一个带有Holling-Ⅳ功能性反应的捕食与被捕食模型的稳定性和分支

研究了一个带Holling-Ⅳ型功能反应的捕食与被捕食模型,讨论了系统解的有界性和各平衡点的存在性,使用Routh-Hurwitz定理得到了平衡点局部渐近稳定的充分条件.引入两个离散时滞,得出了重要的结果:边界平衡点的稳定性随着τ1的增加,由稳定变为不稳定,并且会发生Hopf分支.对正平衡点的稳定性变化,考虑了两个时滞相等的情况,结果是随着分支参数的增加,不仅稳定性会发生变化,产生Hopf分支,甚至可能出现小范围周期解.     

一类具有时滞的疾病感染的捕食-被捕食模型分析

研究了一类具有时滞的疾病感染的捕食-被捕食模型.首先讨论了系统的耗散性;接着分析了系统的平衡点并根据Routh-Hurwitz准则判断其局部稳定性;最后利用Lyapunov方法和Bendixson-Dulac判别法给出了平衡点的全局稳定性.     

分布在非齐次空间中捕食－被捕食系统的锁相振动

本文以经典的带有时滞的捕食－被捕食系统为基础，构造了一类泛函微分方程作为描述分布在由ｎ个岛屿构成的环状区域上的捕食－被捕食种群生长过程的模型．我们假设种群在最相邻的岛屿间相互迁移，以迁移率为分支参数，研究了该系统的锁相振动，给出了离散波分支的存在条件．     

分布在非齐次空间中捕食－被捕食系统的锁相振动

本文以经典的带有时滞的捕食－被捕食系统为基础，构造了一类泛函微分方程作为描述分布在由ｎ个岛屿构成的环状区域上的捕食－被捕食种群生长过程的模型．我们假设种群在最相邻的岛屿间相互迁移，以迁移率为分支参数，研究了该系统的锁相振动，给出了离散波分支的存在条件．     

一类具有时滞的捕食——食饵系统的稳定性与Hopf分支

研究了一类具有时滞的捕食—食饵系统,通过分析正平衡点处的特征方程,讨论了系统正平衡点的稳定性;以时滞作为分支参数,应用Hopf分支理论,得到了系统存在Hopf分支的充分条件.     

疾病在食饵中传播的具有时滞的捕食被捕食模型的分析

研究了一个疾病在食饵中传播的捕食与被捕食模型.在未引入时滞时,利用Routh-Hurwitz定理证明了正平衡点的局部渐近稳定性.在引入时滞后,主要讨论了正平衡点的稳定性,得到了当经过一系列临界条件时发生Hopf分支.     

具有时滞和间接控制的捕食-被捕食模型的分支分析

研究了一类具有时滞和间接控制的捕食-被捕食模型.选择时滞τ为分支参数,证实了系统在一定的时滞范围内是渐近稳定的.当时滞τ通过一系列的临界值时,Hopf分支产生,即当时滞τ通过某些临界值时,从平衡点处产生一簇周期解.最后,用数值模拟验证了理论分析结果的正确性.     

一类捕食与被捕食系统的稳定性与分岔分析

本文利用Schur—Cohn—Jury引理及分岔理论讨论了一类捕食与被捕食系统的动力学性质，分析了其正平衡点的稳定性，并讨论了Neimark—Sacker分岔稳定性与方向。通过数值模拟验证了所得结果的正确性。     

一类带无选择性捕获和时滞的捕食食饵系统的Hopf分支分析

本文主要研究了一个改进的带时滞和无选择捕获函数的捕食-食饵生态经济系统的稳定性和Hopf分支.利用微分代数系统的稳定性理论和分支理论,得到了系统正平衡点稳定性的条件,以及当时滞τ作为分支参数时系统产生Hopf分支的条件.对Leslie-Gower捕食-食饵模型进行了一定程度的完善,使得建立的模型更符合实际情况,因此得到的结论也更加科学.     

具有功能性反应和时滞的扩散捕食-食饵系统

考虑具有功能性反应和时滞的扩散捕食-食饵系统，其中食饵连两个斑块间具有一定的扩散系数，捕食者可以两个斑块中任意走动，我们讨论了系统的一致持久性和周期解的存在性及全局吸引性．     

Hopf bifurcation in Cohen–Grossberg neural network with distributed delays

In this paper, we discuss the stability and bifurcation of the distributed delays Cohen–Grossberg neural networks with two neurons. By choosing the average delay as a bifurcation parameter, we prove that Hopf bifurcation occurs. The stability of bifurcating periodic solutions and the direction of Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Finally, numerical simulation results are given to support the theoretical predictions.     

Stability and Hopf bifurcation for a regulated logistic growth model with discrete and distributed delays

In this paper, we investigate the stability and Hopf bifurcation of a new regulated logistic growth with discrete and distributed delays. By choosing the discrete delay τ as a bifurcation parameter, we prove that the system is locally asymptotically stable in a range of the delay and Hopf bifurcation occurs as τ crosses a critical value. Furthermore, explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived by normal form theorem and center manifold argument. Finally, an illustrative example is also given to support the theoretical results.     

