Hopf analysis of a differential-algebraic predator–prey model with Allee effect and time delay

2015 ◽  
Vol 08 (03) ◽  
pp. 1550041 ◽  
Author(s):  
Xue Zhang ◽  
Qingling Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Normal Form ◽  
Allee Effect ◽  
Functional Differential Equations ◽  
Local Analysis ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Predator Prey Model ◽  
Predator Model

A differential-algebraic prey–predator model with time delay and Allee effect on the growth of the prey population is investigated. Using differential-algebraic system theory, we transform the prey–predator model into its normal form and study its dynamics in terms of local analysis and Hopf bifurcation. By analyzing the associated characteristic equation, it is observed that the model undergoes a Hopf bifurcation at some critical value of time delay. In particular, we study the direction of Hopf bifurcation and the stability of bifurcated periodic solutions, and an explicit algorithm is given by applying the normal form theory and the center manifold reduction for functional differential equations. Finally, numerical simulations supporting the theoretical analysis are also included.

scholarly journals Bifurcation Analysis of a Singular Bioeconomic Model with Allee Effect and Two Time Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Xue Zhang ◽  
Qing-ling Zhang ◽  
Zhongyi Xiang
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Periodic Solutions ◽  
Time Delays ◽  
Allee Effect ◽  
Dynamical Behavior ◽  
Local Analysis ◽  
Normal Form Theory ◽  
Predator Model ◽  
The Stability

A singular prey-predator model with time delays is formulated and analyzed. Allee effect is considered on the growth of the prey population. The singular prey-predator model is transformed into its normal form by using differential-algebraic system theory. We study its dynamics in terms of local analysis and Hopf bifurcation. The existence of periodic solutions via Hopf bifurcation with respect to two delays is established. In particular, we study the direction of Hopf bifurcation and the stability of bifurcated periodic solutions by applying the normal form theory and the center manifold argument. Finally, numerical simulations are included supporting the theoretical analysis and displaying the complex dynamical behavior of the model outside the domain of stability.

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.

scholarly journals Stability and Bifurcation for a Delayed Diffusive Two-Zooplankton One-Phytoplankton Model with Two Different Functions

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-24
Author(s):  
Xin-You Meng ◽  
Li Xiao
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Normal Form ◽  
Center Manifold ◽  
Functional Differential Equations ◽  
Bifurcation Parameter ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Functional Differential

In this paper, a diffusion two-phytoplankton one-zooplankton model with time delay, Beddington–DeAnglis functional response, and Holling II functional response is proposed. First, the existence and local stability of all equilibria of such model are studied. Then, the existence of Hopf bifurcation of the corresponding model without diffusion is given by taking time delay as the bifurcation parameter. Next, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are investigated by using the normal form theory and center manifold theorem. Furthermore, due to the local bifurcation theory of partial functional differential equations, Hopf bifurcation of the model is investigated by considering time delay as the bifurcation parameter. The explicit formulas to determine the properties of Hopf bifurcation are given by the method of the normal form theory and center manifold theorem for partial functional differential equations. Finally, some numerical simulations are performed to check out our theoretical results.

Normal Form of Saddle-Node-Hopf Bifurcation in Retarded Functional Differential Equations and Applications

2016 ◽  
Vol 26 (03) ◽  
pp. 1650040 ◽  
Author(s):  
Heping Jiang ◽  
Jiao Jiang ◽  
Yongli Song
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Differential Equations ◽  
Functional Differential Equations ◽  
Normal Form Theory ◽  
Hopf Bifurcation Point ◽  
Predator Prey Model ◽  
Saddle Node ◽  
Retarded Functional Differential Equations ◽  
Functional Differential

In this paper, we firstly employ the normal form theory of delayed differential equations according to Faria and Magalhães to derive the normal form of saddle-node-Hopf bifurcation for the general retarded functional differential equations. Then, the dynamical behaviors of a Leslie–Gower predator–prey model with time delay and nonmonotonic functional response are considered. Specially, the dynamical classification near the saddle-node-Hopf bifurcation point is investigated by using the normal form and the center manifold approaches. Finally, the numerical simulations are employed to support the theoretical results.

