diffusive delayed predator–prey system with Smith growth and herd behavior subject to the homogeneous Neumann boundary condition. For the analysis of the predator–prey model, we have studied the existence of Hopf bifurcation by analyzing the distribution of the roots of associated characteristic equation. Then we have proved the stability of the periodic solution by calculating the normal form on the center of manifold which is associated to the Hopf bifurcation points. Some numerical simulations are also carried out in order to validate our analysis findings. The implications of our analytical and numerical findings are discussed critically.

scholarly journals The Hopf Bifurcation for a Predator-Prey System with -Logistic Growth and Prey Refuge

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Shaoli Wang ◽  
Zhihao Ge
Keyword(s):  
Hopf Bifurcation ◽  
Logistic Growth ◽  
Positive Equilibrium ◽  
Prey Refuge ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Predator Prey ◽  
Prey System ◽  
Form Theory ◽  
The Stability

The Hopf bifurcation for a predator-prey system with -logistic growth and prey refuge is studied. It is shown that the ODEs undergo a Hopf bifurcation at the positive equilibrium when the prey refuge rate or the index- passed through some critical values. Time delay could be considered as a bifurcation parameter for DDEs, and using the normal form theory and the center manifold reduction, explicit formulae are derived to determine the direction of bifurcations and the stability and other properties of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main results.

scholarly journals Stability and Hopf Bifurcation of Delayed Predator-Prey System Incorporating Harvesting

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Fengying Wei ◽  
Lanqi Wu ◽  
Yuzhi Fang
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Hopf Bifurcations ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Prey System ◽  
Theoretical Predictions ◽  
Form Theory ◽  
The Stability

A kind of delayed predator-prey system with harvesting is considered in this paper. The influence of harvesting and delay is investigated. Our results show that Hopf bifurcations occur as the delayτpasses through critical values. By using of normal form theory and center manifold theorem, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are obtained. Finally, numerical simulations are given to support our theoretical predictions.

Stability and Hopf Bifurcation of a Delayed Density-Dependent Predator–Prey System with Beddington–DeAngelis Functional Response

2016 ◽  
Vol 26 (10) ◽  
pp. 1650165 ◽  
Author(s):  
Haiyin Li ◽  
Gang Meng ◽  
Zhikun She
Keyword(s):  
Hopf Bifurcation ◽  
Density Dependence ◽  
Functional Response ◽  
Positive Equilibrium ◽  
Stability Switches ◽  
Predator Prey ◽  
Geometric Criterion ◽  
Density Dependent ◽  
Prey System ◽  
The Stability

In this paper, we investigate the stability and Hopf bifurcation of a delayed density-dependent predator–prey system with Beddington–DeAngelis functional response, where not only the prey density dependence but also the predator density dependence are considered such that the studied predator–prey system conforms to the realistically biological environment. We start with the geometric criterion introduced by Beretta and Kuang [2002] and then investigate the stability of the positive equilibrium and the stability switches of the system with respect to the delay parameter [Formula: see text]. Especially, we generalize the geometric criterion in [Beretta & Kuang, 2002] by introducing the condition [Formula: see text] which can be assured by the condition [Formula: see text], and adopting the technique of lifting to define the function [Formula: see text] for alternatively determining stability switches at the zeroes of [Formula: see text]s. Afterwards, by the Poincaré normal form for Hopf bifurcation in [Kuznetsov, 1998] and the bifurcation formulae in [Hassard et al., 1981], we qualitatively analyze the properties for the occurring Hopf bifurcations of the system (3). Finally, an example with numerical simulations is given to illustrate the obtained results.

scholarly journals Dynamical Analysis in a Delayed Predator-Prey System with Stage-Structure for Both the Predator and the Prey

2014 ◽  
Vol 2014 ◽  
pp. 1-19
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Dynamical Analysis ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Prey System ◽  
Form Theory ◽  
The Stability

A predator-prey system with two delays and stage-structure for both the predator and the prey is considered. Sufficient conditions for the local stability and the existence of periodic solutions via Hopf bifurcation with respect to both delays are obtained by analyzing the distribution of the roots of the associated characteristic equation. Specially, the direction of the Hopf bifurcation and the stability of the periodic solutions bifurcating from the Hopf bifurcation are determined by applying the normal form theory and center manifold argument. Some numerical simulations for justifying the theoretical analysis are also provided.

STABILITY AND HOPF BIFURCATION FOR A THREE-SPECIES REACTION–DIFFUSION PREDATOR–PREY SYSTEM WITH TWO DELAYS

2013 ◽  
Vol 23 (12) ◽  
pp. 1350194
Author(s):  
GAO-XIANG YANG ◽  
JIAN XU
Keyword(s):  
Hopf Bifurcation ◽  
Reaction Diffusion ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Prey System ◽  
Form Theory ◽  
Two Delays ◽  
The Stability ◽  
Bifurcating Periodic Solution

In this paper, a three-species predator–prey system with diffusion and two delays is investigated. By taking the sum of two delays as a bifurcation parameter, it is found that the spatially homogeneous Hopf bifurcation can occur as the sum of two delays crosses a critical value. The direction of Hopf bifurcation and the stability of the bifurcating periodic solution are obtained by employing the center manifold theorem and the normal form theory. In addition, some numerical simulations are also given to illustrate the theoretical analysis.

Hopf Bifurcation of a Generalized Delay-Induced Predator–Prey System with Habitat Complexity

2020 ◽  
Vol 30 (06) ◽  
pp. 2050082
Author(s):  
Zhihui Ma
Keyword(s):  
Hopf Bifurcation ◽  
Habitat Complexity ◽  
Sufficient Conditions ◽  
Explicit Formulas ◽  
Predator Prey ◽  
Dynamical Behaviors ◽  
Prey System ◽  
The Stability ◽  
Theoretical Results ◽  
Positivity And Boundedness

A delay-induced nonautonomous predator–prey system with variable habitat complexity is proposed based on mathematical and ecological issues, and this system is more realistic than the published models. Firstly, the permanence of the nonautonomous predation system is studied and some sufficient conditions are obtained. Secondly, the dynamical behaviors of the corresponding autonomous predation system are investigated, including the positivity and boundedness, and local and global stabilities. Thirdly, the properties of Hopf bifurcation of the autonomous predation system without/with delay are investigated and the explicit formulas which determine the stability and the direction of periodic solutions are obtained. Finally, a numerical example is given to test our theoretical results.

A diffusive predator–prey system with additional food and intra-specific competition among predators

2018 ◽  
Vol 11 (04) ◽  
pp. 1850060 ◽  
Author(s):  
Ruizhi Yang ◽  
Ming Liu ◽  
Chunrui Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Global Stability ◽  
Neumann Boundary ◽  
Turing Instability ◽  
Delay System ◽  
Additional Food ◽  
Predator Prey ◽  
Prey System ◽  
Boundary Equilibrium ◽  
The Stability

In this paper, a diffusive predator–prey system with additional food and intra-specific competition among predators subject to Neumann boundary condition is investigated. For non-delay system, global stability, Turing instability and Hopf bifurcation are studied. For delay system, instability and Hopf bifurcation induced by time delay and global stability of boundary equilibrium are discussed. By the theory of normal form and center manifold method, the conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution are derived.

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