具有时空时滞的非局部扩散SIR模型的行波解

针对种群中的染病个体在疾病潜伏期内具有自由移动和传染疾病的现象, 研究了一个具有时空时滞的非局部扩散SIR模型的行波解问题.利用基本再生数和最小波速判定行波解是否存在.首先, 通过在有界区域上构造一个初始函数的不变锥, 利用Schauder不动点定理证明在该锥上存在不动点, 然后通过取极限的方法得到行波解的存在性.其次, 利用双边Laplace(拉普拉斯)变换法证明了行波解的不存在性.由于行波解的最小传播速度对控制疾病传播具有重要的指导意义, 最后讨论了非局部扩散、时滞等因素对最小波速的影响.     

Traveling waves in an SEIR epidemic model with a general nonlinear incidence rate

ABSTRACT

We study the traveling waves of reaction-diffusion equations for a diffusive SEIR model with a general nonlinear incidence. The existence of traveling waves is determined by the basic reproduction number of the corresponding ordinary differential equations and the minimal wave speed. Its proof is showed by introducing an auxiliary system, applying Schauder fixed point theorem and then a limiting argument. The non-existence proof is obtained by two-sided Laplace transform when the speed is less than the critical velocity. Finally, we present some examples to support our theoretical results.     

一类带治疗项的非局部扩散SIR传染病模型的行波解

该文主要考虑一类非局部扩散传染病模型的行波解的存在性与不存在性.首先,利用Schauder不动点定理和取极限的方法,得到了行波解的存在性.其次,利用双边拉普拉斯变换和Fubini定理,证明了行波解的不存在性.上述结果表明,最小波速是预测疾病是否传播且以多大速度传播的重要阈值.     

Traveling waves for nonlinear cellular neural networks with distributed delays

In this paper, we will establish the existence and nonexistence of traveling waves for nonlinear cellular neural networks with finite or infinite distributed delays. The dynamics of each given cell depends on itself and its nearest m left or l right neighborhood cells where delays exist in self-feedback and left or right neighborhood interactions. Our approach is to use Schauder?s fixed point theorem coupled with upper and lower solutions of the integral equation in a suitable Banach space. Further, we obtain the exponential asymptotic behavior in the negative infinity and the existence of traveling waves for the minimal wave speed by the limiting argument. Our results improve and cover some previous works.     

度量空间和锥度量空间中的若干不动点定理

给出了度量空间和锥度量空间中的若干不动点定理.利用这些不动点定理,统一并推广了度量空间和锥度量空间中的若干经典的不动点定理.     

时滞格微分方程组的行波解

本文利用Schauder不动点定理的方法和上、下解技巧,研究了时滞格微分方程组的行波解,给出了当系统的非线性项满足“拟单调条件”和“指数拟单调条件”时行波解的存在性.     

A fixed point theorem for the infinite-dimensional simplex

We define the infinite-dimensional simplex to be the closure of the convex hull of the standard basis vectors in R^∞, and prove that this space has the fixed point property: any continuous function from the space into itself has a fixed point. Our proof is constructive, in the sense that it can be used to find an approximate fixed point; the proof relies on elementary analysis and Sperner's lemma. The fixed point theorem is shown to imply Schauder's fixed point theorem on infinite-dimensional compact convex subsets of normed spaces.     

Traveling waves for delayed non-local diffusion equations with crossing-monostability

This paper is concerned with the traveling waves for a class of delayed non-local diffusion equations with crossing-monostability. Based on constructing two associated auxiliary delayed non-local diffusion equations with quasi-monotonicity and a profile set in a suitable Banach space using the traveling wave fronts of the auxiliary equations, the existence of traveling waves is proved by Schauder’s fixed point theorem. The result implies that the traveling waves of the delayed non-local diffusion equations with crossing-monostability are persistent for all values of the delay τ?0.     

