AN ECO-EPIDEMIOLOGICAL MODEL WITH INFECTIOUS DISEASE IN FOOD CHAIN

2011 ◽  
Vol 21 (07) ◽  
pp. 1935-1952 ◽  
Author(s):  
YONGXUE CHEN ◽  
YONG JIANG
Keyword(s):  
Food Chain ◽  
Center Manifold ◽  
Equation Model ◽  
Normal Form Theory ◽  
Model Stability ◽  
Predator Prey ◽  
Form Theory ◽  
Time Required ◽  
Delay Differential ◽  
Infected Prey

In this paper, a model of predator-prey with disease in food chain is investigated — where, prey is infected by bacteria and then the infected prey in turn infects predator, but the disease does not spread among predators. The law for disease development and biodiversity conservation are the focus. Stability and persistence are deduced in terms of system parameters. Next, time required delay is incorporated into the model. Stability and bifurcation analysis of the delay differential equation model are carried out. Furthermore, stability and direction of the bifurcating periodic solutions are performed by the normal form theory and the center manifold argument. Finally, numerical simulations are included for illustrating the theoretical analysis.

Codimension Bifurcation Analysis of a Modified Leslie–Gower Predator–Prey Model with Two Delays

2018 ◽  
Vol 28 (05) ◽  
pp. 1850060 ◽  
Author(s):  
Jianfeng Jiao ◽  
Ruiqi Wang ◽  
Hongcui Chang ◽  
Xia Liu
Keyword(s):  
Time Delays ◽  
Center Manifold ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model ◽  
Prey Model ◽  
Form Theory ◽  
Two Delays ◽  
Delay Differential

The Bogdanov–Takens (B–T) and triple-zero bifurcations of a modified Leslie–Gower predator–prey model with two time delays are studied in this paper. By generalizing and using the normal form theory and center manifold theorem for delay differential equations, the normal forms of the B–T and triple-zero bifurcations of the model at its interior equilibria are obtained. In addition, some numerical simulations are presented to illustrate our main results.

scholarly journals Dynamical Behavior in a Four-Dimensional Neural Network Model with Delay

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Changjin Xu ◽  
Peiluan Li
Keyword(s):  
Neural Network ◽  
Hopf Bifurcation ◽  
Network Model ◽  
Neural Network Model ◽  
Center Manifold ◽  
Dynamical Behavior ◽  
Normal Form Theory ◽  
Form Theory ◽  
Explicit Formulae ◽  
Delay Differential

A four-dimensional neural network model with delay is investigated. With the help of the theory of delay differential equation and Hopf bifurcation, the conditions of the equilibrium undergoing Hopf bifurcation are worked out by choosing the delay as parameter. Applying the normal form theory and the center manifold argument, we derive the explicit formulae for determining the properties of the bifurcating periodic solutions. Numerical simulations are performed to illustrate the analytical results.

scholarly journals Stability and Hopf Bifurcation Analysis of a Vector-Borne Disease with Time Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Yuanyuan Chen ◽  
Ya-Qing Bi
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Bifurcation Theory ◽  
Center Manifold ◽  
Local Analysis ◽  
Normal Form Theory ◽  
Form Theory ◽  
Vector Borne ◽  
The Stability ◽  
Delay Differential

A delay-differential modelling of vector-borne is investigated. Its dynamics are studied in terms of local analysis and Hopf bifurcation theory, and its linear stability and Hopf bifurcation are demonstrated by studying the characteristic equation. The stability and direction of Hopf bifurcation are determined by applying the normal form theory and the center manifold argument.

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.