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 Dynamical analysis of a predator-prey interaction model with time delay and prey refuge

2018 ◽  
Vol 5 (1) ◽  
pp. 138-151 ◽  
Author(s):  
Jai Prakash Tripathi ◽  
Swati Tyagi ◽  
Syed Abbas
Keyword(s):  
Hopf Bifurcation ◽  
Functional Differential Equations ◽  
Interaction Model ◽  
Dynamical Analysis ◽  
Prey Refuge ◽  
Normal Form Theory ◽  
Predator Species ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model

AbstractIn this paper, we study a predator-prey model with prey refuge and delay. We investigate the combined role of prey refuge and delay on the dynamical behaviour of the delayed system by incorporating discrete type gestation delay of predator. It is found that Hopf bifurcation occurs when the delay parameter τ crosses some critical value. In particular, it is shown that the conditions obtained for the Hopf bifurcation behaviour are sufficient but not necessary and the prey reserve is unable to stabilize the unstable interior equilibrium due to Hopf bifurcation. In particular, the direction and stability of bifurcating periodic solutions are determined by applying normal form theory and center manifold theorem for functional differential equations. Mathematically, we analyze the effect of increase or decrease of prey reserve on the equilibrium states of prey and predator species. At the end, we perform some numerical simulations to substantiate our analytical findings.

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.

Hopf Bifurcation Analysis and Stability for a Ratio-Dependent Predator–Prey Diffusive System with Time Delay

2020 ◽  
Vol 30 (03) ◽  
pp. 2050037
Author(s):  
Longyue Li ◽  
Yingying Mei ◽  
Jianzhi Cao
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Periodic Solutions ◽  
Positive Equilibrium ◽  
Dimensional System ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Predator Prey ◽  
Ratio Dependent ◽  
Low Dimensional

In this paper, we are focused on a new ratio-dependent predator–prey system that introduced the diffusive and time delay effect simultaneously. By analyzing the characteristic equations and the distribution of eigenvalues, we examine the stability and boundary of positive equilibrium states, and the existence of spatially homogeneous and spatially inhomogeneous bifurcating periodic solutions, respectively. Further, we prove that when [Formula: see text], the system has Hopf bifurcation at the positive equilibrium state. By using the center manifold reduction, we simplify the system so that we can convert an infinite-dimensional system into a low-dimensional finite-dimensional system. By using the normal form theory, we obtain explicit expressions for the direction, stability and period of Hopf bifurcation periodic solutions. Finally, we have illustrated the main results in this thesis by numerical examples, our work may provide some useful measures to save time or cost and to control the ecosystem.

scholarly journals Stability and Bifurcation Analysis for a Predator-Prey Model with Discrete and Distributed Delay

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Ruiqing Shi ◽  
Junmei Qi ◽  
Sanyi Tang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Three Dimensional ◽  
Distributed Delay ◽  
Dimensional System ◽  
Two Dimensional ◽  
Normal Form Theory ◽  
Predator Prey Model ◽  
Prey Model ◽  
Form Theory

We propose a two-dimensional predatory-prey model with discrete and distributed delay. By the use of a new variable, the original two-dimensional system transforms into an equivalent three-dimensional system. Firstly, we study the existence and local stability of equilibria of the new system. And, by choosing the time delayτas a bifurcation parameter, we show that Hopf bifurcation can occur as the time delayτpasses through some critical values. Secondly, by the use of normal form theory and central manifold argument, we establish the direction and stability of Hopf bifurcation. At last, an example with numerical simulations is provided to verify the theoretical results. In addition, some simple discussion is also presented.

scholarly journals Bifurcation Analysis and Control of a Differential-Algebraic Predator-Prey Model with Allee Effect and Time Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Xue Zhang ◽  
Qing-ling Zhang
Keyword(s):  
Time Delay ◽  
Allee Effect ◽  
Control Method ◽  
Handling Time ◽  
Delayed Feedback Control ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Feedback Control Method ◽  
Predator Model ◽  
And Control

This paper studies systematically a differential-algebraic prey-predator model with time delay and Allee effect. It shows that transcritical bifurcation appears when a variation of predator handling time is taken into account. This model also exhibits singular induced bifurcation as the economic revenue increases through zero, which causes impulsive phenomenon. It can be noted that the impulsive phenomenon can be much weaker by strengthening Allee effect in numerical simulation. On the other hand, at a critical value of time delay, the model undergoes a Hopf bifurcation; that is, the increase of time delay destabilizes the model and bifurcates into small amplitude periodic solution. Moreover, a state delayed feedback control method, which can be implemented by adjusting the harvesting effort for biological populations, is proposed to drive the differential-algebraic system to a steady state. Finally, by using Matlab software, numerical simulations illustrate the effectiveness of the results.

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