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

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.

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

Advances in Artificial Neural Systems ◽

10.1155/2012/397146 ◽

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.

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Rich Dynamics of a Predator-Prey System with Different Kinds of Functional Responses

Complexity ◽

10.1155/2020/4285294 ◽

2020 ◽

Vol 2020 ◽

pp. 1-19

Author(s):

Kankan Sarkar ◽

Subhas Khajanchi ◽

Prakash Chandra Mali ◽

Juan J. Nieto

Keyword(s):

Growth Rate ◽

Hopf Bifurcation ◽

Dynamical Behavior ◽

Prey Population ◽

Uniform Persistence ◽

Functional Responses ◽

Normal Form Theory ◽

Predator Prey ◽

Center Manifold Theorem ◽

Prey System

In this study, we investigate a mathematical model that describes the interactive dynamics of a predator-prey system with different kinds of response function. The positivity, boundedness, and uniform persistence of the system are established. We investigate the biologically feasible singular points and their stability analysis. We perform a comparative study by considering different kinds of functional responses, which suggest that the dynamical behavior of the system remains unaltered, but the position of the bifurcation points altered. Our model system undergoes Hopf bifurcation with respect to the growth rate of the prey population, which indicates that a periodic solution occurs around a fixed point. Also, we observed that our predator-prey system experiences transcritical bifurcation for the prey population growth rate. By using normal form theory and center manifold theorem, we investigate the direction and stability of Hopf bifurcation. The biological implications of the analytical and numerical findings are also discussed in this study.

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

Abstract and Applied Analysis ◽

10.1155/2013/168340 ◽

2013 ◽

Vol 2013 ◽

pp. 1-13 ◽

Cited By ~ 1

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.

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

Abstract and Applied Analysis ◽

10.1155/2014/624162 ◽

2014 ◽

Vol 2014 ◽

pp. 1-12 ◽

Cited By ~ 1

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.

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

Discrete Dynamics in Nature and Society ◽

10.1155/2014/909238 ◽

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.

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

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127413501940 ◽

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.

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Stability and Hopf Bifurcation Analysis for a Gause-Type Predator-Prey System with Multiple Delays

Abstract and Applied Analysis ◽

10.1155/2013/795358 ◽

2013 ◽

Vol 2013 ◽

pp. 1-12 ◽

Cited By ~ 2

Author(s):

Juan Liu ◽

Changwei Sun ◽

Yimin Li

Keyword(s):

Hopf Bifurcation ◽

Sufficient Conditions ◽

Multiple Delays ◽

Normal Form Theory ◽

Predator Prey ◽

Prey System ◽

Form Theory ◽

Two Delays ◽

The Stability ◽

Theoretical Results

This paper is concerned with a Gause-type predator-prey system with two delays. Firstly, we study the stability and the existence of Hopf bifurcation at the coexistence equilibrium by analyzing the distribution of the roots of the associated characteristic equation. A group of sufficient conditions for the existence of Hopf bifurcation is obtained. Secondly, an explicit formula for determining the stability and the direction of periodic solutions that bifurcate from Hopf bifurcation is derived by using the normal form theory and center manifold argument. Finally, some numerical simulations are carried out to illustrate the main theoretical results.

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HOPF BIFURCATION ANALYSIS FOR A RATIO-DEPENDENT PREDATOR–PREY SYSTEM INVOLVING TWO DELAYS

The ANZIAM Journal ◽

10.1017/s1446181114000054 ◽

2014 ◽

Vol 55 (3) ◽

pp. 214-231 ◽

Cited By ~ 4

Author(s):

E. KARAOGLU ◽

H. MERDAN

Keyword(s):

Hopf Bifurcation ◽

Normal Form Theory ◽

Predator Prey ◽

Prey System ◽

Discrete Delays ◽

Form Theory ◽

Two Delays ◽

Ratio Dependent ◽

Centre Manifold Theorem ◽

Analysis Stability

AbstractThe aim of this paper is to give a detailed analysis of Hopf bifurcation of a ratio-dependent predator–prey system involving two discrete delays. A delay parameter is chosen as the bifurcation parameter for the analysis. Stability of the bifurcating periodic solutions is determined by using the centre manifold theorem and the normal form theory introduced by Hassard et al. Some of the bifurcation properties including the direction, stability and period are given. Finally, our theoretical results are supported by some numerical simulations.

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Hopf bifurcation in a delayed predator-prey system with asymmetric functional response and additional food

AIMS Mathematics ◽

10.3934/math.2021708 ◽

2021 ◽

Vol 6 (11) ◽

pp. 12225-12244

Author(s):

Luoyi Wu ◽

◽

Hang Zheng ◽

Keyword(s):

Hopf Bifurcation ◽

Periodic Solutions ◽

Functional Response ◽

Additional Food ◽

Normal Form Theory ◽

Predator Prey ◽

Center Manifold Theorem ◽

Local Hopf Bifurcation ◽

Prey System ◽

Form Theory

<abstract><p>In this paper, a delayed predator-prey system with additional food and asymmetric functional response is investigated. We discuss the local stability of equilibria and the existence of local Hopf bifurcation under the influence of the time delay. By using the normal form theory and center manifold theorem, the explicit formulas which determine the properties of bifurcating periodic solutions are obtained. Further, we prove that global periodic solutions exist after the second critical value of delay via Wu's theory. Finally, the correctness of the previous theoretical analysis is demonstrated by some numerical cases.</p></abstract>

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Dynamical analysis of a giving up smoking model with time delay

Advances in Difference Equations ◽

10.1186/s13662-019-2450-4 ◽

2019 ◽

Vol 2019 (1) ◽

Cited By ~ 4

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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