带有相依重尾冲击的Poisson噪音过程尾的一致渐近性质

本文考虑了带有某种相依重尾冲击的Poisson噪音过程尾的一致渐近性质.当冲击是二元上尾渐近独立的非负随机变量具有长尾和控制变化尾分布且噪音函数具有正的上下界时,得到了过程尾概率的一致渐近公式.进而,当冲击具有连续的一致变化尾分布时,去除了噪音函数具有正的下界的限制.对于噪音函数不一定具有正的上界的情形,当冲击具有两两负象限相依结构时,也得到了一致渐近性结果.     

一类时变非线性系统的一致有界性的注记

主要研究带干扰的广义齐次系统的一致有界和一致最终有界性 ,证明了当干扰项满足一致有界及Lp 可积时系统的一致有界性及一致最终有界性 ,本质推广了最近相关文献中的有关系统一致有界性的结果 .     

马尔科夫过程的强遍历性和一致衰减性

本文证明了黎曼流形上的非爆炸正Harris常返扩散过程的强遍历性等价于某(任)一紧集击中时期望的一致有界性;而马尔科夫过程一致衰减当且仅当爆炸时的期望一致有界.     

具常数输入及饱和发生率的脉冲接种SIQRS传染病模型

研究了具有常数输入及饱和发生率的脉冲接种SIQRS传染病模型,得到了疾病消除与否的阈值R_0=1.证明了当R_01时,系统存在全局渐近稳定的无病周期解;当R_01时,系统一致持久.     

具有一般复发现象的疾病模型的全局稳定性

本文研究了具有一般复发现象和非线性发生率的疾病模型的动力学性质,其中模型是具有无穷分布时滞的微积分方程.该模型描述了包含疱疹等传染病的—般复发现象.利用一致持久性理论和李雅普诺夫函数,我们证明了基本再生数R_0决定的系统的全局动力学性质:当R_0≤1时,疾病灭绝;当R_01时,疾病持久生存,并且正平衡点是全局吸引的.     

基于随机与异质网络共存的SEIRS传染病模型

讨论了随机与异质网络共存的SEIRS传染病模型,通过正平衡点的存在性给出基本再生数R_0=((1-η)Aλ+ηβ)/μ.结果表明,当R_01时,无病平衡点(1,0,0,0)局部稳定;当R_01时,无病平衡点(1,0,0,0)不稳定,此时系统存在唯一的地方病平衡点,并且一致持续存在.最后通过数值仿真,验证了理论结果的正确性.     

有向网络下多个体系统的整体行为

研究非时变有向通讯网络背景下一阶线性多个体动力学系统的整体行为.根据通讯网络的结构,系统可以区分为独立基本子系统和非独立基本子系统.当系统的控制规则为一类平凡的线性类型时,系统的独立基本子系统将趋于自身的一致状态,也即子系统中的每个个体趋于子系统的带权中心.独立基本子系统带权中心由子系统的系数矩阵的零特征根归一化左特征向量确定.非独立子系统中个体将趋于独立基本子系统带权中心的凸集内.当且仅当系统的独立基本子系统唯一时,系统实现一致性行为.     

具有时滞和可变营养消耗率的比率型Chemostat模型稳定性分析

考虑了一类具有时滞和可变营养消耗率、增长函数为比率确定型的微生物连续培养模型.首先,详细地讨论了解的存在性、有界性、平衡点的局部渐近稳定性以及Hopf分支.其次,利用Lyapunov-LaSalle不变性原理证明了边界平衡点的全局渐近性.最后,利用时滞微分系统解的极限集的一些性质,证明了当正平衡点存在时,对任意时滞系统是一致持久的.     

具有一般接触率和治疗的SIS传染病模型的后向分支

建立了具有一般传染率函数和治疗的SIS模型并分析了其动力学性态.通过分析得到,当基本再生数小于1时,系统存在无病平衡点,并且无病平衡点是局部渐近稳定的,当染病者数量较少,发现系统在基本再生数大于1时,系统存在惟一的正平衡点且是局部渐近稳定的;当染病者数量超过医院的最大承受能力时,当基本再生数小于1时,系统可能存在两个正平衡点或无正平衡点.当存在两个正平衡点时,其中染病者数量较小的是鞍点,染病者数量较大的为结点或焦点,且是局部渐近稳定的.当治疗能力较弱时,模型会出现后向分支.     

仅有y-分量耦合的非恒同Lorenz系统的渐近同步

本文考虑外部耦合格式为n×n阶实对称不可约,行和为零且对角线以外的元素非正的矩阵,内部耦合格式为仅有y-分量参与耦合的非恒同Lorenz格点系统的渐近同步.在系统解一致有界耗散的基础上采用常数变易法证明了当耦合强度足够大时仅有y-分量参与耦合的非恒同Lorenz格点系统的解出现渐近同步,即系统解的任意两个对应分量的差在时间趋向于无穷时是一个小的有界量.     

Global dynamics of two phytoplankton-zooplankton models with toxic substances effect

In this paper, we investigate phytoplankton-zooplankton models with toxic substances effect and two different kinds of predator functional responses. For Holling type II predator functional response, it is shown that the local stability of the positive equilibrium implies global stability if there exists a unique positive equilibrium. When there exist multiple positive equilibria, the local stability of the positive equilibrium with small phytoplankton population density implies that the model occurs bistable phenomenon. These results also hold for Holling type III predator functional response under certain conditions.     

