scholarly journals Hopf bifurcation and optimal control in a diffusive predator-prey system with time delay and prey harvesting

2012 ◽  
Vol 17 (4) ◽  
pp. 379-409 ◽  
Author(s):  
Xiaoyuan Chang ◽  
Junjie Wei
Keyword(s):  
Optimal Control ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Functional Differential Equations ◽  
Normal Form Theory ◽  
Flux Boundary Condition ◽  
Predator Prey ◽  
Prey System ◽  
Functional Differential ◽  
The Stability

In this paper, we investigated the dynamics of a diffusive delayed predator-prey system with Holling type II functional response and nozero constant prey harvesting on no-flux boundary condition. At first, we obtain the existence and the stability of the equilibria by analyzing the distribution of the roots of associated characteristic equation. Using the time delay as the bifurcation parameter and the harvesting term as the control parameter, we get the existence and the stability of Hopf bifurcation at the positive constant steady state. Applying the normal form theory and the center manifold argument for partial functional differential equations, we derive an explicit formula for determining the direction and the stability of Hopf bifurcation. Finally, an optimal control problem has been considered.

scholarly journals Time-delay effect on a diffusive predator–prey model with habitat complexity

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yanfeng Li ◽  
Haicheng Liu ◽  
Ruizhi Yang
Keyword(s):  
Time Delay ◽  
Habitat Complexity ◽  
Functional Differential Equations ◽  
Neumann Boundary ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Functional Differential ◽  
The Stability

AbstractBased on the predator–prey system with a Holling type functional response function, a diffusive predator–prey system with digest delay and habitat complexity is proposed. Firstly, the stability of the equilibrium of diffusion system without delay is studied. Secondly, under the Neumann boundary conditions, taking time delay as the bifurcation parameter, by analyzing the eigenvalues of linearized operator of the system and using the normal form theory and center manifold method of partial functional differential equations, the effect of time delay on the stability of the system is studied and the conditions under which Hopf bifurcation occurs are given. In addition, the calculation formulas of the bifurcation direction and the stability of bifurcating periodic solutions are derived. Finally, the accuracy of theoretical analysis results is verified by numerical simulations and the biological explanation is given for the analysis results.

scholarly journals Global Hopf Bifurcation on Two-Delays Leslie-Gower Predator-Prey System with a Prey Refuge

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Qingsong Liu ◽  
Yiping Lin ◽  
Jingnan Cao
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Functional Differential Equations ◽  
Global Hopf Bifurcation ◽  
Bifurcation Theorem ◽  
Predator Prey ◽  
Prey System ◽  
Two Delays ◽  
Functional Differential ◽  
The Stability

A modified Leslie-Gower predator-prey system with two delays is investigated. By choosingτ1andτ2as bifurcation parameters, we show that the Hopf bifurcations occur when time delay crosses some critical values. Moreover, we derive the equation describing the flow on the center manifold; then we give the formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the theoretical results and chaotic behaviors are observed. Finally, using a global Hopf bifurcation theorem for functional differential equations, we show the global existence of the periodic solutions.

scholarly journals Time delay induced Hopf bifurcation in a diffusive predator–prey model with prey toxicity

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ruizhi Yang ◽  
Yuxin Ma ◽  
Chiyu Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Functional Differential Equations ◽  
Positive Equilibrium ◽  
Center Manifold Reduction ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Functional Differential ◽  
The Stability

AbstractIn this paper, we consider a diffusive predator–prey model with a time delay and prey toxicity. The effect of time delay on the stability of the positive equilibrium is studied by analyzing the eigenvalue spectrum. Delay-induced Hopf bifurcation is also investigated. By utilizing the normal form method and center manifold reduction for partial functional differential equations, the formulas for determining the property of Hopf bifurcation are given.

STABILITY AND HOPF BIFURCATIONS IN A DELAYED PREDATOR–PREY SYSTEM WITH A DISTRIBUTED DELAY

2009 ◽  
Vol 19 (07) ◽  
pp. 2283-2294 ◽  
Author(s):  
CUN-HUA ZHANG ◽  
XIANG-PING YAN
Keyword(s):  
Functional Differential Equations ◽  
Distributed Delay ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Predator Prey ◽  
Prey System ◽  
Functional Differential ◽  
The Stability

This paper is concerned with a delayed Lotka–Volterra two-species predator–prey system with a distributed delay. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of positive equilibrium is investigated and Hopf bifurcations are demonstrated. It is found that the positive equilibrium of the system is always locally asymptotically stable when the delay kernel is the weak kernel while there is a stability switch of positive equilibrium when the delay kernel is the strong kernel and the system can undergo a Hopf bifurcation at the positive equilibrium when the average time delay in the delay kernel crosses certain critical values. In particular, by applying the normal form theory and center manifold reduction to functional differential equations (FDEs), the explicit formula determining the direction of Hopf bifurcations and the stability of bifurcated periodic solutions is given. Finally, some numerical simulations are also included to support the analytical results obtained.

HOPF BIFURCATION ANALYSIS FOR A DELAYED LESLIE–GOWER PREDATOR–PREY SYSTEM WITH DIFFUSION EFFECTS

2014 ◽  
Vol 07 (01) ◽  
pp. 1450007
Author(s):  
LIN-LIN WANG ◽  
BEI-BEI ZHOU ◽  
YONG-HONG FAN
Keyword(s):  
Periodic Solution ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Functional Differential Equations ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Predator Prey ◽  
Homogeneous Neumann Boundary Condition ◽  
The Stability ◽  
The Impact

A delayed predator–prey diffusion system with homogeneous Neumann boundary condition is considered. In order to study the impact of the time delay on the stability of the model, the delay τ is taken as the bifurcation parameter, the results show that when the time delay across some critical values, the Hopf bifurcations may occur. In particular, by using the normal form theory and the center manifold reduction for partial functional differential equations, the direction of the Hopf bifurcation and the stability of the bifurcated periodic solution have been established. The effect of the diffusion on the bifurcated periodic solution is also considered. A numerical example is given to support the main result.

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.

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.

GLOBAL STABILITY OF A STAGE-STRUCTURED PREDATOR-PREY SYSTEM

2008 ◽  
Vol 01 (03) ◽  
pp. 313-326 ◽  
Author(s):  
XINYU SONG ◽  
HONGJIAN GUO
Keyword(s):  
Time Delay ◽  
Functional Differential Equations ◽  
Stage Structure ◽  
Immature Stage ◽  
Mature Stage ◽  
Predator Prey ◽  
Oscillatory Dynamics ◽  
Prey System ◽  
Functional Differential ◽  
Two Stages

In this paper, we consider a predator-prey system with stage structure. The populations have two stages, an immature stage and a mature stage, the age to maturity is represented by a time delay, which leads to systems of retarded functional differential equations. We obtain the conditions for global asymptotic stability of two nonnegative equilibria of this system. The numerical simulation suggests that time delay has both oscillatory dynamics and stabilizing effects.

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