Global Asymptotic Stability and Hopf Bifurcation in a Homogeneous Diffusive Predator-Prey System with Holling Type II Functional Response

In this paper, we considered a homogeneous reaction-diffusion predator-prey system with Holling type II functional response subject to Neumann boundary conditions. Some new sufficient conditions were analytically established to ensure that this system has globally asymptotically stable equilibria and Hopf bifurcation surrounding interior equilibrium. In the analysis of Hopf bifurcation, based on the phenomenon of Turing instability and well-done conditions, the system undergoes a Hopf bifurcation and an example incorporating with numerical simulations to support the existence of Hopf bifurcation is presented. We also derived a useful algorithm for determining direction of Hopf bifurcation and stability of bifurcating periodic solutions correspond to j ≠0 and j = 0, respectively. Finally, all these theoretical results are expected to be useful in the future study of dynamical complexity of ecological environment.

Stability Analysis and Hopf Bifurcation for ODE System of Predator-Prey Model with Mutual Interference

In this paper, the temporal and spatial patterns of a diffusive predator-prey model with mutual interference under homogeneous Neumann boundary conditions were studied. Specifically, first, taking the intrinsic growth rate of the predator as the parameter, we give a computational and theoretical analysis of Hopf bifurcation on the positive equilibrium for the ODE system. As well, we have discussed the conditions for determining the bifurcation direction and the stability of the bifurcating periodic solutions.

Global Stability and Hopf Bifurcation for a Virus Dynamics Model with General Incidence Rate and Delayed CTL Immune Response

In this work, we investigate an HIV-1 infection model with a general incidence rate and delayed CTL immune response. The model admits three possible equilibria, an infection-free equilibrium E*0, CTL-inactivated infection equilibrium E*1 and CTL-activated infection equilibrium E*2. We prove that in the absence of CTL immune delay, the model has exactly the basic behaviour model, for all positive intracellular delays, the global dynamics are determined by two threshold parameters R0 and R1, if R0 ≤ 1, E*0 is globally asymptotically stable, if R1 ≤ 1 < R0, E*1 is globally asymptotically stable and if R1 >1, E*2 is globally asymptotically stable. But if the CTL immune response delay is different from zero, then the behaviour of the model at E*2 changes completely, although R1 > 1, a Hopf bifurcation at E*2 is established. In the end, we present some numerical simulations.

Bifurcation and Turing Pattern Formation in a Diffusion Modified Leslie-Gower Predator-Prey Model with Crowley-Martin Functional Response

In this paper, we study a modified Leslie-Gower predator-prey model with Smith growth subject to homogeneous Neumann boundary condition, in which the functional response is the Crowley-Martin functional response term. Firstly, for ODE model, the local stability of equilibrium point is given. And by using bifurcation theory and selecting suitable bifurcation parameters, we find many kinds of bifurcation phenomena, including Transcritical bifurcation and Hopf bifurcation. For the reaction-diffusion model, we find that Turing instability occurs. Besides, it is proved that Hopf bifurcation exists in the model. Finally, numerical simulations are presented to verify and illustrate the theoretical results.

Generalization of Hopf Functional Equation

This paper generalizes the Hopf functional equation in order to apply it to a wider class of not necessarily incompressible fluid flows. We start by defining characteristic functionals of the velocity field, the density field and the temperature field of a compressible field. Using the continuity equation, the Navier-Stokes equations and the equation of energy we derive a functional equation governing the motion of an ideal gas flow and a van der Waals gas flow, and then give some general methods of deriving a functional equation governing the motion of any compressible fluid flow. These functional equations can be considered as the generalization of the Hopf functional equation.

HOPF BIFURCATION OF AN OSCILLATOR WITH QUADRATIC AND CUBIC NONLINEARITIES AND WITH DELAYED VELOCITY FEEDBACK

This paper studies the local dynamics of an SDOF system with quadratic and cubic stiffness terms,and with linear delayed velocity feedback.The analysis indicates that for a sufficiently large velocity feedback gain,the equilibrium of the system may undergo a number of stability switches with an increase of time delay,and then becomes unstable forever.At each critical value of time delay for which the system changes its stability,a generic Hopf bifurcation occurs and a periodic motion emerges in a one-sided neighbourhood of the critical time delay.The method of Fredholm alternative is applied to determine the bifurcating periodic motions and their stability.It stresses on the effect of the system parameters on the stable regions and the amplitudes of the bifurcating periodic solutions.

