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.     

多时滞捕食-食饵系统正平衡点的稳定性及全局Hopf分支

本文首先用Cooke等人建立的关于超越函数的零点分布定理,研究了一类多时滞捕食-食饵系统正平衡点的稳定性及局部Hopf分支,在此基础上再结合吴建宏等人用等变拓扑度理论建立起的一般泛函微分方程的全局Hopf分支定理,进一步研究了该系统的全局Hopf分支.     

Global Hopf bifurcation in the Leslie-Gower predator-prey model with two delays

In this paper, the Leslie-Gower predator-prey system with two delays is investigated. By choosing the delay as a bifurcation parameter, we show that Hopf bifurcations can occur as the delay crosses some critical values. In addition, special attention is paid to the global continuation of local Hopf bifurcations. Using a global Hopf bifurcation theorem for functional differential equations, we show the global existence of periodic solutions.     

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.     

中立型微分方程零解的稳定性与全局Hopf分支

本文用Ｒｏｕｃｈｅ定理建立起关于一般的超越函数的零点分布定理，以此定理为基础，结合应用吴建宏等用等变拓扑度理论建立起的一般泛函微分方程的Ｈｏｐｆ分支定理，研究了描述无损传输网络线路的中立型微分方程的零解的稳定性和全局Ｈｏｐｆ分支．     

Synchronized Hopf bifurcation analysis in a neural network model with delays

We consider the synchronized periodic oscillation in a ring neural network model with two different delays in self-connection and nearest neighbor coupling. Employing the center manifold theorem and normal form method introduced by Hassard et al., we give an algorithm for determining the Hopf bifurcation properties. Using the global Hopf bifurcation theorem for FDE due to Wu and Bendixson's criterion for high-dimensional ODE due to Li and Muldowney, we obtain several groups of conditions that guarantee the model have multiple synchronized periodic solutions when the transfer coefficient or time delay is sufficiently large.     

Stability,bifurcation and global existence of a Hopf-bifurcating periodic solution for a class of three-neuron delayed network models

In this paper a system of three delay differential equations representing a Hopfield type general model for three neurons with two-way (bidirectional) time delayed connections between the neurons and time delayed self-connection from each neuron to itself is studied. Delay independent and delay dependent sufficient conditions for linear stability, instability and the occurrence of a Hopf bifurcation about the trivial equilibrium are addressed. The partition of the resulting parametric space into regions of stability, instability, and Hopf bifurcation in the absence of self-connection is realized. To extend the local Hopf branches for large delay values a particular bidirectional delayed tri-neuron model without self-connection is investigated. Sufficient conditions for global existence of multiple non-constant periodic solutions are obtained for such a model using the global Hopf-bifurcation theorem for functional differential equations due to J. Wu and the Bendixson criterion for higher dimensional ordinary differential equations due to Li and Muldowney, and following the approach developed by Wei and Li.     

Asymptotic behaviour and bifurcation in competitive Lotka-Volterra Systems

A conjecture about global attraction in autonomous competitive Lotka-Volterra systems is clarified by investigating a special system with a circular matrix. Under suitable assumptions, this system meets the condition of the conjecture but Hopf bifurcation occurs in a particular instance. This shows that the conjecture is not true in general and the condition of the conjecture is too weak to guarantee global attraction of an equilibrium. Sufficient conditions for global attraction are also obtained for this system.     

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.     

多时滞合作系统的稳定性与全局Hopf分支

以时滞为参数,研究了一类多时滞合作系统的正平衡点的稳定性及局部Hopf分支的存在性.在此基础上结合一般泛函微分方程的全局Hopf分支定理,讨论了该系统全局Hopf分支的存在性.     

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.     

Analysis of stability and Hopf bifurcation for HIV-1 dynamics with PI and three intracellular delays

Mathematical model for the effects of protease inhibitor on the dynamics of HIV-1 infection model with three delays is proposed and analyzed. Some analytical results on the global stability of viral free steady state and infected steady state are obtained. The stability/instability of the positive steady state and associated Hopf bifurcation are investigated by analyzing the characteristic equations.     

