一类具扩散和时滞Belousov-Zhabotinskii系统的行波解

研究一般的带时滞的反应扩散方程组的行波解,这儿反应项具混拟单调性质,我们定义了相应的行波解的耦合上下解,以耦合上下解为初始迭代函数构造了耦合迭代序列,并且证明了在一定的单调性条件下该耦合序列收敛于行波解.以一个具体的带时滞的Belousov-Zhabotinskii模型为例,建立了有序的拟上解和拟下解并且得到行波解的存在性.     

具时滞的人口模型的行波解

研究具时滞的人口模型的行波解存在性问题,利用[5]中的方法,鹕行波解的存在性问题转化为寻找上下解的问题。     

一类具有时空时滞的单种群模型行波解的存在性

通过单调迭代和上下解技术,研究了一类具有时空时滞的单物种种群模型行波解的存在性,证明了当时滞充分小时,方程具有连接两个平衡点的波前解,并得到了一些新的结果.     

具有分布时滞的一类反应扩散方程的波前解的存在性（英文）

本文考虑了一类广义分布时滞下的反应扩散方程的行波解的存在性问题。运用几何奇异摄动理论和线性链方法,我们研究了反应扩散方程若在没有时滞情形下具有行波解,则只要平均时滞充分小,所给的广义时滞核下这个行波解可以保持存在.     

时滞Nicholson＇s Blowflies方程中行波解的Hopf分支

主要利用时滞微分方程中Hopf分支理论探讨时滞Nicholson＇s Blowflies方程中行波解随时滞量τ大小变化的分支行为.结果发现时滞量经过某一数值τ_0=1/（cω_0） arcsin-cω_0/p时,原系统会产生分支现象,最终导致形成周期性行波解.     

带时滞互惠模型的稳定性和行波解的存在性

本文建立了一类空间非局部带时滞影响的互惠生物种群系统模型.前部分利用线性化方法证明了该模型的简单动力学行为,即证明了零平衡点和两个边界平衡点都是不稳定的,唯一的正平衡点是稳定的,同时还用Redlinger上下解方法得出了该模型的初边值问题存在唯一的正则解;后部分则证明了该反应扩散系统连接零平衡点和正平衡点的行波解的存在性.     

一类SI传染病模型的平衡点稳定性及行波解的存在性（英文）

本文考虑了一类SI传染病模型,并引入了扩散和时滞的影响,得到一类捕食型的反应扩散模型.运用线性化方法得到了该系统平衡点的稳定性,由此指出了控制传染病传播的有效措施.然后运用上下解单调迭代的方法证明了行波解的存在性.     

一类含时滞的Logistic反应扩散方程的波前解

研究一类含时滞的Logistic滞反应扩散方程的波前解．通过构造合适的上下解,证明了当时滞充分小时,方程存在波前解．用线性化方法,给出了存在波前解的时滞τ取值范围的一个估计．     

一类S-I传染病模型行波解的存在性

本文研究了一类具有扩散且是非线性传染率的SI传染病模型,分析了模型的行波解的存在性条件,给出了最小波速与产生单调和振荡行波解的条件,并且进行了计算机仿真．     

混合拟单调系统行波解的存在性

运用单调迭代方法,证明了混合拟单调系统的行波解的存在性．当反应扩散系。统的反应函数是混合拟单调函数时,如果选取一对合适的耦合上下解作为迭代初值,则迭代序列将收敛到一对拟解．而且在这对拟解之间存在系统的行波解．     

Travelling waves of a delayed SIR epidemic model with nonlinear incidence rate and spatial diffusion

This paper is concerned with the existence of travelling waves to a SIR epidemic model with nonlinear incidence rate, spatial diffusion and time delay. By analyzing the corresponding characteristic equations, the local stability of a disease-free steady state and an endemic steady state to this system under homogeneous Neumann boundary conditions is discussed. By using the cross iteration method and the Schauder's fixed point theorem, we reduce the existence of travelling waves to the existence of a pair of upper-lower solutions. By constructing a pair of upper-lower solutions, we derive the existence of a travelling wave connecting the disease-free steady state and the endemic steady state. Numerical simulations are carried out to illustrate the main results.     

Steady states and outbreaks of two-phase nonlinear age-structured model of population dynamics with discrete time delay

In this paper we study the nonlinear age-structured model of a polycyclic two-phase population dynamics including delayed effect of population density growth on the mortality. Both phases are modelled as a system of initial boundary values problem for semi-linear transport equation with delay and initial problem for nonlinear delay ODE. The obtained system is studied both theoretically and numerically. Three different regimes of population dynamics for asymptotically stable states of autonomous systems are obtained in numerical experiments for the different initial values of population density. The quasi-periodical travelling wave solutions are studied numerically for the autonomous system with the different values of time delays and for the system with oscillating death rate and birth modulus. In both cases it is observed three types of travelling wave solutions: harmonic oscillations, pulse sequence and single pulse.     

Pursuit-evasion wave trains in prey-predator systems with diffusionally coupled delays

The Lotka-Volterra system of prey-predator equations is considered with a special type of continuous time delay. In the case of equal diffusion coefficients Hopf’s bifurcation technique is used to show the existence of travelling wave train solutions for the prey-predator system.     