Dynamics of a Delayed Predator-prey System with Stage Structure for Predator and Prey

In this paper, a predator-prey system with two discrete delays and stage structure for both the predator and the prey is investigated. The dynamical behaviors such as local stability and local Hopf bifurcation are analyzed by regarding the possible combinations of the two delays as bifurcating parameter. Some explicit formulae determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form method and the center manifold theory. Finally, numerical simulations are presented to support the theoretical analysis.     

Dynamics of a congestion control model in a wireless access network

In this paper, a congestion control algorithm with heterogeneous delays in a wireless access network is considered. We regard the communication time delay as a bifurcating parameter to study the dynamical behaviors, i.e., local asymptotical stability, Hopf bifurcation and resonant codimension-two bifurcation. By analyzing the associated characteristic equation, the Hopf bifurcation occurs when the delay passes through a sequence of critical value. Furthermore, the direction and stability of the bifurcating periodic solutions are derived by applying the normal form theory and the center manifold theorem. In the meantime, the resonant codimension-two bifurcation is also found in this model. Some numerical examples are finally performed to verify the theoretical results.     

Stability and bifurcation of a Cohen-Grossberg neural network with discrete delays

A Cohen-Grossberg neural network with discrete delays is investigated in this paper. The qualitative analysis is given for the system and it is found that the system undergoes a sequence of Hopf bifurcations by choosing the discrete time delay as a bifurcation parameter. Moreover, by applying the normal form theory and the center manifold theorem, the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions are obtained. Numerical simulations are given to illustrate the obtained results.     

Stability and Hopf bifurcation analysis on a four-neuron BAM neural network with distributed delays

In this paper, a four-neuron BAM neural network with distributed delays is considered, where kernels are chosen as weak kernels. Its dynamics is studied in terms of local stability analysis and Hopf bifurcation analysis. By choosing the average delay as a bifurcation parameter and analyzing the associated characteristic equation, Hopf bifurcation occurs when the bifurcation parameter passes through some exceptive values. The stability of bifurcating periodic solutions and a formula for determining the direction of Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Finally, numerical simulation results are given to validate the theorem obtained.     

Stability Switches, Hopf Bifurcations, and Spatio-temporal Patterns in a Delayed Neural Model with Bidirectional Coupling

The dynamical behavior of a delayed neural network with bi-directional coupling is investigated by taking the delay as the bifurcating parameter. Some parameter regions are given for conditional/absolute stability and Hopf bifurcations by using the theory of functional differential equations. As the propagation time delay in the coupling varies, stability switches for the trivial solution are found. Conditions ensuring the stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. We also discuss the spatio-temporal patterns of bifurcating periodic oscillations by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. In particular, we obtain that the spatio-temporal patterns of bifurcating periodic oscillations will alternate according to the change of the propagation time delay in the coupling, i.e., different ranges of delays correspond to different patterns of neural activities. Numerical simulations are given to illustrate the obtained results and show the existence of bursts in some interval of the time for large enough delay.     

Hopf bifurcation and multiple periodic solutions in Lotka–Volterra systems with symmetries

The purpose of this paper is to study Hopf bifurcations in a delayed Lotka–Volterra system with dihedral symmetry. By treating the response delay as bifurcation parameter and employing equivariant degree method, we obtain the existence of multiple branches of nonconstant periodic solutions through a local Hopf bifurcation around an equilibrium. We find that competing coefficients and the response delay in the system can affect the spatio-temporal patterns of bifurcating periodic solutions. According to their symmetric properties, a topological classification is given for these periodic solutions. Furthermore, an estimation is presented on minimal number of bifurcating branches. These theoretical results are helpful to better understand the complex dynamics induced by response delays and symmetries in Lotka–Volterra systems.     

Bifurcation analysis for a three-species predator-prey system with two delays

In this paper, a three-species predator-prey system with two delays is investigated. By choosing the sum τ of two delays as a bifurcation parameter, we first show that Hopf bifurcation at the positive equilibrium of the system can occur as τ crosses some critical values. Second, we obtain the formulae determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions by using the normal form theory and center manifold theorem. Finally, numerical simulations supporting our theoretical results are also included.     

Dynamical analysis of a delayed six‐neuron BAM network

In this article, a simplified bidirectional associative memory network with delays involving six neurons is considered. By analyzing the distribution of roots of the characteristic equation for linearized system regarding the sum of delay as a bifurcation parameter, we obtain the condition of the occurrence for Hopf bifurcation. It reveals that Hopf bifurcation occurs when the sum of the delay passes through a critical value. The direction and the stability of the bifurcating periodic solution are determined by applying the normal form theory and center manifold theorem. Finally, a numerical example is used to illustrate the validity of theoretical results obtained. © 2015 Wiley Periodicals, Inc. Complexity 21: 9–28, 2016