一类随机时滞捕食者-食饵模型的动力学行为（英文）

研究了一类含时滞的Harrison型捕食者-食饵模型在随机扰动环境下的动力学行为.对于非时滞和时滞模型分别给出了局部和全局稳定性条件.通过白噪声分别对食饵人口增长率的和捕食者人口死亡率进行随机扰动,构建相应的随机时滞微分方程模型讨论环境噪声对其作用的动力学行为.在一定条件下,随机时滞模型存在随机最终有界的唯一全局正解且解的二阶均值是有界的.最后通过数值模拟对给出的分析结果进行了验证.     

具有比率依赖型功能性反应的捕食脉冲系统的持久性（英文）

本文基于经典的阶段结构模型和Lotka-Volterra捕食模型,提出和研究了具有比率依赖型功能性反应模式的脉冲非自治两维时滞微分方程的周期性释放天敌在固定时刻对害虫控制的过程。得到了害虫灭绝周期解的全局吸引性和依赖于时滞和脉冲的人口模型持久性的条件.     

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.     

含连续时滞的二维竞争系统的周期解和循环解

本文利用HOPf分支定理和Birkhoff定理给出了含连续时滞的二维Lotka-Volterra竞争系统存在周期解和循环解的条件。     

具有脉冲和时滞的Lotka-Volterra系统的正周期解的存在性和全局渐近稳定性

主要研究具有脉冲和时滞的Lotka-Volterra系统的正周期解的存在性和全局渐近稳定性．     

具有扩散和放养的时滞竞争系统的正周期解

主要研究缀块环境下具有扩散和放养的时滞Lotka-Volterra竞争系统,得到了系统的周期解存在性,唯一性和全局渐近稳定性的充分条件．     

具有三时滞Lotka-Volterra互惠系统的Hopf分支

建立了具有三个时滞的Lotka-Volterra互惠系统;获得了正平衡点和Hopf分支存在的条件等;并对所获得的结果进行了数值模拟．     

一类具有时滞的微生物连续培养数学模型研究

研究了具有强核函数时滞的微生物连续培养数学模型,利用泛函微分方程理论和数值解法得到系统在一定操作条件下存在Hopf分叉以及分叉值随操作参数变化的规律,并研究了Hopf分叉产生的方向及周期解的稳定性,绘制了周期解的图形和相图.该模型定性地描述了实验中的振荡和过渡现象.最后与弱核函数时滞模型、离散时滞模型进行比较,分析了它们对多态、振荡等动态行为的影响。     

具有时滞的随机捕食-食饵系统的研究

本文研究了一类具有时滞的随机捕食-食饵系统,证明了系统全局正解的存在性和系统的解的随机最终有界性,确定了系统灭绝和平均持续生存的充分条件.最后,用数值模拟验证理论结果的正确性.     

一类时滞非自治Lotka-Volterra竞争系统的灭绝性

讨论一类具有离散时滞和连续分布时滞的Lotka-Volterra系统,通过构造Lyapunov函数并引入上下平均的概念,将[3]和[6]的方法结合在一起,得到比[6]种群灭绝条件弱的充分条件,同时把文献[3]的结果推广到了时滞非自治系统上.     

The effect of delay in viral production in within-host models during early infection

ABSTRACT

Delay in viral production may have a significant impact on the early stages of infection. During the eclipse phase, the time from viral entry until active production of viral particles, no viruses are produced. This delay affects the probability that a viral infection becomes established and timing of the peak viral load. Deterministic and stochastic models are formulated with either multiple latent stages or a fixed delay for the eclipse phase. The deterministic model with multiple latent stages approaches in the limit the model with a fixed delay as the number of stages approaches infinity. The deterministic model framework is used to formulate continuous-time Markov chain and stochastic differential equation models. The probability of a minor infection with rapid viral clearance as opposed to a major full-blown infection with a high viral load is estimated from a branching process approximation of the Markov chain model and the results are confirmed through numerical simulations. In addition, parameter values for influenza A are used to numerically estimate the time to peak viral infection and peak viral load for the deterministic and stochastic models. Although the average length of the eclipse phase is the same in each of the models, as the number of latent stages increases, the numerical results show that the time to viral peak and the peak viral load increase.     

