Stability and Hopf bifurcation in a diffusive logistic population model with nonlocal delay effect

A reaction-diffusion model with logistic type growth, nonlocal delay effect and Dirichlet boundary condition is considered, and combined effect of the time delay and nonlocal spatial dispersal provides a more realistic way of modeling the complex spatiotemporal behavior. The stability of the positive spatially nonhomogeneous positive equilibrium and associated Hopf bifurcation are investigated for the case of near equilibrium bifurcation point and the case of spatially homogeneous dispersal kernel.     

Stability and bifurcation analysis in a viral model with delay

In this paper, we consider a three‐dimensional viral model with delay. We first investigate the linear stability and the existence of a Hopf bifurcation. It is shown that Hopf bifurcations occur as the delay τ passes through a sequence of critical values. Then, using the normal form theory and center manifold reduction, we derive the explicit formulaes that determine the stability, the direction, and the period of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the validity of the main results. Finally, some brief conclusions are given. Copyright © 2012 John Wiley & Sons, Ltd.     

Stability and Hopf bifurcation of a delayed predator-prey system with nonlocal competition and herd behaviour

In this paper, we investigate the stability and Hopf bifurcation of a diffusive predator-prey system with herd behaviour. The model is described by introducing both time delay and nonlocal prey intraspecific competition. Compared to the model without time delay, or without nonlocal competition, thanks to the together action of time delay and nonlocal competition, we prove that the first critical value of Hopf bifurcation may be homogenous or non-homogeneous. We also show that a double-Hopf bifurcation occurs at the intersection point of the homogenous and non-homogeneous Hopf bifurcation curves. Furthermore, by the computation of normal forms for the system near equilibria, we investigate the stability and direction of Hopf bifurcation. Numerical simulations also show that the spatially homogeneous and non-homogeneous periodic patters.     

Travelling waves of a predator–prey model with a spatiotemporal delay

This paper is concerned with the existence of traveling waves to a predator–prey model with a spatiotemporal delay. By analyzing the corresponding characteristic equations, the local stability of a positive steady state and each of boundary steady states are established, and the existence of Hopf bifurcation at the positive steady state is also discussed. By constructing a pair of upper–lower solutions and by using the cross‐iteration method as well as the Schauder's fixed‐point theorem, the existence of a traveling wave solution connecting the semi‐trivial steady state and the positive steady state is proved. Numerical simulations are carried out to illustrate the main theoretical results. Copyright © 2013 John Wiley & Sons, Ltd.     

Bifurcation analysis of a modified Holling-Tanner predator-prey model with time delay

In this paper, a modified Holling-Tanner predator-prey model with time delay is considered. By regarding the delay as the bifurcation parameter, the local asymptotic stability of the positive equilibrium is investigated. Meanwhile, we find that the system can also undergo a Hopf bifurcation of nonconstant periodic solution at the positive equilibrium when the delay crosses through a sequence of critical values. In particular, we study the direction of Hopf bifurcation and the stability of bifurcated periodic solutions, 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.     

Dynamics of a ratio‐dependent stage‐structured predator‐prey model with delay

In this paper, we investigate the dynamics of a time‐delay ratio‐dependent predator‐prey model with stage structure for the predator. This predator‐prey system conforms to the realistically biological environment. The existence and stability of the positive equilibrium are thoroughly analyzed, and the sufficient and necessary conditions for the stability and instability of the positive equilibrium are obtained for the case without delay. Then, the influence of delay on the dynamics of the system is investigated using the geometric criterion developed by Beretta and Kuang. 26 We show that the positive steady state can be destabilized through a Hopf bifurcation and there exist stability switches under some conditions. The formulas determining the direction and the stability of Hopf bifurcations are explicitly derived by using the center manifold reduction and normal form theory. Finally, some numerical simulations are performed to illustrate and expand our theoretical results.     

A delayed prey–predator model with Crowley–Martin‐type functional response including prey refuge

In this paper, we have studied a prey–predator model living in a habitat that divided into two regions: an unreserved region and a reserved (refuge) region. The migration between these two regions is allowed. The interaction between unreserved prey and predator is Crowley–Martin‐type functional response. The local and global stability of the system is discussed. Further, the system is extended to incorporate the effect of time delay. Then the dynamical behavior of the system is analyzed, taking delay as a bifurcation parameter. The direction of Hopf bifurcation and the stability of the bifurcated periodic solution are determined with the help of normal form theory and centre manifold theorem. We have also discussed the influence of prey refuge on densities of prey and predator species. The analytical results are supplemented with numerical simulations. Copyright © 2017 John Wiley & Sons, Ltd.     

Properties of stability and Hopf bifurcation for a HIV infection model with time delay

In this paper, we consider the classical mathematical model with saturation response of the infection rate and time delay. By stability analysis we obtain sufficient conditions for the global stability of the infection-free steady state and the permanence of the infected steady state. Numerical simulations are carried out to explain the mathematical conclusions.     

