scholarly journals Dynamical Behaviour of a Nutrient-Plankton Model with Holling Type IV, Delay, and Harvesting

2018 ◽  
Vol 2018 ◽  
pp. 1-19
Author(s):  
Xin-You Meng ◽  
Jiao-Guo Wang ◽  
Hai-Feng Huo
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Local Stability ◽  
Dynamical Behaviour ◽  
Optimal Harvesting ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Type Iv ◽  
Local Hopf Bifurcation ◽  
Plankton Model

In this paper, a Holling type IV nutrient-plankton model with time delay and linear plankton harvesting is investigated. The existence and local stability of all equilibria of model without time delay are given. Regarding time delay as bifurcation parameter, such system around the interior equilibrium loses its local stability, and Hopf bifurcation occurs when time delay crosses its critical value. In addition, the properties of the bifurcating periodic solutions are investigated based on normal form theory and center manifold theorem. What is more, the global continuation of the local Hopf bifurcation is discussed by using a global Hopf bifurcation result. Furthermore, the optimal harvesting is obtained by the Pontryagin’s Maximum Principle. Finally, some numerical simulations are given to confirm our theoretical analysis.

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.

scholarly journals Stability and Hopf Bifurcation in a Computer Virus Model with Multistate Antivirus

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Tao Dong ◽  
Xiaofeng Liao ◽  
Huaqing Li
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Computer Virus ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Local Hopf Bifurcation ◽  
Dynamical Behaviors ◽  
Form Theory ◽  
Virus Model ◽  
Theoretical Results

By considering that people may immunize their computers with countermeasures in susceptible state, exposed state and using anti-virus software may take a period of time, a computer virus model with time delay based on an SEIR model is proposed. We regard time delay as bifurcating parameter to study the dynamical behaviors which include local asymptotical stability and local Hopf bifurcation. By analyzing the associated characteristic equation, Hopf bifurcation occurs when time delay passes through a sequence of critical value. The linerized model and stability of the bifurcating periodic solutions are also derived by applying the normal form theory and the center manifold theorem. Finally, an illustrative example is also given to support the theoretical results.

scholarly journals Hopf Bifurcation in a Delayed SEIQRS Model for the Transmission of Malicious Objects in Computer Network

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Juan Liu
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Local Stability ◽  
Center Manifold ◽  
Computer Network ◽  
Positive Equilibrium ◽  
Center Manifold Theorem ◽  
Local Hopf Bifurcation ◽  
Normal Form Method ◽  
Theoretical Results

A delayed SEIQRS model for the transmission of malicious objects in computer network is considered in this paper. Local stability of the positive equilibrium of the model and existence of local Hopf bifurcation are investigated by regarding the time delay due to the temporary immunity period after which a recovered computer may be infected again. Further, the properties of the Hopf bifurcation are studied by using the normal form method and center manifold theorem. Numerical simulations are also presented to support the theoretical results.

scholarly journals Stability and Hopf Bifurcation in a Delayed SEIRS Worm Model in Computer Network

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Hopf Bifurcation ◽  
Computer Network ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Local Hopf Bifurcation ◽  
Form Theory ◽  
Worm Model ◽  
The Stability

A delayed SEIRS epidemic model with vertical transmission in computer network is considered. Sufficient conditions for local stability of the positive equilibrium and existence of local Hopf bifurcation are obtained by analyzing distribution of the roots of the associated characteristic equation. Furthermore, the direction of the local Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by using the normal form theory and center manifold theorem. Finally, a numerical example is presented to verify the theoretical analysis.

scholarly journals Hopf Bifurcation of a Delayed Ecoepidemic Model with Ratio-Dependent Transmission Rate

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Wanjun Xia ◽  
Soumen Kundu ◽  
Sarit Maitra
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Center Manifold ◽  
Transmission Rate ◽  
Sufficient Conditions ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Ratio Dependent ◽  
Theoretical Results

A delayed ecoepidemic model with ratio-dependent transmission rate has been proposed in this paper. Effects of the time delay due to the gestation of the predator are the main focus of our work. Sufficient conditions for local stability and existence of a Hopf bifurcation of the model are derived by regarding the time delay as the bifurcation parameter. Furthermore, properties of the Hopf bifurcation are investigated by using the normal form theory and the center manifold theorem. Finally, numerical simulations are carried out in order to validate our obtained theoretical results.