Travelling wave solutions for a nonlocal dispersal HIV infection dynamical model

This paper is devoted to developing a nonlocal dispersal HIV infection dynamical model. The existence of travelling wave solutions is investigated by employing Schauder's fixed point theorem. That is, we study the existence of travelling wave solutions for $R_0>1$ and each wave speed $c>c^{?}$. In addition, the boundary asymptotic behaviour of travelling wave solutions at +∞ is obtained by constructing suitable Lyapunov functions and employing Lebesgue dominated convergence theorem. By employing a limiting argument, we investigate the existence of travelling wave solutions for $R_0>1$ and $c=c^{?}$. The main difficulties are that the semiflow generated by the model does not have the order-preserving property and the solutions lack of regularity.     

Propagation dynamics of a three-species nonlocal competitive–cooperative system

This paper is concerned with the existence, monotonicity, asymptotic behavior and uniqueness of traveling wave solutions for a three-species competitive–cooperative system with nonlocal dispersal and bistable dynamics. By considering a related truncated problem, we first establish the existence and strict monotonicity of traveling waves by means of a limiting argument and a comparative lemma. Then the asymptotic behavior of traveling waves is investigated by using Ikehara’s lemma and bilateral Laplace transform. Finally, we obtain the uniqueness of wave speed and traveling wave by sliding method.     

Traveling waves for a reaction–diffusion–advection predator–prey model

In this paper we study a reaction–diffusion–advection predator–prey model in a river. The existence of predator-invasion traveling wave solutions and prey-spread traveling wave solutions in the upstream and downstream directions is established and the corresponding minimal wave speeds are obtained. While some crucial improvements in theoretical methods have been established, the proofs of the existence and nonexistence of such traveling waves are based on Schauder’s fixed-point theorem, LaSalle’s invariance principle and Laplace transform. Based on theoretical results, we investigate the effect of the hydrological and biological factors on minimal wave speeds and hence on the spread of the prey and the invasion of the predator in the river. The linear determinacy of the predator–prey Lotka–Volterra system is compared with nonlinear determinacy of the competitive Lotka–Volterra system to investigate the mechanics of linear and nonlinear determinacy.     

On Basic Semiconductor Equations with Heat Conduction

We prove the global existence of solution to basic semiconductor equations with heat conduction; If the domain is narrow in one direction, then the basic equations has a unique steady-state which is locally asymptotically stable.     

TRAVELING WAVES FOR A NONLOCAL DISPERSAL EPIDEMIC MODEL

This paper is concerned with traveling wave solutions to a nonlocal dispersal epide- mic model. Combining the upper and lower solutions and monotone iteration method, we establish the existence of nondecreasing traveling wave fronts for the speed being larger than the critical one. Furthermore, by the approximation method, the existence of traveling wave fronts for the critical speed is established as well. Finally, we discuss the nonexistence of traveling wave fronts for the speed being smaller than critical one by Laplace transform.     

Existence and uniqueness of traveling waves for non-monotone integral equations with application

This paper is concerned with the traveling waves in a class of non-monotone integral equations. First we establish the existence of traveling waves. The approach is based on the construction of two associated auxiliary monotone integral equations and a profile set in a suitable Banach space. Then we show that the traveling waves are unique up to translations under some reasonable assumptions. The exact asymptotic behavior of the profiles as ξ→−∞ and the existence of minimal wave speed are also obtained. Finally, we apply our results to an epidemic model with non-monotone “force of infection”.     

Positive Solutions for Second-Order m-Point Boundary Value Problems on Time Scales

Let T be a time scale such that 0, T ∈ T. By means of the Schauder fixed point theorem and analysis method, we establish some existence criteria for positive solutions of the m-point boundary value problem on time scales where α ∈ Ctd（（O,T,[0,∞））,f∈ Ckd（[0,∞）×[0,∞））,β,γ ∈[0,∞）,ξi ∈（0,p（T）.b,ai∈ （0,∞） （for i = 1,..., m - 2） are given constants satisfying some suitable hypotheses. We show that this problem has at least one positive solution for sufficiently small b 〉 0 and no solution for sufficiently large b. Our results are new even for the corresponding differential equation （T = R） and difference equation （T = Z）.     

非线性Volterra-Stieltjes积分方程的解

利用Schauder不动点定理和饱和解的理论,研究下列非线性Volterrastieltjes积分方程x(t)=h(t) ∫t0u(t,s,x(s))dsg(t,s).在适当的条件下,证明了上述方程在[0, ∞)上有连续解.