一个具有非局部效应的非线性周期反应扩散方程的渐近形态

研究一个具有非线性-非局部反应的周期反应扩散系统.利用周期半流的渐近理论来讨论渐近波速c~*和周期行波解的存在性,证明参数c~*也是周期行波解的最小波速,并清晰描述解传播的阈值性质.最后给出渐近波速和最小波速c~*的估计.     

带分布时滞的时间周期S-Ⅰ型反应扩散传染病模型的传播性质

利用渐近传播速度理论研究了一类带分布时滞的时间周期S-Ⅰ型反应扩散传染病模型具有紧支集初值的解的演化性质,由此可以解释新发传染病的地理传播现象.首先,对于疾病已入侵区域,利用一致持久性思想结合比较技巧分三步验证了模型系统的一致持久性,在此过程中,通过构造截断区间初边值问题解决了模型系数的周期性和时滞共同导致的关键困难.其次,通过构造单调方程并利用单调系统的传播速度理论和比较原理分析了宿主种群在疾病未入侵区域的演化性质.     

一类时间周期的时滞竞争系统行波解的存在性

该文研究了一类时间周期的时滞Lotka-Volterra竞争系统的行波解.首先,通过构造适当的上、下解,结合单调迭代的方法证明了当cc~*时,存在连接两个半正周期平衡点的行波解,并且利用比较原理得到了周期行波解关于z的单调性.其次,通过单调性证明了行波解在正、负无穷远处的渐近行为.最后,证明了当c=c~*时周期行波解的存在性.     

一类K-单调系统产生的K-单调算子

本文研究了K-单调系统的解的渐近性质.应用K-单调算子的性质,得到了保证K-单调系统的正周期解的存在性、唯一性、全局渐近稳定性的充分条件.     

格微分方程的行波解研究简介(英文)

本文介绍了格微分方程关于行波解及其相关渐近性态方面的最新研究进展,如行波解的存在性、唯一性和稳定性,格微分方程的传播失败现象、最小波速和传播速度.本文力求展示相关问题的研究背景和线索脉络,阐述相关结论和研究方法,以期待相关主题研究的进一步深入.     

带有外部输入项的时间周期SIR传染病模型的周期行波解北大核心CSCD

研究了一类带有外部输入项的时间周期SIR传染病模型周期行波解的存在性和不存在性.首先,通过构造辅助系统适当的上下解并定义闭凸锥,将周期行波解的存在性转化为定义在这个闭凸锥上的非单调算子的不动点问题,利用Schauder不动点定理建立辅助系统周期解的存在性,并利用Arzela-Ascoli定理证明了原模型周期行波解的存在性.其次,借助分析技术得到了周期行波解的不存在性.     

具有非局部效应的时滞SEIR系统的周期行波解

该文研究了一类具有非局部效应和非线性发生率的时滞SEIR系统的周期行波解.首先,定义基本再生数R₀并构造适当的上下解,将周期行波解的存在性转化为闭凸集上非单调算子的不动点问题,利用Schauder不动点定理结合极限理论建立该系统周期行波解的存在性.其次,利用反证法结合比较原理,建立当基本再生数R₀<1时该系统周期行波解的不存在性.     

Vakhnenko方程的光滑周期波的波长

主要研究Vakhnenko方程的光滑周期行波解的波长.通过变量变换,Vakhnenko方程可以转化为一个平面多项式微分系统.利用动力系统的临界周期分支方法研究这个多项式微分系统,其主要结果是给出了周期函数T(h)或波长函数λ(a)的单调性质.与Kd V方程比较,波长函数λ(a)单调递减到一个有限的数,而不是单调递增到无穷.结果表明,对于固定波速c,Vakhnenko方程不存在任意小或任意大波长的光滑周期行波解.     

一类K-单调系统产生的K-单调算子

本文研究了Ｋ－单调系统的解的渐近性质．应用Ｋ－单调算子的性质,得到了保证Ｋ－单调系统的正周期解的存在性、唯一性、全局渐近稳定性的充分条件．     

一类一阶和二阶微分方程的渐近概周期解

对于一阶微分系统u′+F(u)=h(t),其中F为R~n上的严格单调算子,本文给出了其渐近概周期解存在和唯一的一个充分条件和一个必要条件.特别,对于一阶微分系统u′+▽Φ(u)=h(t),其中▽Φ代表R~N上凸函数Φ的梯度,讨论了其渐近概周期解存在和唯一的充分必要条件,并且把一些结果推广到了一类二阶方程.     

Asymptotic pattern for a periodic lattice model with age structure and global interaction

In this paper, we derive a time-periodic lattice model for a single species in a patchy environment, which has age structure and an infinite number of patches connected locally by diffusion. By appealing to the theory of asymptotic speed of propagation and monotonic periodic semiflows for travelling waves, we establish the existence of periodic travelling wave and spreading speed of the model.     

