scholarly journals Stability and Bifurcation in a Delayed Holling-Tanner Predator-Prey System with Ratio-Dependent Functional Response

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Juan Liu ◽  
Zizhen Zhang ◽  
Ming Fu
Keyword(s):  
Hopf Bifurcation ◽  
Asymptotic Stability ◽  
Numerical Simulations ◽  
Functional Response ◽  
Center Manifold ◽  
Predator Prey ◽  
Local Asymptotic Stability ◽  
Prey System ◽  
Ratio Dependent ◽  
The Stability

We analyze a delayed Holling-Tanner predator-prey system with ratio-dependent functional response. The local asymptotic stability and the existence of the Hopf bifurcation are investigated. Direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are studied by deriving the equation describing the flow on the center manifold. Finally, numerical simulations are presented for the support of our analytical findings.

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 Hopf Bifurcation and Global Periodic Solutions in a Predator-Prey System with Michaelis-Menten Type Functional Response and Two Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-16
Author(s):  
Yunxian Dai ◽  
Yiping Lin ◽  
Huitao Zhao
Keyword(s):  
Hopf Bifurcation ◽  
Functional Response ◽  
Center Manifold ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Local Hopf Bifurcation ◽  
Prey System ◽  
Model Study ◽  
Two Delays ◽  
Normal Form Method

We consider a predator-prey system with Michaelis-Menten type functional response and two delays. We focus on the case with two unequal and non-zero delays present in the model, study the local stability of the equilibria and the existence of Hopf bifurcation, and then obtain explicit formulas to determine the properties of Hopf bifurcation by using the normal form method and center manifold theorem. Special attention is paid to the global continuation of local Hopf bifurcation when the delaysτ1≠τ2.

scholarly journals Hopf Bifurcation Control in a Delayed Predator-Prey System with Prey Infection and Modified Leslie-Gower Scheme

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Hopf Bifurcation ◽  
State Feedback ◽  
Center Manifold ◽  
Controlled System ◽  
Predator Prey ◽  
Hybrid Controller ◽  
Prey System ◽  
The Stability ◽  
Manifold Theory ◽  
Bifurcation And Stability

Hopf bifurcation of a delayed predator-prey system with prey infection and the modified Leslie-Gower scheme is investigated. The conditions for the stability and existence of Hopf bifurcation of the system are obtained. The state feedback and parameter perturbation are used for controlling Hopf bifurcation in the system. In addition, direction of Hopf bifurcation and stability of the bifurcated periodic solutions of the controlled system are obtained by using normal form and center manifold theory. Finally, numerical simulation results are presented to show that the hybrid controller is efficient in controlling Hopf bifurcation.

A RATIO-DEPENDENT PREDATOR-PREY MODEL WITH DELAY AND HARVESTING

2010 ◽  
Vol 18 (02) ◽  
pp. 437-453 ◽  
Author(s):  
A. K. MISRA ◽  
B. DUBEY
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Functional Response ◽  
Bifurcation Parameter ◽  
Discrete Delay ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Ratio Dependent

In this paper a predator-prey model with discrete delay and harvesting of predator is proposed and analyzed by considering ratio-dependent functional response. Conditions of existence of various equilibria and their stability have been discussed. By taking delay as a bifurcation parameter, the system is found to undergo a Hopf bifurcation. Numerical simulations are also performed to illustrate the results.

Spatiotemporal Dynamics of a Diffusive Leslie–Gower Predator–Prey Model with Ratio-Dependent Functional Response

2015 ◽  
Vol 25 (05) ◽  
pp. 1530014 ◽  
Author(s):  
Hong-Bo Shi ◽  
Shigui Ruan ◽  
Ying Su ◽  
Jia-Fang Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Functional Response ◽  
Neumann Boundary ◽  
Turing Instability ◽  
Spatiotemporal Dynamics ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Ratio Dependent ◽  
Theoretical Results

This paper is devoted to the study of spatiotemporal dynamics of a diffusive Leslie–Gower predator–prey system with ratio-dependent Holling type III functional response under homogeneous Neumann boundary conditions. It is shown that the model exhibits spatial patterns via Turing (diffusion-driven) instability and temporal patterns via Hopf bifurcation. Moreover, the existence of spatiotemporal patterns is established via Turing–Hopf bifurcation at the degenerate points where the Turing instability curve and the Hopf bifurcation curve intersect. Various numerical simulations are also presented to illustrate the theoretical results.

Stability and Bifurcation Analysis in a Stage-Structured Predator-Prey Model with Delay

2014 ◽  
Vol 513-517 ◽  
pp. 3723-3727
Author(s):  
Hong Yan Wang ◽  
Hong Mei Wang
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
The Past ◽  
Prey System ◽  
Characteristic Values ◽  
The Stability ◽  
Manifold Theory

Hopf bifurcation occurs in most of dynamics systems when the influence from the past state varies. In modeling population dynamics, it is more reasonable taking into account the time delays. In this paper, a stage-structured predator-prey system with delay is considered. The existence of Hopf bifurcations at the positive equilibrium is established by analyzing the distribution of the characteristic values. An explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form and the center manifold theory. Numerical simulations to support the analytical conclusions are carried out.

Stability and Bifurcation in a State-Dependent Delayed Predator–Prey System

2016 ◽  
Vol 26 (04) ◽  
pp. 1650060 ◽  
Author(s):  
Aiyu Hou ◽  
Shangjiang Guo
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Local Stability ◽  
Perturbation Methods ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Prey System ◽  
State Dependent ◽  
The Stability ◽  
Theoretical Results

In this paper, we consider a class of predator–prey equations with state-dependent delayed feedback. Firstly, we investigate the local stability of the positive equilibrium and the existence of the Hopf bifurcation. Then we use perturbation methods to determine the sub/supercriticality of Hopf bifurcation and hence the stability of Hopf bifurcating periodic solutions. Finally, numerical simulations supporting our theoretical results are also provided.

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.

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