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.     

格上时滞Lotka-Volterra合作系统的波前解

研究了定义在格上并具有时滞的Lotka-Volterra合作系统的波前解.通过构造上下解得到了波前解的存在性,借助于比较原理和渐近传播理论得到了波前解的不存在性,进而在得到了波前解最小波速的充分条件.     

一类具有分布时滞的扩散种群模型行波解的存在性

研究了一类具有分布时滞的扩散种群模型行波解的存在性,证明了当平均时滞充分小时,方程具有连接两个平衡点的单调行波解.     

含有两个时滞项的退化时滞微分方程周期解

讨论了含有两个时滞项退化时滞微分方程的周期解的问题,特别的,给出了此类方程存在非常数周期解的充要条件,并对二维退化微分方程给出了非常数周期解存在性的代数判据,并在最后给出一个例子验证了判据的有效性.     

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.     

一类具非单调泛函响应的状态-依赖型时滞捕食者-食饵系统的周期性

考虑一类具非单调泛函响应的状态-依赖型时滞捕食者-食饵系统,利用重合度和延拓定理,得到了一系列易检验的类方程至少存在一个严格正周期解的充分条件．     

一个具有年龄结构的单种群离散反应扩散模型的波前解

利用上下解方法研究了一个具有年龄结构的单种群离散反应扩散模型波前解的存在性,并证明了存在具有临界波速的波前解.     

一类带时滞竞争模型的周期解

研究了来源于水生种群植化相克的模型,提出了带时滞的半线性抛物系统．用上下解方法讨论了抛物方程组周期解存在性的原理,利用特征函数构造所提出抛物系统的上解,给出了正周期解存在的充分条件．     

时滞逻辑斯谛方程的稳定性和振动性

本文研究时滞逻辑斯谛方程其中得到方程（E）关于正常数平衡点为渐近稳定和振动的条件,也得到方程（E）存在正周期解的条件,所得结果发展和推广了文献［1］,［2］的结果。     

非线性非自治中立型时滞微分方程的周期解

本文研究非线性非自治中立型时滞微分方程N（t）=r（t）N（t）[a（t）-N（t-1）-c（t）N（t-1）],（E）得到方程（E）存在正周期解的充分条件．所得结果解决了Y·Kuang和A·Feldstein所提出的一个公开问题．     

Neutral Genetic Patterns for Expanding Populations with Nonoverlapping Generations

We investigate the inside dynamics of solutions to integrodifference equations to understand the genetic consequences of a population with nonoverlapping generations undergoing range expansion. To obtain the inside dynamics, we decompose the solution into neutral genetic components. The inside dynamics are given by the spatiotemporal evolution of the neutral genetic components. We consider thin-tailed dispersal kernels and a variety of per capita growth rate functions to classify the traveling wave solutions as either pushed or pulled fronts. We find that pulled fronts are synonymous with the founder effect in population genetics. Adding overcompensation to the dynamics of these fronts has no impact on genetic diversity in the expanding population. However, growth functions with a strong Allee effect cause the traveling wave solution to be a pushed front preserving the genetic variation in the population. In this case, the contribution of each neutral fraction can be computed by a simple formula dependent on the initial distribution of the neutral fractions, the traveling wave solution, and the asymptotic spreading speed.     

Modeling Spatial Spread of Infectious Diseases with a Fixed Latent Period in a Spatially Continuous Domain

In this paper, with the assumptions that an infectious disease in a population has a fixed latent period and the latent individuals of the population may diffuse, we formulate an SIR model with a simple demographic structure for the population living in a spatially continuous environment. The model is given by a system of reaction-diffusion equations with a discrete delay accounting for the latency and a spatially non-local term caused by the mobility of the individuals during the latent period. We address the existence, uniqueness, and positivity of solution to the initial-value problem for this type of system. Moreover, we investigate the traveling wave fronts of the system and obtain a critical value c ^* which is a lower bound for the wave speed of the traveling wave fronts. Although we can not prove that this value is exactly the minimal wave speed, numeric simulations seem to suggest that it is. Furthermore, the simulations on the PDE model also suggest that the spread speed of the disease indeed coincides with c ^*. We also discuss how the model parameters affect c ^*.     