NUMERICAL HOPF BIFURCATION OF DELAY-DIFFERENTIAL EQUATIONS

In this paper we consider the numerical solution of some delay differential equations undergoing a Hopf bifurcation. We prove that if the delay differential equations have a Hopf bifurcation point atλ=λ*, then the numerical solution of the equation also has a Hopf bifurcation point atλh =λ* + O(h).

Hopf Bifurcation of a Positive Feedback Delay Differential Equation

Under some minor technical hypotheses, for each T larger than a certain rS > 0, Krisztin, Walther and Wu showed the existence of a periodic orbit for the positive feedback delay differential equation x(t) = -rμx(t) + rf(x(t-1)), where r and μ are positive constants and f : R → R satisfies f(0) = 0 and f' > 0. Combining this with a unique result of Krisztin and Walther, we know that this periodic orbit is the one branched out from 0 through Hopf bifurcation. Using the normal form theory for delay differential equations, we show the same result under the condition that f ∈ C3(R,R) is such that f''(0) = 0 and f'''(0) < 0, which is weaker than those of Krisztin and Walther.

A perturbation-incremental scheme for studying Hopf bifurcation in delayed differential systems

A new method, called perturbation-incremental scheme (PIS), is presented to investigate the periodic solution derived from Hopf bifurcation due to time delay in a system of first-order delayed differential equations. The method is summarized as three steps, namely linear analysis at critical value, perturba- tion and increment for continuation. The PIS can bypass and avoid the tedious calculation of the center manifold reduction (CMR) and normal form. Meanwhile, the PIS not only inherits the advantages of the method of multiple scales (MMS) but also overcomes the disadvantages of the incremental harmonic balance (IHB) method. Three delayed systems are used as illustrative examples to demonstrate the validity of the present method. The periodic solution derived from the delay-induced Hopf bifurcation is obtained in a closed form by the PIS procedure. The validity of the results is shown by their consis- tency with the numerical simulation. Furthermore, an approximate solution can be calculated in any required accuracy.

具有Holling-Ⅱ型功能反应函数的双时滞捕食者-食饵系统的Hopf分支

在具有Holling-Ⅱ型功能反应函数的捕食者-食饵系统中引入2个时滞参数,用来刻画捕食者和食饵的生长时滞,研究了系统平衡点的局部稳定性.结果表明,随着参数的变化,系统平衡点发生了扰动,进而出现了周期解.给出了Hopf分支存在条件的显示表达式,并通过数值实验验证了结论.

Delayed Hopf bifurcation in time-delayed slow-fast systems

This paper presents an investigation on the phenomenon of delayed bifurcation in time-delayed slow-fast differential systems.Here the two delayed's have different meanings.The delayed bifurcation means that the bifurcation does not happen immediately at the bifurcation point as the bifurcation parameter passes through some bifurcation point,but at some other point which is above the bifurcation point by an obvious distance.In a time-delayed system,the evolution of the system depends not only on the present state but also on past states.In this paper,the time-delayed slow-fast system is firstly simplified to a slow-fast system without time delay by means of the center manifold reduction,and then the so-called entry-exit function is defined to characterize the delayed bifurcation on the basis of Neishtadt's theory.It shows that delayed Hopf bifurcation exists in time-delayed slow-fast systems,and the theoretical prediction on the exit-point is in good agreement with the numerical calculation,as illustrated in the two illustrative examples.

A multi-delay model for pest control with awareness induced interventions — Hopf bifurcation and optimal control analysis

Farming awareness is an important measure for pest controlling in agricultural practice.Time delay in controlling pest may affect the system.Time delay occurs in organizing awareness campaigns,also time delay may takes place in becoming aware of the control strategies or implementing suitable controlling methods informed through social media.Thus we have derived a mathematical model incorporating two time delays into the system and Holling type-II functional response.The existence and the stability criteria of the equilibria are obtained in terms of the basic reproduction number and time delays.Stability changes occur through Hopf-bifurcation when time delays cross the critical values.Optimal control theory has been applied for cost-effectiveness of the delayed system.Numerical simulations are carried out to justify the analytical results.This study shows that optimal farming awareness through radio,TV etc.can control the delay induced bifurcation in a cost-effective way.

Bifurcationing Analysis of Predator-Prey Diffusive System Based on Bazykin Functional Response

A predator-prey diffusion system with a Bazykin functional response is studied. The existence of equilibrium points, the stability of normal number equilibrium points and the existence of Hopf bifurcationes are investigated for the proposed system, the existence of positive solutions in the system is discussed under Neumann boundary conditions, and the stability of constant equilibrium points is focused on under the condition of Hurwitz criterion. The results show that there exist positive equilibrium points in the system under Neumann boundary conditions, and the normal number equilibrium points are stable when specific conditions are satisfied, and the bifurcation points of Hopf bifurcationes and their orders are given.