Bogdanov–Takens bifurcation in a predator–prey model

In this paper, we investigate a class of predator–prey model with age structure and discuss whether the model can undergo Bogdanov–Takens bifurcation. The analysis is based on the normal form theory and the center manifold theory for semilinear equations with non-dense domain combined with integrated semigroup theory. Qualitative analysis indicates that there exist some parameter values such that this predator–prey model has an unique positive equilibrium which is Bogdanov–Takens singularity. Moreover, it is shown that under suitable small perturbation, the system undergoes the Bogdanov–Takens bifurcation in a small neighborhood of this positive equilibrium.     

一类具有Watt型功能性反应的捕食系统的极限环与稳定性

研究一类具有Watt型功能性反应的捕食模型.讨论了该系统正平衡点的存在性以及非负平衡点的性态,应用Poincare-Bendixson定理和张芷芬定理,证明了极限环的存在性和唯一性,并采用构造Dulac函数的方法,获得了正平衡点全局渐近稳定性的一个充分条件.     

Stability and Hopf bifurcation in a ratio-dependent predator–prey system with stage structure

A ratio-dependent predator–prey model with stage structure for the predator and time delay due to the gestation of the predator is investigated. By analyzing the characteristic equations, the local stability of a positive equilibrium and a boundary equilibrium is discussed, respectively. Further, it is proved that the system undergoes a Hopf bifurcation at the positive equilibrium when τ = τ₀. By using an iteration technique, sufficient conditions are derived for the global attractivity of the positive equilibrium. By comparison arguments, sufficient conditions are obtained for the global stability of the boundary equilibrium. Numerical simulations are carried out to illustrate the main results.     

Global Analysis of Predator–Prey System with Hawk and Dove Tactics

Functional response of the Holling type II is incorporated into a predator–prey model with predators using hawk‐dove tactics to consider combination effects of nonlinear functional response and individual tactics. By mathematical analysis, it is shown that the model undergoes a sequence of bifurcations including saddle‐node bifurcation, supercritical Hopf bifurcation and homoclinic bifurcation. New phenomena are found that include the bistable coexistence of prey and predators in the form of a stable limit cycle and a stable positive equilibrium, the bistable coexistence of prey and predators in a large stable limit cycle that encloses three positive equilibria and a stable positive equilibrium within the cycle, and the bistable coexistence of two stable limit cycles.     

Stability of equilibrium points of demand-inventory model with stock-dependent demand

A model of demand and inventory of a product in one echelon of supply chain is considered. The model is formulated as a system of difference equations, in which every equilibrium point is nonhyperbolic. A positive invariant set of the system is constructed. An analysis of properties of equilibrium points of the system is based on the Lyapunov method or reducing it to the family of systems of difference equations with hyperbolic equilibrium points.     

Cross-diffusion induced Turing instability for a three species food chain model

In this paper, we study a strongly coupled reaction–diffusion system describing three interacting species in a food chain model, where the third species preys on the second one and simultaneously the second species preys on the first one. We first show that the unique positive equilibrium solution is globally asymptotically stable for the corresponding ODE system. The positive equilibrium solution remains linearly stable for the reaction–diffusion system without cross-diffusion, hence it does not belong to the classical Turing instability scheme. We further proved that the positive equilibrium solution is globally asymptotically stable for the reaction–diffusion system without cross-diffusion by constructing a Lyapunov function. But it becomes linearly unstable only when cross-diffusion also plays a role in the reaction–diffusion system, hence the instability is driven solely from the effect of cross-diffusion. Our results also exhibit some interesting combining effects of cross-diffusion, intra-species competitions and inter-species interactions.     

具有非线性传染率和时滞的随机SIQR计算机病毒模型的渐近行为

研究了一类具有非线性发生率和时滞的随机SIQR计算机病毒模型.首先证明了该系统具有唯一的全局正解,然后通过构造适当的Lyapunov函数并利用伊藤公式,分析了该模型的解在无病平衡点附近及地方病平衡点附近的渐近行为,最后通过数值模拟对随机系统解的渐近行为做了进一步的分析并给出了结论.     

Global asymptotic behaviour for a Nicholson model with patch structure and multiple delays

For a Nicholson’s blowflies model with patch structure and multiple discrete delays, we study some aspects of its global dynamics. Conditions for the absolute global asymptotic stability of both the trivial equilibrium and a positive equilibrium (when it exists) are given. The existence of positive heteroclinic solutions connecting the two equilibria is also addressed. We further consider a diffusive Nicholson-type model with patch structure, and establish a criterion for the existence of positive travelling wave solutions, for large wave speeds. Several applications illustrate the results, improving some criteria in the recent literature.     

一类比率型功能性反应捕食模型的稳定性分析

研究了一类具有比率型功能性反应的捕食模型,对模型进行了定性和稳定性分析,讨论了模型唯一正平衡点的存在条件,以及模型各个平衡点的性态.得到了各个平衡点全局渐近稳定的充分条件.通过绘制模型的相轨线,分析轨线的走向得到了原点全局渐近稳定的条件,并证明了模型不存在非平凡正周期解的条件,通过构造Lyapunov函数得到了模型的唯一正平衡点是全局渐近稳定的结论.