Spatiotemporal propagation of a time-periodic reaction–diffusion SI epidemic model with treatment

This work is concerned with the spatiotemporal propagation phenomena for a time-periodic reaction-diffusion susceptible-infectious (SI) epidemic model with treatment in terms of the asymptotic speed of spread and periodic traveling waves. First, the asymptotic speed of spread $c^{\ast }$ is characterized and the spreading properties of the model are analyzed by combining the periodic principal eigenvalue problem, comparison method, and the uniform persistence idea for a dynamical system. Second, by constructing suitable super- and subsolutions for truncation problems corresponding to the traveling wave system, the existence of periodic traveling waves is established via the fixed point theorem twice. It turned out that the asymptotic speed of spread coincides with the minimum wave speed of periodic traveling waves. Finally, via numerical simulation, the effects of some important parameters (such as diffusion rate, treatment rate, etc.) on the spreading speed are discussed, and the asymptotic properties of the periodic traveling waves are explored.     

Propagation dynamics of a nonlocal periodic organism model with non-monotone birth rates

This work is concerned with a nonlocal reaction–diffusion system modeling the propagation dynamics of organisms owning mobile and stationary states in periodic environments. We establish the existence of the asymptotic speed of spreading for the model system with monotone birth function via asymptotic propagation theory of monotone semiflow, and then discuss the case for non-monotone birth function by using the squeezing technique. In terms of the truncated problem on a finite interval, we apply the method of super- and sub-solutions and the fixed point theorem combined with regularity estimation and limit arguments to obtain the existence of time periodic traveling waves for the model system without quasi-monotonicity. The non-existence proof is to use the results of the spreading speed. Finally, as an application, we study the spatial dynamics of the model with the birth rate function of Ricker type and numerically demonstrate analytic results.     

Propagation dynamics for a spatially periodic integrodifference competition model

In this paper, we study the propagation dynamics for a class of integrodifference competition models in a periodic habitat. An interesting feature of such a system is that multiple spreading speeds can be observed, which biologically means different species may have different spreading speeds. We show that the model system admits a single spreading speed, and it coincides with the minimal wave speed of the spatially periodic traveling waves. A set of sufficient conditions for linear determinacy of the spreading speed is also given.     

Spreading speeds and traveling waves for a time periodic DS-I-A epidemic model

This paper is devoted to studying the speed of asymptotic spreading and minimal wave speed of traveling wave solutions for a time periodic and diffusive DS-I-A epidemic model, which describes the propagation threshold of disease spreading. The main feature of this model is the possible deficiency of the classical comparison principle such that many known results do not directly work. The speed of asymptotic spreading is estimated by constructing auxiliary equations and applying the classical theory of asymptotic spreading for Fisher type equation. The minimal wave speed is established by proving the existence and nonexistence of the nonconstant traveling wave solutions. Moreover, some numerical examples are presented to model the propagation dynamics of this system.     

Traveling waves for a periodic Lotka–Volterra predator–prey system

This paper is concerned with the time periodic traveling wave solutions for a periodic Lotka–Volterra predator–prey system, which formulates that both species synchronously invade a new habitat. We first establish the existence of periodic traveling wave solutions by combining the upper and lower solutions with contracting mapping principle and Schauder’s fixed point theorem. The asymptotic behavior of nontrivial solution is given precisely by the stability of the corresponding kinetic system that has been widely investigated. Then, the nonexistence of periodic traveling wave solutions is confirmed by applying the theory of asymptotic spreading. We show the conclusion for all positive wave speed and obtain the minimal wave speed.     

Propagation dynamics of a time periodic diffusion equation with degenerate nonlinearity

This paper is on study of traveling wave solutions and asymptotic spreading of a class of time periodic diffusion equations with degenerate nonlinearity. The asymptotic behavior of traveling wave solutions is investigated by using auxiliary equations and a limit process. In addition, the monotonicity and uniqueness, up to translation, of traveling wave solution with critical speed are determined by sliding method. Finally, combining super and sub-solutions and the stability of steady states, some sufficient conditions on asymptotic spreading are given, which indicates that the success or failure of asymptotic spreading are dependent on the degeneracy of nonlinearity as well as the size of compact support of initial value.     

Spreading speed and traveling waves for a multi-type SIS epidemic model

The theory of asymptotic speeds of spread and monotone traveling waves for monotone semiflows is applied to a multi-type SIS epidemic model to obtain the spreading speed c^∗, and the nonexistence of traveling waves with wave speed c<c^∗. Then the method of upper and lower solutions is used to establish the existence of monotone traveling waves connecting the disease-free and endemic equilibria for c?c^∗. This shows that the spreading speed coincides with the minimum wave speed for monotone traveling waves. We also give an affirmative answer to an open problem presented by Rass and Radcliffe [L. Rass, J. Radcliffe, Spatial Deterministic Epidemics, Math. Surveys Monogr. 102, Amer. Math. Soc., Providence, RI, 2003].     

Travelling waves in lattice dynamical systems

For a class of one‐dimensional lattice dynamical systems we prove the existence of periodic travelling waves with prescribed speed and arbitrary period. Then we study asymptotic behaviour of such waves for big values of period and show that they converge, in an appropriate topology, to a solitary travelling wave. Copyright © 2000 John Wiley & Sons. Ltd.     

Traveling Waves of a Reaction-Diffusion SIRQ Epidemic Model with Relapse

This paper is concerned with the traveling waves of a reaction-diffusion SIRQ epidemic model with relapse. We find that the existence and nonexistence of traveling waves are determined by the basic reproduction number of the system and the minimal wave speed. This threshold dynamics is proved by Schauder''s fixed-point theorem combining Lyapunov functional with the theory of asymptotic spreading. Moreover, the numerical simulations are provided to illustrate our analytical results and the effect of the relapse is also discussed.