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．     

Spreading speeds and traveling waves in competitive recursion systems

This paper is concerned with the spreading speeds and traveling wave solutions of discrete time recursion systems, which describe the spatial propagation mode of two competitive invaders. We first establish the existence of traveling wave solutions when the wave speed is larger than a given threshold. Furthermore, we prove that the threshold is the spreading speed of one species while the spreading speed of the other species is distinctly slower compared to the case when the interspecific competition disappears. Our results also show that the interspecific competition does affect the spread of both species so that the eventual population densities at the coexistence domain are lower than the case when the competition vanishes.     

Spreading speed, traveling waves, and minimal domain size in impulsive reaction-diffusion models

How growth, mortality, and dispersal in a species affect the species' spread and persistence constitutes a central problem in spatial ecology. We propose impulsive reaction-diffusion equation models for species with distinct reproductive and dispersal stages. These models can describe a seasonal birth pulse plus nonlinear mortality and dispersal throughout the year. Alternatively, they can describe seasonal harvesting, plus nonlinear birth and mortality as well as dispersal throughout the year. The population dynamics in the seasonal pulse is described by a discrete map that gives the density of the population at the end of a pulse as a possibly nonmonotone function of the density of the population at the beginning of the pulse. The dynamics in the dispersal stage is governed by a nonlinear reaction-diffusion equation in a bounded or unbounded domain. We develop a spatially explicit theoretical framework that links species vital rates (mortality or fecundity) and dispersal characteristics with species' spreading speeds, traveling wave speeds, as well as minimal domain size for species persistence. We provide an explicit formula for the spreading speed in terms of model parameters, and show that the spreading speed can be characterized as the slowest speed of a class of traveling wave solutions. We also give an explicit formula for the minimal domain size using model parameters. Our results show how the diffusion coefficient, and the combination of discrete- and continuous-time growth and mortality determine the spread and persistence dynamics of the population in a wide variety of ecological scenarios. Numerical simulations are presented to demonstrate the theoretical results.     

Traveling wave solutions in a plant population model with a seed bank

We propose an integro-difference equation model to predict the spatial spread of a plant population with a seed bank. The formulation of the model consists of a nonmonotone convolution integral operator describing the recruitment and seed dispersal and a linear contraction operator addressing the effect of the seed bank. The recursion operator of the model is noncompact, which poses a challenge to establishing the existence of traveling wave solutions. We show that the model has a spreading speed, and prove that the spreading speed can be characterized as the slowest speed of a class of traveling wave solutions by using an asymptotic fixed point theorem. Our numerical simulations show that the seed bank has the stabilizing effect on the spatial patterns of traveling wave solutions.     

Traveling wave solutions of a reaction diffusion model for competing pioneer and climax species

Presented is a reaction-diffusion model for the interaction of pioneer and climax species. For certain parameters the system exhibits bistability and traveling wave solutions. Specifically, we show that when the climax species diffuses at a slow rate there are traveling wave solutions which correspond to extinction waves of either the pioneer or climax species. A leading order analysis is used in the one-dimensional spatial case to estimate the wave speed sign that determines which species becomes extinct. Results of these analyses are then compared to numerical simulations of wave front propagation for the model on one and two-dimensional spatial domains. A simple mechanism for harvesting is also introduced.     

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 ^*.     

Traveling waves in the discrete fast buffered bistable system

We study the existence and uniqueness of traveling wave solutions of the discrete buffered bistable equation. Buffered excitable systems are used to model, among other things, the propagation of waves of increased calcium concentration, and discrete models are often used to describe the propagation of such waves across multiple cells. We derive necessary conditions for the existence of waves, and, under some restrictive technical assumptions, we derive sufficient conditions. When the wave exists it is unique and stable.      

Multi-scale modeling of a wound-healing cell migration assay

A continuum model and a discrete model are developed to capture the population-scale and cell-scale behavior in a wound-healing cell migration assay created from a scrape wound in a confluent cell monolayer. During wound closure, the cell population forms a sustained traveling wave, with close contact between cells behind the wavefront. Cells exhibit contact inhibition of migration and contact-limited proliferation. The continuum model includes the two dominant mechanisms and characteristics of cell migration and proliferation, using a cell diffusivity function that decreases with cell density and a logistic proliferative growth term. The discrete model arises naturally from the continuum model. Individual cells are simulated as continuous-time random walkers with nearest-neighbor transitions, together with a birth/death process. The migration and proliferation parameters are determined by analysing individual mice 3T3 fibroblast cell trajectories obtained during the development of a confluent cell monolayer and in a wound healing assay. The population-scale model successfully predicts the shape and speed of the traveling wave, while the discrete model is also successful in capturing the contact inhibition of migration effects.     