三维退化时滞微分系统的Hopf分支

讨论了三维退化时滞微分系统的Hopf分支.通过分析其特征方程,发现当时滞穿越某些值时出现了分支.给出了寻找Hopf分支点的计算方法.     

Bifurcations in a diffusive predator–prey model with predator saturation and competition response

A diffusive predator–prey model with predator saturation and competition response subject to homogeneous Neumann boundary conditions is considered in this paper. We find that the spatially homogeneous and non‐homogeneous periodic solutions through all parameters of the system are spatially homogeneous. To verify our theoretical results, some numerical simulations are also carried out. Copyright © 2014 John Wiley & Sons, Ltd.     

Turing instability and Hopf bifurcation in a diffusive Leslie–Gower predator–prey model

In this paper, a reaction‐diffusion predator–prey system that incorporates the Holling‐type II and a modified Leslie‐Gower functional responses is considered. For ODE, the local stability of the positive equilibrium is investigated and the specific conditions are obtained. For partial differential equation, we consider the dissipation and persistence of solutions, the Turing instability of the equilibrium solutions, and the Hopf bifurcation. By calculating the normal form, we derive the formulae, which can determine the direction and the stability of Hopf bifurcation according to the original parameters of the system. We also use some numerical simulations to illustrate our theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.     

Dynamical analysis of a delayed singular prey–predator economic model with stochastic fluctuations

This article studies a delayed singular prey–predator economic model with stochastic fluctuations, which is described by differential‐algebraic equations due to a economic theory. Local stability and Hopf bifurcation condition are described on the delayed singular prey–predator economic model within deterministic environment. It reveals the sensitivity of the model dynamics on gestation time delay. A phenomenon of Hopf bifurcation occurs as the gestation time delay increases through a certain threshold. Subsequently, a singular stochastic prey–predator economic model with time delay is obtained by introducing Gaussian white noise terms to the above deterministic model system. The fluctuation intensity of population and harvest effort are calculated by Fourier transforms method. Numerical simulations are carried out to substantiate these theory analysis. © 2013 Wiley Periodicals, Inc. Complexity 19: 23–29, 2014     

Bifurcation of periodic solution in a three-unit neural network with delay

1. IntroductionDynamical characteristics of neural networks have become recelltly a subject of intenseresearch activity. J. B6lair and S. Dufou.[1] investigated a system of neural networks introduced by Hopfield[2]. Especially' they studied the three-uult network system with noself connectiondxi(t) 3dt = --xi(t) Z Ti,f; (x;(t -- T)), i = 1, 2, 3, (l)J = 1where f;(0) = 0, j = 1, 2, 3 and T, = 0, i = 1, 2, 3, give the stability properties of the nullsolution. [2] discussed the lineaJr stab…     

Dynamical analysis in a diffusive predator-prey system with a delay and strong Allee effect

In this paper, we consider the dynamics of a diffusive predator-prey system with strong Allee effect and delayed Ivlev-type functional response. At first, we apply the method of upper-lower solutions and the comparison principle in proving the nonnegativity of the solutions. Then by analyzing the distribution of the eigenvalues, we obtain the bistability of the system and the existence of Hopf bifurcation. Furthermore, by using the center manifolds theory and normal form method, we study the properties of the Hopf bifurcations. Finally, some numerical simulations are carried out for illustrating the theoretical results.     

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.     

Stability and bifurcation analysis of a nutrient-phytoplankton model with time delay

We proposed a nutrient-phytoplankton interaction model with a discrete and distributed time delay to provide a better understanding of phytoplankton growth dynamics and nutrient-phytoplankton oscillations induced by delay. Standard linear analysis indicated that delay can induce instability of a positive equilibrium via Hopf bifurcation. We derived the conditions guaranteeing the existence of Hopf bifurcation and tracked its direction and the stability of the bifurcating periodic solutions. We also obtained the sufficient conditions for the global asymptotic stability of the unique positive steady state. Numerical analysis in the fully nonlinear regime showed that the stability of the positive equilibrium is sensitive to changes in delay values under select conditions. Numerical results were consistent with results predicted by linear analysis.     

一类具有时滞的病菌-免疫力模型的Hopf分歧

考虑病菌的一种信息交流机制,建立一类病菌与免疫系统竞争的时滞传染病模型.分析正平衡点的存在性、渐近稳定性、Hopf分歧的存在性及方向.运用计算机数值模拟验证所得理论结果,为传染病的控制和预防提供了理论基础和数值依据.     

Global Bifurcation for a Class of Lotka-Volterra Competitive Systems

For a class of Lotka-Volterra competitive systems including both diffusion and advection, a global bifurcation result of positive steady states is established via a bifurcation approach. Also, the phenomenon of multiple positive steady states is discussed.     

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.