Monotone travelling fronts in an age-structured reaction-diffusion model of a single species

 We consider a partially coupled diffusive population model in which the state variables represent the densities of the immature and mature population of a single species. The equation for the mature population can be considered on its own, and is a delay differential equation with a delay-dependent coefficient. For the case when the immatures are immobile, we prove that travelling wavefront solutions exist connecting the zero solution of the equation for the matures with the delay-dependent positive equilibrium state. As a perturbation of this case we then consider the case of low immature diffusivity showing that the travelling front solutions continue to persist. Our findings are contrasted with recent studies of the delayed Fisher equation. Travelling fronts of the latter are known to lose monotonicity for sufficiently large delays. In contrast, travelling fronts of our equation appear to remain monotone for all values of the delay. Received: 1 November 2001 / Revised version: 10 May 2002 / Published online: 23 August 2002 Mathematics Subject Classification (2000): 35K57, 92D25 Key words or phrases: Age-structure – Time-delay – Travelling Fronts – Reaction-diffusion     

A Resonance Approach to Cochlear Mechanics

Background

How does the cochlea analyse sound into its component frequencies? In the 1850s Helmholtz thought it occurred by resonance, whereas a century later Békésy''s work indicated a travelling wave. The latter answer seemed to settle the question, but with the discovery in 1978 that the cochlea emits sound, the mechanics of the cochlea was back on the drawing board. Recent studies have raised questions about whether the travelling wave, as currently understood, is adequate to explain observations.

Approach

Applying basic resonance principles, this paper revisits the question. A graded bank of harmonic oscillators with cochlear-like frequencies and quality factors is simultaneously excited, and it is found that resonance gives rise to similar frequency responses, group delays, and travelling wave velocities as observed by experiment. The overall effect of the group delay gradient is to produce a decelerating wave of peak displacement moving from base to apex at characteristic travelling wave speeds. The extensive literature on chains of coupled oscillators is considered, and the occurrence of travelling waves, pseudowaves, phase plateaus, and forced resonance in such systems is noted.

Conclusion and significance

This alternative approach to cochlear mechanics shows that a travelling wave can simply arise as an apparently moving amplitude peak which passes along a bank of resonators without carrying energy. This highlights the possible role of the fast pressure wave and indicates how phase delays and group delays of a set of driven harmonic oscillators can generate an apparent travelling wave. It is possible to view the cochlea as a chain of globally forced coupled oscillators, and this model incorporates fundamental aspects of both the resonance and travelling wave theories.     

The logistic equation with a diffusionally coupled delay

The asymptotic behaviour of a logistic equation with diffusion on a bounded region and a diffusionally coupled delay is investigated. An equivelent parabolic system is derived for certain types of delays. Using a Layapunov functional, sufficient conditions for the global asymptotic stability of the constant steady state are obtained. When the global stability is lost, using Hopf's bifurcation theory, existence of travelling waves is shown for ring-like and periodic one dimensional habitats.     

Temporal parameters of male—female sound communication in Leptophyes punctatissima

ABSTRACT. Quantitative information is presented about acoustic communication between the sexes of the speckled bush cricket Leptophyes punctatissima Bosc (Orthoptera: Tettigoniidae). The male song elicits an acoustic response from a receptive female with a net delay (excluding the travelling time of the sound) of about 25 ms depending upon ambient temperature. Only when the female responses fall within the narrow temporal window between 20 and 50 ms after the onset of his song does the male perform phonotaxis. The precise timing of this duet is described, and its advantages and disadvantages are discussed.     

Exponential Versus Hyperbolic Discounting of Delayed Outcomes: Risk and Waiting Time

Frequently, animals must choose between more immediate, smallerrewards and more delayed, but larger rewards. For example, theyoften must decide between accepting a smaller prey item versuscontinuing to search for a larger one, or between entering aleaner patch versus travelling to a richer patch that is furtheraway. In both situations, choice of the more immediate, butsmaller reward may be interpreted as implying that the valueof the later reward is discounted; that is, the value of thelater reward decreases as the delay to its receipt increases.This decrease in value may occur because of the increased riskinvolved in waiting for rewards, or because of the decreasedrate of reward associated with increased waiting time. The presentresearch attempts to determine the form of the relation betweenvalue and delay, and examines implications of this relationfor mechanisms underlying risk-sensitive foraging. Two accounts of the relation between value and delay have beenproposed to describe the decrease in value resulting from increasesin delay: an exponential model and a hyperbolic model. Our researchdemonstrates that, of the two, a hyperbola-like discountingmodel consistently explains more of the variance in temporaldiscounting data at the group level and, importantly, at theindividual level as well. We show mathematically that the hyperbolicmodel shares fundamental features with models of prey and patchchoice. In addition, the present review highlights the implicationsof a psychological perspective for the behavioral biology ofrisksensitive foraging, as well as the implications of an ecologicalperspective for the behavioral psychology of risk-sensitivechoice and decision-making.     

A model of herd grazing as a travelling wave,chemotaxis and stability

Possible constitutive models are examined for the formation of a herd, under the assumption that a herd forms a travelling wave while grazing. Under quite general conditions, it is found that the only possibility for a travelling wave is a balance between food seeking and natural dispersion, such as in chemotaxis. The stability of the travelling wave previously conjectured, is shown both for one- and two-dimensional perturbations.