Hopf Bifurcation of a Gene-Protein Network Module with Reaction Diffusion and Delay Effects

The infinite dimensional partial delay differential equation is set forth and delay difference state feedback control is considered to describe the cell cycle growth in eukaryotic cell cycles. Hopf bifurcation occurs as varying free parameters and time delay continuously and the multi-layer oscillation phenomena of the homogeneous steady state of a simple gene-protein network module is investigated. Normal form is derived based on normal formal analysis technique combined with center manifold theory, which is further to compute the bifurcating direction and the stability of bifurcation periodical solutions underlying Hopf bifurcation. Finally, the numerical simulation oscillation phenomena is in coincidence with the theoretical analysis results.

Bifurcation Analysis of a Nonlinear Genetic Network Model with Time Delay

This paper presents a bifurcation study of a mRNA-protein network with negative feedback and time delay. The network is modeled as a coupled system of N ordinary differential equations (ODEs) and N delay differential equations (DDEs). Linear analysis of the stable equilibria shows the existence of a critical time delay beyond which limit cycle oscillations are born in a Hopf bifurcation. The Poincaré-Lindstedt perturbation method is applied to the nonlinear system, resulting in closed form approximate expressions for the amplitude and frequency of oscillation. We confirm our perturbation analysis results by numerically simulating the full 2N-dimensional nonlinear system for N = 2, 7, 15, and 30. Our results show that for small perturbations the equilibrium undergoes a supercritical Hopf and the system exhibits stable periodic solutions. Furthermore, our closed form numerical expressions exhibit the importance of the network’s negative feedback and nonlinear effects.

具时滞物价瑞利方程的Hopf分支

研究具时滞物价瑞利方程模型的动力学性质,利用指数多项式的-τD划分讨论平衡点的稳定性和Hopf分支的存在性,利用具有限时滞Liénard方程的Hopf分支公式获得了Hopf分支方向和周期解的稳定性计算公式,并给出了在r-γ参数平面上的Hopf分支图,得到了"时滞反映出价格对供给具有滞后作用"的结论,合理地解释了经济生活中的价格振荡现象.

跨音速操纵面嗡鸣Hopf分叉分析及结构参数对嗡鸣特性影响的研究

采用Ｈｏｐｆ分叉分析方法，对跨音速操纵面嗡鸣问题进行了研究。用二维守恒型非定常Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ 方程计算了舵面振动时的非定常气动力，计算了出现Ｈｏｐｆ分叉的Ｍａ 数，研究了一些参数对嗡鸣的影响。将Ｈｏｐｆ分叉的计算结果与嗡鸣的时间历程计算做了对比，两者相当吻合，与飞行试验结果相比，也比较接近。

van der Pol-Duffing时滞系统的稳定性和Hopf分岔

研究了具有三次项的ｖａｎｄｅｒＰｏｌ－Ｄｕｆｆｉｎｇ非线性时滞系统的稳定性和Ｈｏｐｆ分岔，分析了当线性化特征方程随两参数（时滞量和增益系数）变化时特征根的分布；证明Ｈｏｐｆ分岔的存在性；通过构造中心流形并且使用范式方法给出Ｈｏｐｆ分岔的方向以及周期解的稳定性；讨论时滞量对该系统的Ｈｏｐｆ分岔的影响．

van der Pol型时滞系统的两参数余维一Hopf分岔及其稳定性

研究具有三次非线性时滞项的van der Pol 型时滞系统随两参数( 时滞量和增益系数) 余维一Hopf 分岔,说明了线性化特征方程随两参数变化时的根的分布和Hopf 分岔存在性;通过构造中心流形并且使用范式方法确定出Hopf 分岔的方向以及周期解的稳定性;分析了时滞量对所论系统发生Hopf

一类具时滞和收获的捕食模型的稳定性与Hopf分支

本文研究一类具有常数收获率和时滞的捕食模型,其中时滞描述了捕食种群的妊娠期。通过分析特征方程,得到了正平衡点局部稳定的条件。当时滞τ增加时,正平衡点失去稳定性,当τ跨过临界值时系统将出现Hopf分支。应用中心流形定理和规范型理论,得到了确定Hopf分支方向和分支周期解的稳定性的计算公式。最后对所得理论结果进行了数值模拟。