Local Hopf bifurcation and global periodic solutions in a delayed predator-prey system

We consider a delayed predator-prey system. We first consider the existence of local Hopf bifurcations, and then derive explicit formulas which enable us to determine the stability and the direction of periodic solutions bifurcating from Hopf bifurcations, using the normal form theory and center manifold argument. Special attention is paid to the global existence of periodic solutions bifurcating from Hopf bifurcations. By using a global Hopf bifurcation result due to Wu [Trans. Amer. Math. Soc. 350 (1998) 4799], we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of delay. Finally, several numerical simulations supporting the theoretical analysis are also given.     

Stability and global Hopf bifurcation in a delayed food web consisting of a prey and two predators

This paper is concerned with a predator–prey system with Holling II functional response and hunting delay and gestation. By regarding the sum of delays as the bifurcation parameter, the local stability of the positive equilibrium and the existence of Hopf bifurcation are investigated. We obtained explicit formulas to determine the properties of Hopf bifurcation by using the normal form method and center manifold theorem. Special attention is paid to the global continuation of local Hopf bifurcation. Using a global Hopf bifurcation result of Wu [Wu JH. Symmetric functional differential equations and neural networks with memory, Trans Amer Math Soc 1998;350:4799–4838] for functional differential equations, we may show the global existence of the periodic solutions. Finally, several numerical simulations illustrating the theoretical analysis are also given.     

Hopf bifurcation in a three-species system with delays

A kind of three-species system with Holling II functional response and two delays is introduced. Its local stability and the existence of Hopf bifurcation are demonstrated by analyzing the associated characteristic equation. By using the normal form method and center manifold theorem, explicit formulas to determine the direction of the Hopf bifurcation and the stability of bifurcating periodic solution are also obtained. In addition, the global existence results of periodic solutions bifurcating from Hopf bifurcations are established by using a global Hopf bifurcation result. Numerical simulation results are also given to support our theoretical predictions.     

Hopf bifurcation analysis of a food-limited population model with delay

The dynamics of a food-limited population model with delay are investigated. We prove that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay increases. An explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived, using the theory of normal form and center manifold. Global existence of periodic solutions is established by using a global Hopf bifurcation result due to [J. Wu, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc. 350 (1998) 4799–4838].     

一类时滞造血模型的全局Hopf分支

对一类具有时滞的造血模型,通过讨论线性部分超越特征方程根的分布情况,得到了正平衡点的稳定性及局部Hopf分支的存在性.进而利用吴建宏建立的全局分支理论,将周期解的存在性由局部延拓到全局.     

Dynamics of a delayed business cycle model with general investment function

The aim of this paper is to study the dynamics of a delayed business cycle model with general investment function. The model describes the interaction of the gross product and capital stock. Furthermore, the delay represents the time between the decision of investment and implementation. Firstly, we show that the model is well posed by proving the global existence and boundedness of solutions. Secondly, we determine the economic equilibrium of the model. By analyzing the characteristic equation, we investigate the stability of the economic equilibrium and the local existence of Hopf bifurcation. Also, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by applying the normal form method and center manifold theory. Moreover, the global existence of bifurcating periodic solutions is established by using the global Hopf bifurcation theory. Finally, our theoretical results are illustrated with some numerical simulations.     

Bifurcations for a predator-prey system with two delays

In this paper, a predator-prey system with two delays is investigated. By choosing the sum τ of two delays as a bifurcation parameter, we show that Hopf bifurcations can occur as τ crosses some critical values. By deriving the equation describing the flow on the center manifold, we can determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions. In addition, special attention is paid to the global continuation of local Hopf bifurcations. Using a global Hopf bifurcation result of [J. Wu, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc. 350 (1998) 4799-4838], we may show the global existence of periodic solutions.     

Local and global Hopf bifurcation in a neutral population model with age structure

In this paper, an age‐structured population model with the form of neutral functional differential equation is studied. We discuss the stability of the positive equilibrium by analyzing the characteristic equation. Local Hopf bifurcation results are also obtained by choosing the mature delay as bifurcation parameter. On the center manifold, the normal form of the Hopf bifurcation is derived, and explicit formulae for determining the criticality of bifurcation are theoretically given. Moreover, the global continuation of Hopf bifurcating periodic solutions is investigated by using the global Hopf bifurcation theory of neutral equations. Finally, some numerical examples are carried out to support the main results.     

Hopf Bifurcation Analysis of a Class of Abstract Delay Differential Equation

The dynamics of a class of abstract delay differential equations are investigated. We prove that a sequence of Hopf bifurcations occur at the origin equilibrium as the delay increases. By using the theory of normal form and centre manifold, the direction of Hopf bifurcations and the stability of the bifurcating periodic solutions is derived. Then, the existence of the global Hopf bifurcation of the system is discussed by applying the global Hopf bifurcation theorem of general functional differential equation.     

Bifurcation analysis in a neutral differential equation

The dynamics of a neural network model in neutral form is investigated. We prove that a sequence of Hopf bifurcations occurs at the origin as the delay increases. The direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are determined by using normal form method and center manifold theory. Global existence of periodic solutions is established using a global Hopf bifurcation result of Krawcewicz et al. and a Bendixson's criterion for higher dimensional ordinary differential equations due to Li and Muldowney.     

