关于具时滞的捕食-被捕食系统的稳定性与Hopf分支

研究了一类具时滞的捕食—被捕食系统正平衡解的稳定性和Hopf分支的存在性,即以转化率l为参数,得到存在一列l_k(k=0,1,2…),使当l∈(0,l_0)时正平衡解是渐近稳定的;当l>l_0时,正平衡解是不稳定的,而每一个l_k都是该系统的Hopf分支值,改进了一些已知的工作。     

Dynamics of excitable elements with time-delayed coupling

Motivated by recent experiments on intracellular calcium release we study the effects of different types of coupling on the dynamics of arrays of excitable elements. We intend to find a mechanism that produces a sustained activity of the elements following a spike. While instantaneous diffusive coupling does not exhibit this property, we show that, for a coupling term with temporal delay, signals from adjacent elements can serve as mutual excitations and thus prolong the duration of the signal. We propose that time delayed coupling is generated by diffusion between isolated clusters of calcium channels. Our model could thus provide an explanation for two different release modes observed in the Ca²⁺ system.     

Hopf bifurcation and transition to chaos in Lotka-Volterra equation

It is shown that in a suitable class of Lotka-Volterra systems it is possible to characterize the centre-critical case of the Hopf bifurcation of the multipopulation equilibrium. Moreover, for three populations, it is shown that, in the non-critical case, Hopf bifurcation is supercritical. Numerical evidence of transition to chaotic dynamics, via period-doubling cascades, from the limit cycle is reported.     

Multiparameter bifurcation diagrams in predator-prey models with time lag

A predator-prey model is considered in which prey is limited by the carrying capacity of the environment, and predator growth rate depends on past quantities of prey. Conditions for stability of an equilibrium, and its bifurcation are established taking into account all the parameters.     

具有三时滞Lotka-Volterra互惠系统的Hopf分支

建立了具有三个时滞的Lotka-Volterra互惠系统;获得了正平衡点和Hopf分支存在的条件等;并对所获得的结果进行了数值模拟．     

The role of Hopf bifurcation dynamics in sensory processes

We emphasize here the role of the Hopf bifurcation in detection of stimuli in sensory processes—we discuss in particular chemosensors. It is shown that the essential nonlinearities inherent in the signal transduction mechanism can take advantage of the noise from the environment the system is subject to, to display a highly amplified response to stimuli in a frequency-selective manner. It is shown that in the absence of any externally applied stimulus, the feedback mechanisms playing a regulatory role in the transduction mechanism can give rise, in the presence of noise, to peaks in the spectral power density, suggesting enhanced spontaneous activity in sensory cells. The power law in this spectrum is determined.     

Global Hopf bifurcation analysis of an susceptible-infective-removed epidemic model incorporating media coverage with time delay

An susceptible-infective-removed epidemic model incorporating media coverage with time delay is proposed. The stability of the disease-free equilibrium and endemic equilibrium is studied. And then, the conditions which guarantee the existence of local Hopf bifurcation are given. Furthermore, we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of delay. The obtained results show that the time delay in media coverage can not affect the stability of the disease-free equilibrium when the basic reproduction number is less than unity. However, the time delay affects the stability of the endemic equilibrium and produces limit cycle oscillations while the basic reproduction number is greater than unity. Finally, some examples for numerical simulations are included to support the theoretical prediction.     

Stability and Hopf bifurcation for a prey-predator model with prey-stage structure and diffusion

In this paper, we first propose a prey-predator model with prey-stage structure and diffusion. Then we discuss the following three problems: (1) stability of non-negative constant steady states for the reduced ODE system and the corresponding reaction diffusion system with homogeneous Neumann boundary conditions; (2) Hopf bifurcation for the ODE system; (3) Hopf bifurcation created by diffusion.     

Quantitative analysis of the Hopf bifurcation in the Goodwin n-dimensional metabolic control system

We study, from a quantitative point of view, the Hopf bifurcation in an ODE model of feedback control type introduced by Goodwin (1963) to describe the dynamics of end-product inhibition of gene activity. We formally prove that the exchange of linear stability of the positive equilibrium in the n-dimensional Goodwin system with equal reaction constants coexists with a Hopf bifurcation of nontrivial periodic solutions emanating from this equilibrium, without any further restriction on the dimension n 3 or on the Hill coefficient . The direction of the bifurcation, and the stability and the period of the bifurcating orbits are estimated by means of the algorithm proposed by Hassard et al. (1981).Supported by MURST 40/60%     

一具有非线性发生率传染病模型的稳定性和霍普夫分支

在这篇文章中,我们研究了一具有非线性发生率的传染病模型．该模型经历了鞍结点分支和霍普夫分支．我们对模型的霍普夫分支进行了详细的分析,得知该霍普夫分支是超临界的．此外,我们给出了支持理论分析的数值模拟．     