scholarly journals Stability and Hopf Bifurcation of ann-Neuron Cohen-Grossberg Neural Network with Time Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Qiming Liu ◽  
Sumin Yang
Keyword(s):  
Neural Network ◽  
Hopf Bifurcation ◽  
Center Manifold ◽  
Sufficient Conditions ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Local Hopf Bifurcation ◽  
Discrete Delays ◽  
Form Theory ◽  
Distribution Of Roots

A Cohen-Grossberg neural network with discrete delays is investigated in this paper. Sufficient conditions for the existence of local Hopf bifurcation are obtained by analyzing the distribution of roots of characteristic equation. Moreover, the direction and stability of Hopf bifurcation are obtained by applying the normal form theory and the center manifold theorem. Numerical simulations are given to illustrate the obtained results.

scholarly journals Local Hopf Bifurcation in a Competitive Model of Market Structure with Consumptive Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Xuhui Li
Keyword(s):  
Hopf Bifurcation ◽  
Market Structure ◽  
Local Stability ◽  
Center Manifold ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Local Hopf Bifurcation ◽  
Competitive Model ◽  
Form Theory ◽  
The Stability

A competitive model of market structure with consumptive delays is considered. The local stability of the positive equilibrium and the existence of local Hopf bifurcation are investigated by analyzing the distribution of the roots of the associated characteristic equation. The explicit formulas determining the stability and other properties of bifurcating periodic solutions are derived by using normal form theory and center manifold argument. Finally, numerical simulations are given to support the analytical results.

Stability and Bifurcation of a Fishery Model with Crowley–Martin Functional Response

2017 ◽  
Vol 27 (11) ◽  
pp. 1750174 ◽  
Author(s):  
Atasi Patra Maiti ◽  
B. Dubey
Keyword(s):  
Hopf Bifurcation ◽  
Functional Response ◽  
Equilibrium Points ◽  
Control Variable ◽  
Dynamical Variable ◽  
Optimal Harvesting ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Fishery Model

To understand the dynamics of a fishery system, a nonlinear mathematical model is proposed and analyzed. In an aquatic environment, we considered two populations: one is prey and another is predator. Here both the fish populations grow logistically and interaction between them is of Crowley–Martin type functional response. It is assumed that both the populations are harvested and the harvesting effort is assumed to be dynamical variable and tax is considered as a control variable. The existence of equilibrium points and their local stability are examined. The existence of Hopf-bifurcation, stability and direction of Hopf-bifurcation are also analyzed with the help of Center Manifold theorem and normal form theory. The global stability behavior of the positive equilibrium point is also discussed. In order to find the value of optimal tax, the optimal harvesting policy is used. To verify our analytical findings, an extensive numerical simulation is carried out for this model system.

scholarly journals Dynamical Behavior Analysis of a Time-Delay SIRS-L Model in Rechargeable Wireless Sensor Networks

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 2007
Author(s):  
Guiyun Liu ◽  
Junqiang Li ◽  
Zhongwei Liang ◽  
Zhimin Peng
Keyword(s):  
Hopf Bifurcation ◽  
Sensor Networks ◽  
Time Delay ◽  
Dynamical Behavior ◽  
Propagation Model ◽  
Low Energy ◽  
Energy Status ◽  
Virus Propagation ◽  
Normal Form Theory ◽  
Center Manifold Theorem

The traditional SIRS virus propagation model is used to analyze the malware propagation behavior of wireless rechargeable sensor networks (WRSNs) by adding a new concept: the low-energy status nodes. The SIRS-L model has been developed in this article. Furthermore, the influence of time delay during the charging behavior of the low-energy status nodes needs to be considered. Hopf bifurcation is studied by discussing the time delay that is chosen as the bifurcation parameter. Finally, the properties of the Hopf bifurcation are explored by applying the normal form theory and the center manifold theorem.

scholarly journals Delay Induced Hopf Bifurcation of an Epidemic Model with Graded Infection Rates for Internet Worms

2017 ◽  
Vol 2017 ◽  
pp. 1-10
Author(s):  
Tao Zhao ◽  
Dianjie Bi
Keyword(s):  
Hopf Bifurcation ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Propagation Model ◽  
Infection Rates ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Internet Worms ◽  
Form Theory ◽  
The Stability

A delayed SEIQRS worm propagation model with different infection rates for the exposed computers and the infectious computers is investigated in this paper. The results are given in terms of the local stability and Hopf bifurcation. Sufficient conditions for the local stability and the existence of Hopf bifurcation are obtained by using eigenvalue method and choosing the delay as the bifurcation parameter. In particular, the direction and the stability of the Hopf bifurcation are investigated by means of the normal form theory and center manifold theorem. Finally, a numerical example is also presented to support the obtained theoretical results.

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