Deterministic limits to stochastic spatial models of natural enemies

Stochastic spatial models are becoming an increasingly popular tool for understanding ecological and epidemiological problems. However, due to the complexities inherent in such models, it has been difficult to obtain any analytical insights. Here, we consider individual-based, stochastic models of both the continuous-time Lotka-Volterra system and the discrete-time Nicholson-Bailey model. The stability of these two stochastic models of natural enemies is assessed by constructing moment equations. The inclusion of these moments, which mimic the effects of spatial aggregation, can produce either stabilizing or destabilizing influences on the population dynamics. Throughout, the theoretical results are compared to numerical models for the full distribution of populations, as well as stochastic simulations.     

Niche overlap and invasion of competitors in random environments I. Models without demographic stochasticity

The relationship between persistent, small to moderate levels of random environmental fluctuations and limits to the similarity of competing species is studied. The analytical theory hinges on deriving conditions under which a rare invading species will tend to increase when faced with an array of resident competitors in a fluctuating environment. A general approximation scheme predicts that the effects of low levels of stochasticity will typically be small. The technique is applied explicitly to a class of symmetric, discrete-time stochastic analogs of the Lotka-Volterra equations that incorporate cross-correlation but no autocorrelation. The random environment limits to similarity are always very close to the corresponding constant environment limits. However, stochasticity can either facilitate or hinder invasion. The exact limits to similarity are extremely model-dependent. In addition to the symmetric models, an analytically tractable class of models is presented that incorporates both auto- and cross-correlation and no symmetry assumptions. For all of the models investigated, the analytical theory predicts that small-scale stochasticity does little, if anything, to limit similarity. Extensive Monte Carlo results are presented that confirm the analytical results whenever the dynamics of the discretetime models are biologically reasonable in the sense that trajectories do not exhibit unrealistic crashes. Interestingly, the class of stochastic models that is well behaved in this sense includes models whose deterministic analogs are chaotic. The qualitative conclusion, supported by both the analytical and simulation results, is that for competitive guilds adequately modeled by Lotka-Volterra equations including small to moderate levels of random fluctuations, practical limits to similarity can be obtained by ignoring the stochastic terms and performing a deterministic analysis. The mathematical and biological robustness of this conclusion is discussed.     

Numerical integration of the master equation in some models of stochastic epidemiology

The processes by which disease spreads in a population of individuals are inherently stochastic. The master equation has proven to be a useful tool for modeling such processes. Unfortunately, solving the master equation analytically is possible only in limited cases (e.g., when the model is linear), and thus numerical procedures or approximation methods must be employed. Available approximation methods, such as the system size expansion method of van Kampen, may fail to provide reliable solutions, whereas current numerical approaches can induce appreciable computational cost. In this paper, we propose a new numerical technique for solving the master equation. Our method is based on a more informative stochastic process than the population process commonly used in the literature. By exploiting the structure of the master equation governing this process, we develop a novel technique for calculating the exact solution of the master equation--up to a desired precision--in certain models of stochastic epidemiology. We demonstrate the potential of our method by solving the master equation associated with the stochastic SIR epidemic model. MATLAB software that implements the methods discussed in this paper is freely available as Supporting Information S1.     

一类具分段常数变量的捕食-食饵系统的Neimark-Sacker分支

讨论了具时滞与分段常数变量的捕食-食饵生态模型的稳定性及Neimark-Sacker分支;通过计算得到连续模型对应的差分模型,基于特征值理论和Schur-Cohn判据得到正平衡态局部渐进稳定的充分条件;以食饵的内禀增长率为分支参数,运用分支理论和中心流形定理分析了Neimark-Sacker分支的存在性与稳定性条件;通过举例和数值模拟验证了理论的正确性。     

Influence of stochastic perturbation on prey-predator systems

We analyse the influence of various stochastic perturbations on prey-predator systems. The prey-predator model is described by stochastic versions of a deterministic Lotka-Volterra system. We study long-time behaviour of both trajectories and distributions of the solutions. We indicate the differences between the deterministic and stochastic models.     

Invariant distributions for multi-population models in random environments

We obtain the existence of a solution and invariant distribution for systems of stochastic differential equations which represent populations in random environments. The method used is a stochastic Lyapunov function, based on a theorem of Kushner. The method is applied to a system of two populations exchainging individuals through migration, and to a generalized n-dimensional Lotka-Volterra system.     

随机Lotka-Volterra系统的稳定性

利用Lyapunov方法与K．lto公式及鞅的理论,研究了随机Lotka-Volterra系统正平衡点的全局渐近稳定性．得到了随机全局渐近稳定的主要定理,并以确定性系统的全局稳定性作为定理的推论．     

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.     

具无限时滞的Lotka-Volterra 方程的持续性

本文研究了具无限时滞的Lotka-Volterra方程的持久性问题,给出了保证系统持久性的充分条件．