HOPF BIFURCATION INDUCED BY NEUTRAL DELAY IN A PREDATOR–PREY SYSTEM

2013 ◽  
Vol 23 (11) ◽  
pp. 1350174 ◽  
Author(s):  
BEN NIU ◽  
WEIHUA JIANG
Keyword(s):  
Hopf Bifurcation ◽  
Dynamical Behavior ◽  
Neutral Delay ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Prey System ◽  
Neutral Delay Differential Equations ◽  
Form Theory ◽  
Delay Differential ◽  
Quasiperiodic Oscillations

A predator–prey system with neutral delay is investigated from the viewpoint of bifurcation analysis on neutral delay differential equations. Stability and Hopf bifurcation of the inner equilibrium are given, by which we show how the neutral terms affect the dynamical behavior of the prey and the predator. To give more detailed information on the periodic oscillations, the direction and stability of Hopf bifurcation are studied by using the normal form theory of neutral equation. We find neutral delay makes the predator–prey system more complicated and usually induces stability switches or double Hopf bifurcations. Near the double Hopf bifurcation we give the detailed bifurcation set by calculating the universal unfoldings. It is shown that the population of prey or predator may exhibit transient quasiperiodic oscillations driven by the neutral delay. Finally, we carry out several groups of illustrations.

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 Local Hopf Bifurcation for a Predator-Prey Model with Delay

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Yakui Xue ◽  
Xiaoqing Wang
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Local Analysis ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Local Hopf Bifurcation ◽  
Form Theory ◽  
Explicit Formulae ◽  
The Stability

A predator-prey system with disease in the predator is investigated, where the discrete delayτis regarded as a parameter. Its dynamics are studied in terms of local analysis and Hopf bifurcation analysis. By analyzing the associated characteristic equation, it is found that Hopf bifurcation occurs whenτcrosses some critical values. Using the normal form theory and center manifold argument, the explicit formulae which determine the stability, direction, and other properties of bifurcating periodic solutions are derived.

scholarly journals Dynamic Analysis and Hopf Bifurcation of a Lengyel–Epstein System with Two Delays

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Long Li ◽  
Yanxia Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Sufficient Conditions ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Two Delays ◽  
Stability Method ◽  
The Stability ◽  
Delay Differential

In this paper, a Lengyel–Epstein model with two delays is proposed and considered. By choosing the different delay as a parameter, the stability and Hopf bifurcation of the system under different situations are investigated in detail by using the linear stability method. Furthermore, the sufficient conditions for the stability of the equilibrium and the Hopf conditions are obtained. In addition, the explicit formula determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are obtained with the normal form theory and the center manifold theorem to delay differential equations. Some numerical examples and simulation results are also conducted at the end of this paper to validate the developed theories.

scholarly journals Bogdanov-Takens and Triple Zero Bifurcations of a Delayed Modified Leslie-Gower Predator Prey System

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Xia Liu ◽  
Jinling Wang
Keyword(s):  
Center Manifold ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Prey System ◽  
Form Theory ◽  
Nonlinear Harvesting ◽  
Existence Conditions ◽  
Manifold Reduction

A delayed modified Leslie-Gower predator prey system with nonlinear harvesting is considered. The existence conditions that an equilibrium is Bogdanov-Takens (BT) or triple zero singularity of the system are given. By using the center manifold reduction, the normal form theory, and the formulae developed by Xu and Huang, 2008 and Qiao et al., 2010, the normal forms and the versal unfoldings for this singularity are presented. The Hopf bifurcation of the system at another interior equilibrium is analyzed by taking delay (small or large) as bifurcation parameter.

scholarly journals Hopf bifurcation of a delayed worm model with two latent periods

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Juan Liu ◽  
Zizhen Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Sensor Network ◽  
Center Manifold ◽  
Sufficient Conditions ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Worm Model ◽  
Distribution Of Roots ◽  
Influential Parameters

Abstract We investigate a delayed epidemic model for the propagation of worm in wireless sensor network with two latent periods. We derive sufficient conditions for local stability of the worm-induced equilibrium of the system and the existence of Hopf bifurcation by regarding different combination of two latent time delays as the bifurcation parameter and analyzing the distribution of roots of the associated characteristic equation. In particular, we investigate the direction and stability of the Hopf bifurcation by means of the normal form theory and center manifold theorem. To verify analytical results, we present numerical simulations. Also, the effect of some influential parameters of sensor network is properly executed so that the oscillations can be reduced and removed from the network.

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