一类带毒生化反应竞争模型中的极限环

研究两个微生物竞争同一营养,而其中一个竞争者会产生毒素抑制另一竞争者且产物系数为γ1(S)=A1+B1Sn和γ2(S)=A2+B2Sm(n和m是自然数)函数时的生化反应模型.分析了平衡点的稳定性,并证明了三维系统经历Hopf分支后产生的周期解的稳定性,进一步又证明极限环存在于有关的二维稳定流形之上,并用实例验证了结果.     

Bifurcation for a functional yield chemostat when one competitor produces a toxin

A model, with general yield functions: Fi(S), i=1,2, of competition in the chemostat of two competitors for a single nutrient when one of the competitors produces toxin against its opponent is studied in this paper. The conditions in terms of the relevant parameters for the Hopf bifurcation of the three-dimensional system have been proved, which implies the existence of limit cycles in the 3-D system.     

Hopf bifurcation in a two-competitor, one-prey system with time delay

In this paper, we consider a delayed two-competitor/one-prey system in which both two competitors exhibit Holling II functional response. By choosing the time delay as a bifurcation parameter, it is found that the Hopf bifurcation occurs when the delay passes through a certain critical value. Numerical simulations are performed to illustrate the analytical results.     

Limit cycles in chemostat with constant yields

Using a 3-D Hopf bifurcation, we prove the existence of limit cycles in the chemostat of two competitors for a single nutrient with constant yields. This proof is done analytically with a corollary to the center manifold theorem. Therefore, the nonlinear oscillatory phenomena exist even in the chemostat with constant yields.     

Hopf bifurcation in a delayed reaction–diffusion–advection population model

In this paper, we investigate a reaction–diffusion–advection model with time delay effect. The stability/instability of the spatially nonhomogeneous positive steady state and the associated Hopf bifurcation are investigated when the given parameter of the model is near the principle eigenvalue of an elliptic operator. Our results imply that time delay can make the spatially nonhomogeneous positive steady state unstable for a reaction–diffusion–advection model, and the model can exhibit oscillatory pattern through Hopf bifurcation. The effect of advection on Hopf bifurcation values is also considered, and our results suggest that Hopf bifurcation is more likely to occur when the advection rate increases.     

Delayed feedback-based bifurcation control in an Internet congestion model

In this paper, the control of Hopf bifurcation in an Internet congestion model with a single link accessed by a single source is presented. By choosing the gain parameter as a bifurcation parameter, it is found that the system without control cannot guarantee a stationary sending rate. Furthermore, Hopf bifurcation occurs when the positive gain parameter of the system exceeds a critical value. For Internet congestion model, a control model based on delayed feedback is proposed and analyzed for delaying the onset of undesirable Hopf bifurcation. Numerical simulations are given to justify the validity of delayed feedback controller in bifurcation control.     

Brusselator模型的扩散引起不稳定性和Hopf分支

研究了Brusselator常微分系统和相应的偏微分系统的Hopf分支,并用规范形理论和中心流形定理讨论了当空间的维数为1时Hopf分支解的稳定性．证明了:当参数满足某些条件时,Brusselator常微分系统的平衡解和周期解是渐近稳定的,而相应的偏微分系统的空间齐次平衡解和空间齐次周期解是不稳定的;如果适当选取参数,那么Brusselator常微分系统不出现Hopf分支,但偏微分系统出现Hopf分支,这表明,扩散可以导致Hopf分支．     

HOPF BIFURCATION AND OTHER DYNAMICAL BEHAVIORS FOR A FOURTH ORDER DIFFERENTIAL EQUATION IN MODELS OF INFECTIOUS DISEASE

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Hopf bifurcation of the stochastic model on HAB nonlinear stochastic dynamics

A nonlinear stochastic dynamical model on a typical HAB algae diatom and dianoflagellate densities was created and presented in this paper. Simplifying the model through a stochastic averaging method, we obtained a two-dimensional diffusion process of averaged amplitude and phase. The singular boundary theory of diffusion process and the invariant measure theory were applied in analyzing the bifurcation of stability and the Hopf bifurcation of the stochastic system. The critical value of the stochastic Hopf bifurcation parameter was obtained and the conclusion that the position of Hopf bifurcation drifting with the parameter increase is presented as a result.     

Codimension two bifurcation in a delayed neural network with unidirectional coupling

In this paper, a delayed neural network model with unidirectional coupling is considered. Zero–Hopf bifurcation is studied by using the center manifold reduction and the normal form method for retarded functional differential equation. We get the versal unfolding of the norm form at the zero–Hopf singularity and show that the model can exhibit pitchfork, Hopf bifurcation, and double Hopf bifurcation is also found to occur in this model. Some numerical simulations are given to support the analytic results.