Hopf bifurcation for neutral functional differential equations

In this paper, we extend the computation of the properties of Hopf bifurcation, such as the direction of bifurcation and stability of bifurcating periodic solutions, of DDE introduced by Kazarinoff et al. [N.D. Kazarinoff, P. van den Driessche, Y.H. Wan, Hopf bifurcation and stability of periodic solutions of differential–difference and integro-differential equations, J. Inst. Math. Appl. 21 (1978) 461–477] to a kind of neutral functional differential equation (NFDE). As an example, a neutral delay logistic differential equation is considered, and the explicit formulas for determining the direction of bifurcation and the stability of bifurcating periodic solutions are derived. Finally, some numerical simulations are carried out to support the analytic results.     

A stage-structured predator-prey system with time delay

A stage-structured predator-prey system with time delay is considered. By analyzing the corresponding characteristic equation, the local stability of a positive equilibrium is investigated. The existence of Hopf bifurcations is established. Formulae are derived to determine the direction of bifurcations and the stability of bifurcating periodic solutions by using the normal form theory and center manifold theorem. Numerical simulations are carried out to illustrate the theoretical results. Based on the global Hopf bifurcation theorem for general functional differential equations, the global existence of periodic solutions is established.     

Hopf bifurcation analysis in a tri-neuron network with time delay

A neural network model with three neurons and a single delay is considered. The existence of local Hopf bifurcations is first considered and then explicit formulas are derived by using the normal form method and center manifold theory to determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions. A global Hopf bifurcation theorem due to Wu and a Bendixson's criterion for high-dimensional ODE due to Li and Muldowney are used to obtain a group of conditions for the system to have multiple periodic solutions when the delay is sufficiently large. Finally, numerical simulations are carried out to support the theoretical analysis of the research.     

Hopf bifurcation analysis in a delayed Nicholson blowflies equation

The dynamics of a Nicholson's blowflies equation with a finite delay are investigated. We prove that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay increases. Explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived, using the theory of normal form and center manifold. Global existence of periodic solutions are established using a global Hopf bifurcation result of Wu (Trans. Amer. Math. Soc. 350 (1998) 4799), and a Bendixson criterion for higher dimensional ordinary differential equations due to Li and Muldowney (J. Differential Equations 106 (1994) 27).     

Dynamic behaviour of a delayed predator-prey model with harvesting

In this paper, we analyze the dynamics of a delayed predator-prey system in the presence of harvesting. This is a modified version of the Leslie-Gower and Holling-type II scheme. The main result is given in terms of local stability, global stability, influence of harvesting and bifurcation. Direction of Hopf bifurcation and the stability of bifurcating periodic solutions are also studied by using the normal form method and center manifold theorem.     

具有捕食者相互残杀项时滞系统的Hopf分支

研究了具有捕食者相互残杀项的时滞系统的Hopf分支,通过选择时滞作为一个分支参数,研究了正平衡点的稳定性和正周期解的Hopf分支.而且通过应用规范型和中心流形的理论,得出了确定分支方向的明确的算法.     

Local and global bifurcations in an SIRS epidemic model

This paper studies the existence and stability of the disease-free equilibrium and endemic equilibria for the SIRS epidemic model with the saturated incidence rate, considering the factor of population dynamics such as the disease-related, the natural mortality and the constant recruitment of population. Analytical techniques are used to show, for some parameter values, the periodic solutions can arise through the Hopf bifurcation, which is important to carry different strategies for the controlling disease. Then the codimension-two bifurcation, i.e. BT bifurcation, is investigated by using a global qualitative method and the curves of saddle-node bifurcation, Hopf bifurcation and homoclinic bifurcation are obtained at the degenerate equilibrium. Moreover, several numerical simulations are given to support the theoretical analysis.     

Global existence of periodic solutions on a simplified BAM neural network model with delays

A simplified n-dimensional BAM neural network model with delays is considered. Some results of Hopf bifurcations occurring at the zero equilibrium as the delay increases are exhibited. Global existence of periodic solutions are established using a global Hopf bifurcation result of Wu [Wu J. Symmetric functional-differential equations and neural networks with memory. Trans Am Math Soc 1998;350:4799–838], and a Bendixson criterion for higher dimensional ordinary differential equations due to Li and Muldowney [Li MY, Muldowney J. On Bendixson’s criterion. J Differ Equations 1994;106:27–39]. Finally, computer simulations are performed to illustrate the analytical results found.     

Global existence of periodic solutions in a special neural network model with two delays

A simple neural network model with two delays is considered. By analyzing the associated characteristic transcendental equation, it is found that Hopf bifurcation occurs when the sum of two delays passes through a sequence of critical values. Using a global Hopf bifurcation theorem for FDE due to Wu [Wu J. Symmetric functional differential equations and neural networks with memory. Trans Amer Math Soc 1998;350:4799–838], a group of sufficient conditions for this model to have multiple periodic solutions are obtained when the sum of delays is sufficiently large. Numerical simulations are presented to support the obtained theoretical results.     

Local Hopf bifurcation and global existence of periodic solutions in a kind of physiological system

The dynamics of a physiological control systems described by a first-order nonlinear delay differential equations are investigated. we proved that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay increases. Explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived, using the theory of normal form and center manifold. Global existence of periodic solutions are established using a global Hopf bifurcation result due to Wu [Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc. 350 (1998) 4799–4838].