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     

A Discrete Velocity Kinetic Model with Food Metric: Chemotaxis Traveling Waves

We introduce a mesoscopic scale chemotaxis model for traveling wave phenomena which is induced by food metric. The organisms of this simplified kinetic model have two discrete velocity modes, \(\pm s\) and a constant tumbling rate. The main feature of the model is that the speed of organisms is constant \(s\,{>}\,0\) with respect to the food metric, not the Euclidean metric. The uniqueness and the existence of the traveling wave solution of the model are obtained. Unlike the classical logarithmic model case there exist traveling waves under super-linear consumption rates and infinite population pulse-type traveling waves are obtained. Numerical simulations are also provided.     

A Traveling Wave Model for Invasion by Precursor and Differentiated Cells

We develop and investigate a continuum model for invasion of a domain by cells that migrate, proliferate and differentiate. The model is applicable to neural crest cell invasion in the developing enteric nervous system, but is presented in general terms and is of broader applicability. Two cell populations are identified and modeled explicitly; a population of precursor cells that migrate and proliferate, and a population of differentiated cells derived from the precursors which have impaired migration and proliferation. The equation describing the precursor cells is based on Fisher’s equation with the addition of a carrying-capacity limited differentiation term. Two variations of the proliferation term are considered and compared. For most parameter values, the model admits a traveling wave solution for each population, both traveling at the same speed. The traveling wave solutions are investigated using perturbation analysis, phase plane methods, and numerical techniques. Analytical and numerical results suggest the existence of two wavespeed selection regimes. Regions of the parameter space are characterized according to existence, shape, and speed of traveling wave solutions. Our observations may be used in conjunction with experimental results to identify key parameters determining the invasion speed for a particular biological system. Furthermore, our results may assist experimentalists in identifying the resource that is limiting proliferation of precursor cells.     

Traveling waves in a simple population model involving growth and death

The structure of solutions to a simple spatially dependent population model involving growth and death is investigated. Two forms of motility of the population are considered: (1) random motion only modeled by a Fickian law, and (2) a directed component of motion (chemotaxis), included in addition to the random motion. Under certain growth conditions a traveling wave of constant speed is approached. This speed can be increased by the addition of the chemotaxis with a corresponding increase in the asymptotic population. Development of initial conditions into a wave is illustrated numerically.     

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.     

Speeds of invasion in a model with strong or weak Allee effects

We study an invasion model based on a reaction-diffusion equation with an Allee effect. We use a special, piecewise-linear, population growth rate. This function allows us to obtain traveling wave solutions and to compute wave speeds for a full range of Allee effects, including weak Allee effects. Some investigators claim that linearization fails to give the correct speed of invasion if there is an Allee effect. We show that the minimum speed for a sufficiently weak Allee may, in fact, be the same as that derived by means of linearization.     

Integrodifference equations, Allee effects, and invasions

 Models that describe the spread of invading organisms often assume no Allee effect. In contrast, abundant observational data provide evidence for Allee effects. We study an invasion model based on an integrodifference equation with an Allee effect. We derive a general result for the sign of the speed of invasion. We then examine a special, linear–constant, Allee function and introduce a numerical scheme that allows us to estimate the speed of traveling wave solutions. Received: 14 April 2000 / Revised version: 23 December 2000 / Published online: 8 February 2002     

Effects of long-range taxis and population pressure on the range expansion of invasive species in heterogeneous environments

We consider a new model for biological invasions in periodic patchy environments, in which long-range taxis and population pressure are incorporated in the framework of reaction-diffusion-advection equations. We assume that long-range taxis is induced by a weighted integral of stimuli within a certain sensing range. Population pressure is incorporated in the diffusion coefficient that linearly increases with population density. We first analyze the model in the absence of population pressure and demonstrate how the sensing length of long-range taxis influences the range expansion pattern of invasive species and its rate of spread. The effects of population pressure are examined for both homogeneous and periodic patchy environments. For the homogeneous environment, an exact and explicit traveling wave solution and the spreading speed are obtained. For the periodic patchy environment, we find numerically that a population starting from any localized distribution evolves to a traveling periodic wave if the null solution of the RDA equation is locally unstable, and that the traveling wave speed significantly increases with increasing population pressure. Furthermore, the population pressure and taxis intensity synergistically enhance the spreading speed when they are increased together.     

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.     

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.     

Repulsion Effect on Superinfecting Virions by Infected Cells

In this paper, the repulsion effect of superinfecting virion by infected cells is studied by a reaction diffusion equation model for virus infection dynamics. In this model, the diffusion of virus depends not only on its concentration gradient but also on the concentration of infected cells. The basic reproduction number, linear stability of steady states, spreading speed and existence of traveling wave solutions for the model are discussed. It is shown that viruses spread more rapidly with the repulsion effect of infected cells on superinfecting virions, than with random diffusion only. For our model, the spreading speed of free virus is not consistent with the minimal traveling wave speed. With our general model, numerical computations of the spreading speed show that the repulsion of superinfecting virion promotes the spread of virus, which confirms, not only qualitatively but also quantitatively, the experimental result of Doceul et al. (Science 327:873–876, 2010).