Dynamics of prostate cancer stem cells with diffusion and organism response

We develop a systems based model for prostate cancer, as a sub-system of the organism. We accomplish this in two stages. We first start with a general ODE that includes organism response terms. Then, to account for normally observed spatial diffusion of cell populations, the ODE is extended to a PDE that includes spatial terms. Numerical solutions of the full PDE are provided, and are indicative of traveling wave fronts. This motivates the use of a well known transformation to derive a canonically related (non-linear) system of ODEs for traveling wave solutions. For biological feasibility, we show that the non-negative cone for the traveling wave system is time invariant. We also prove that the traveling waves have a unique global attractor. Biologically, the global attractor would be the limit for the avascular tumor growth. We conclude with comments on clinical implications of the model.     

Sea-urchin feeding fronts

Sea-urchin feeding fronts are a striking example of spatial pattern formation in an ecological system. If it is assumed that urchins are asocial, and that they move randomly, then the formation of these dense fronts is an apparent paradox. The key lies in observations that urchins move further in areas where their algal food is less plentiful. This naturally leads to the accumulation of urchins in areas with abundant algae. If urchin movement is represented as a random walk, with a step size that depends on algal concentration, then their movement may be described by a Fokker–Planck diffusion equation. For certain combinations of algal growth and urchin grazing, traveling wave solutions are obtained. Two-dimensional simulations of urchin algal dynamics show that an initially uniformly distributed urchin population, grazing on an alga with a smoothly varying density, may form a propagating front separating two sharply delineated regions. On one side of the front algal density is uniformly low, and on the other side of the front algal density is uniformly high. Bounds on when stable fronts will form are obtained in terms of urchin density and grazing, and algal growth.     

The long-time behavior of the solution to a non-linear diffusion problem in population genetics and combustion

The Fisher (1937) or Kolmogoroff-Petrovsky-Piscounoff (1937) equation exemplifies wave-like phenomena occurring in population genetics and combustion. In an earlier paper, we proposed an extension of this equation and obtained closed form traveling wave, stationary, and “symmetric” solutions. Employing the transformation properties of the extended equation, two integral invariants for the problem are obtained and two Lyapunov functionals, which characterize the evolution of the profile to a uniformly propagating traveling wave, are constructed. A generalization of this modified Fisher equation is proposed and we obtain its integral invariants, traveling wave solutions and wave speeds, as well as the Lyapunov functionals which describe its asymptotic evolution.     

Traveling wave solutions of a nonlinear reaction–advection equation

 We establish the existence of traveling wave solutions for a nonlinear partial differential equation that models a logistically growing population whose movement is governed by an advective process. Conditions are presented for which traveling wave solutions exist and for which they are stable to small semi-finite domain perturbations. The wave is of mathematical interest because its behavior is determined by a singular differential equation and those with small speed of propagation steepen into a shock-like solutions. Finally, we indicate that the smoothing presence of diffusion allows wave persistence when an advective slow moving wave may collapse. Received: 24 November 1997 / Revised version: 13 July 1998     

Are buffers boring? Uniqueness and asymptotical stability of traveling wave fronts in the buffered bistable system

Traveling waves of calcium are widely observed under the condition that the free cytosolic calcium is buffered. Thus it is of physiological interest to determine how buffers affect the properties of calcium waves. Here we summarise and extend previous results on the existence, uniqueness and stability of traveling wave solutions of the buffered bistable equation, which is the simplest possible model of the upstroke of a calcium wave. Taken together, the results show that immobile buffers do not change the existence, uniqueness or stability of the traveling wave, while mobile buffers can eliminate a traveling wave. However, if a wave exists in the latter case, it remains unique and stable.