Dynamical behavior of an epidemic model with a nonlinear incidence rate

In this paper, we study the global dynamics of an epidemic model with vital dynamics and nonlinear incidence rate of saturated mass action. By carrying out global qualitative and bifurcation analyses, it is shown that either the number of infective individuals tends to zero as time evolves or there is a region such that the disease will be persistent if the initial position lies in the region and the disease will disappear if the initial position lies outside this region. When such a region exists, it is shown that the model undergoes a Bogdanov-Takens bifurcation, i.e., it exhibits a saddle-node bifurcation, Hopf bifurcations, and a homoclinic bifurcation. Existence of none, one or two limit cycles is also discussed.     

Persistence and global stability of Bazykin predator–prey model with Beddington–DeAngelis response function

In this article, a predator–prey model of Beddington–DeAngelis type with discrete delay is proposed and analyzed. The essential mathematical features of the proposed model are investigated in terms of local, global analysis and bifurcation theory. By analyzing the associated characteristic equation, it is found that the Hopf bifurcation occurs when the delay parameter τ crosses some critical values. In this article, the classical Bazykin’s model is modified with Beddington–DeAngelis functional response. The parametric space under which the system enters into Hopf bifurcation for both delay and non-delay cases are investigated. Global stability results are obtained by constructing suitable Lyapunov functions for both the cases. We also derive the explicit formulae for determining the stability, direction and other properties of bifurcating periodic solutions by using normal form and central manifold theory. Our analytical findings are supported by numerical simulations. Biological implication of the analytical findings are discussed in the conclusion section.     

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.     

Dynamics of an SIR epidemic model with limited medical resources revisited

The dynamics of an SIR epidemic model is explored in this paper in order to understand how the limited medical resources and their supply efficiency affect the transmission of infectious diseases. The study reveals that, with varying amount of medical resources and their supply efficiency, the target model admits both backward bifurcation and Hopf bifurcation. Sufficient criteria are established for the existence of backward bifurcation, the existence, the stability and the direction of Hopf bifurcation. The mechanism of backward bifurcation and its implication for the control of the infectious disease are also explored. Numerical simulations are presented to support and complement the theoretical findings.     

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.     

具有时滞的生态流行病模型的稳定性和Hopf分支

该文考虑一类食饵染病的时滞捕食被捕食模型. 作者分析了系统的非负不变性, 边界平衡点的性质和全局稳定性. 证明了当时滞τ=τ\-1+τ\-2适当小时, 正平衡点是局部渐近稳定的,随着时滞的增加, 正平衡点由稳定变为不稳定, 系统在正平衡点附近发生Hopf分支.     

Analysis of stability and Hopf bifurcation for a delayed logistic equation

The dynamics of a logistic equation with discrete delay are investigated, together with the local and global stability of the equilibria. In particular, the conditions under which a sequence of Hopf bifurcations occur at the positive equilibrium are obtained. Explicit algorithm for determining the stability of the bifurcating periodic solutions and the direction of the Hopf bifurcation are derived by using the theory of normal form and center manifold [Hassard B, Kazarino D, Wan Y. Theory and applications of Hopf bifurcation. Cambridge: Cambridge University Press; 1981.]. Global existence of periodic solutions is also established by using a global Hopf bifurcation result of Wu [Symmetric functional differential equations and neural networks with memory. Trans Amer Math Soc 350:1998;4799–38.]     

Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system

A diffusive predator-prey system with Holling type-II predator functional response subject to Neumann boundary conditions is considered. Hopf and steady state bifurcation analysis are carried out in details. In particular we show the existence of multiple spatially non-homogeneous periodic orbits while the system parameters are all spatially homogeneous. Our results and global bifurcation theory also suggest the existence of loops of spatially non-homogeneous periodic orbits and steady state solutions. These results provide theoretical evidences to the complex spatiotemporal dynamics found by numerical simulation.