一类具有时滞的传染病模型的稳定性分析

研究了一类具有时滞的传染病生物模型．首先研究了该模型的线性稳定性,并给出了一列Hopf分支值,然后利用中心流形定理和正规型方法,给出了确定分支周期解的分支方向与稳定性的计算公式．     

一类时滞神经网络模型的稳定性与全局Hopf分支

研究了一类由两个神经元构成的时滞神经网络模型的稳定性和局部Hopf分支,并结合一般泛函微分方程的全局Hopf分支定理,利用度理论研究了全局Hopf分支的存在性．     

Algae-herbivore interactions with Allee effect and chemical defense

Macroalgae exhibit a variety of characteristics that provide a degree of protection from herbivores. One characteristic is the production of chemicals that are toxic to herbivores. The toxic effect of macroalgae on herbivorous reef fish is studied by means of a spatiotemporal model of population dynamics with a nonmonotonic toxin-determined functional response of herbivores. It is assumed that the growth rate of macroalgae is mediated by Allee effect. We see that under certain conditions the system is uniformly persistent. Conditions for local stability of the system is obtained with weak and strong Allee effects. We observe that in presence of Allee effect on macroalgae, the system exhibits complex dynamics including Hopf bifurcation and saddle-node bifurcation. The obtained results show that the spatiotemporal system does not exhibit diffusion-driven instability. Computer simulations have been carried out to illustrate different analytical results.     

延迟遗传调控网络稳定性及分支

本文主要研究了延迟遗传调控网络的局部稳定性和该网络的Hopf分支存在条件．延迟遗传调控网络是无穷维系统,此类系统在平衡点线性化后的特征方程为超越方程。通过对此超越方程进行研究,得到了系统系数不同时的系统稳定的条件及相关结论,又进一步说明了此系统的Hopf分支存在条件．最后,举一个例子进行了数值仿真验证了所得到的结论．     

时滞BAM神经网络模型的稳定性分析

本文研究了一个含有n+1个神经元的多时滞BAM型神经网络模型.利用儒歇定理及其推论分析了该动力特征方程根的分布情况,进而得到动力系统稳定和Hopf分支存在的条件,画出了分支图.研究了该动力系统的Pitchfork分支,得到了Pitchfork分支曲线,讨论了在分支曲线的不同区域里平衡点的稳定性.     

An SIS epidemic model with variable population size and a delay

The SIS epidemiological model has births, natural deaths, disease-related deaths and a delay corresponding to the infectious period. The thresholds for persistence, equilibria and stability are determined. The persistence of the disease combined with the disease-related deaths can cause the population size to decrease to zero, to remain finite, or to grow exponentially with a smaller growth rate constant. For some parameter values, the endemic infective-fraction equilibrium is asymptotically stable, but for other parameter values, it is unstable and a surrounding periodic solution appears by Hopf bifurcation.Research Supported in part by NSERC grant A-8965 and the University of Victoria Committee on Faculty Research & Travel     

Dynamics and bifurcation of a model for hormonal control of the menstrual cycle with inhibin delay

A system of 13 ordinary differential equations with 42 parameters is presented to model hormonal regulation of the menstrual cycle. For an excellent fit to clinical data, the model requires a 36 h time delay for the effect of inhibin on the synthesis of follicle stimulating hormone. Biological and mathematical reasons for this delay are discussed. Bifurcations with respect to changes in three important parameters are examined. One parameter represents the level of estradiol adequate for significant synthesis of luteinizing hormone. Bifurcation diagrams with respect to this parameter reveal an interval of parameter values for which a unique stable periodic solution exists and this solution represents a menstrual cycle during which ovulation occurs. The second parameter measures mass transfer between the first two stages of ovarian development and is indicative of healthy follicular growth. The third parameter is the time delay. Changes in the second parameter and the time delay affect the size of the uniqueness interval defined with respect to the first parameter. Saddle-node, transcritical and degenerate Hopf bifurcations are studied.     

Complex dynamics in biological systems arising from multiple limit cycle bifurcation

In this paper, we study complex dynamical behaviour in biological systems due to multiple limit cycles bifurcation. We use simple epidemic and predator–prey models to show exact routes to new types of bistability, that is, bistability between equilibrium and periodic oscillation, and bistability between two oscillations, which may more realistically describe the real situations. Bifurcation theory and normal form theory are applied to investigate the multiple limit cycles bifurcating from Hopf critical point.     

一个具有时滞的媒介传播流行病模型中的Hopt分支（英文）

文章研究的是一个具有时滞的媒介传播流行病模型.假定长期的发病率是双线性大规模行动的方式,确定了疾病是否流行的阈值R_0.当R_0≤1时,得到无病平衡点是全局稳定的,即疾病消失;当R_0〉1时,得到地方病平衡点.在具有时滞的微分模型中,时滞与载体转变成传染源的孵化期有关。我们研究了时滞对平衡点稳定性的影响,研究表明,在从寄生源到载体的传播过程中,时滞可以破坏动力系统并且得到了Hopt分支的周期解.