Lotka–Volterra equations with chemotaxis walls, barriers and travelling waves

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lotka-volterra模型的假设

lotka-volterra模型的假设

lotka-volterra模型的假设
Lotka-Volterra模型,又称为Lotka-Volterra方程或LV方程,是一组描述两个或两个以上相互竞争或相互捕食的种群动态的微分方程。

这个模型由意大利科学家Vito Volterra和Albert Lotka在20世纪初独立提出,用于分析生态学中的种群增长问题。

Lotka-Volterra模型基于以下几个基本假设:
1. 种群恒定:假设每个种群的个体数量在短时间内保持恒定,即出生率和死亡率在短期内平衡。

2. 密度无关:假设种群的增长率与种群密度无关,即种群的增长不受密度效应的影响。

3. 资源充足:假设生态系统中的资源(如食物、空间等)是充足的,不会成为限制种群增长的因素。

4. 没有迁移:假设种群之间没有个体的迁移,每个种群都是封闭的。

5. 没有疾病和天敌:假设没有疾病和天敌的影响,即种群的生存率是100%。

6. 指数增长:假设种群的增长遵循指数增长规律,即每代的增长率是恒定的。

7. 二维生态位:假设种群之间存在生态位分化,每个种群占据一个生态位,相互之间不存在竞争。

Lotka-Volterra模型简化了实际的生态过程,因此在应用时需要谨慎,并考虑到模型假设与实际情况之间的差异。

在现实世界的生态系统中,这些假设往往并不完全成立,因此Lotka-Volterra模型通常需要通过实验数据进行校正,或者与其他生态模型结合使用,以更准确地描述种群动态。

通过线性状态反馈使四种群Lotka-Volterra模型永久持续生存(英文)

通过线性状态反馈使四种群Lotka-Volterra模型永久持续生存(英文)

通过线性状态反馈使四种群Lotka-Volterra模型永久持续生
存(英文)
赵立纯;荆海英;陶凤梅
【期刊名称】《生物数学学报》
【年(卷),期】2001(16)1
【摘要】本文研究了四种群Lotka-Volterra模型的永久持续生存问题,给出了通过线性状态反馈使四种群Lotka-Volterra模型永久持续生存的一些充分和必要条件.
【总页数】7页(P9-15)
【关键词】Lotka-Volterra模型;永久持续生存;线性状态反馈
【作者】赵立纯;荆海英;陶凤梅
【作者单位】鞍山师范学院数学系;东北大学理学院
【正文语种】中文
【中图分类】Q141
【相关文献】
1.带时滞的离散Lotka-Volterra型食物链模型的永久持续生存性 [J], 李宝麟;韩凤娟
2.一个具有状态反馈控制的二维种群模型的永久持续生存 [J], 陶凤梅; 夏立显
3.n阶Lotka-Volterra系统永久持续生存的新判据(英文) [J], 荆海英
4.具反馈控制的两种群竞争系统的持续生存性与周期解(英文) [J], 刘志军
5.通过非线性控制使一个捕食——被捕食系统永久持续生存 [J], 赵立纯;张庆灵;等因版权原因,仅展示原文概要,查看原文内容请购买。

Lotka-Volterra 方程

Lotka-Volterra 方程

度娘不是万能的!
维多· 沃尔泰拉(Vito Volterra, 1860年5月3日-1940年10月11日)是 一位意大利数学家与物理学家,著名 于生物数学研究。
20世纪20年代,意大利生物学家达柯纳对第一 次世界大战期间亚得里亚海湾的渔业作了研究。他 发现,在战争年代,鲨鱼等大鱼的捕量占总量的百 分比急剧增加,而战后又趋于正常。
求助
百思不得其解的生物学家达柯纳就这个生 物领域的问题向数学家沃尔泰拉请教,希望 他建立一个数学模型来解释这种现象。
模型一 不考虑捕获
沃尔泰拉在分析了这些数据后,忽略了一 些次要因素,仅考虑自然环境中食饵与捕食者 之间的制约关系,建立了以下简单模型。
模型二 考虑捕获
e为捕获能力系数
当达到平衡时,即x[(a-e)-by]=0,y[-(c+e)+dx]=0, 解得
年份 百分比 年份 百分比
1914 1915 1916 1917 1918 11.9 21.4 22.1 21.2 36.4
1919 1920 1921 1922 1923 27.3 16.0 15.9 14.8 19.7
困惑
战争期间,捕鱼量减少,鲨鱼和小鱼的数量 应同时增加,其比例应基本保持不变。而实际 上,鲨鱼的比例明显增大,小鱼的比例明显减 少,无法理解!
的发史
Lotka-Volterra方程别称掠食者-猎物,由两 条一阶非线性微分方程组成。经常用来描述生物 系统中,掠食者与猎物进行互动时的动力学,也 就是两者族群规模的消长。
1925年,阿弗雷德· 洛特卡发表 1926年,维多· 沃尔泰拉发表
阿弗雷德· 洛特卡(Alfred James Lotka,1880年3月2日-1949年12月5日) 是一位美国统计学家、数学家与物理化 学家,著名于生态学上的族群动力学与 能量学研究。

基于3种群lotka-volterra模型的种群动力学函数优化算法

基于3种群lotka-volterra模型的种群动力学函数优化算法

基于3种群lotka-volterra模型的种群动力学函数优化算法种群动力学是指研究种群数量随时间变化的数学模型。

Lotka-Volterra模型是一种经典的种群动力学模型,它基于两个物种的互动关系来描述种群数量的变化。

然而,实际上很多生态系统中存在多种物种的互动,因此将Lotka-Volterra模型扩展到三种物种是一种有趣和重要的研究方向。

为了优化三种群Lotka-Volterra模型的种群动力学函数,可以采用多种方法。

下面将介绍三种常用的优化算法。

1. 粒子群算法(Particle Swarm Optimization,PSO)粒子群算法是一种启发式优化算法,它模拟了鸟群或鱼群等生物的群体行为。

在PSO中,每个个体被看作是粒子,个体的位置表示解空间中的一个解,粒子的速度表示方向和速度。

通过更新速度和位置,粒子群逐渐收敛到最优解。

在三种群Lotka-Volterra模型中,可以将每个粒子的位置看作是物种数量,通过更新速度和位置,找到最优的物种数量组合。

2. 遗传算法(Genetic Algorithm,GA)遗传算法是一种模拟自然界生物进化过程的优化算法。

在遗传算法中,每个个体被编码为一串基因,通过选择、交叉和变异等操作,不断优化个体的适应度。

在三种群Lotka-Volterra模型中,可以将每个个体的基因编码为物种数量,通过选择、交叉和变异等操作,寻找最优的物种数量组合。

3. 蚁群算法(Ant Colony Optimization,ACO)蚁群算法是一种模拟蚁群行为的优化算法。

在ACO中,每个蚂蚁通过释放信息素和选择路径的方式寻找最优解。

信息素表示路径的好坏程度,蚂蚁通过信息素的引导选择路径,并更新信息素浓度。

在三种群Lotka-Volterra模型中,可以将信息素浓度看作是物种数量的评价,蚂蚁在过程中通过更新信息素浓度,找到最优的物种数量组合。

以上三种优化算法都可以应用于优化三种群Lotka-Volterra模型的种群动力学函数,通过不断迭代和更新寻找最优的物种数量组合。

Lotka—Volterra竞争扩散系统连接边界平衡点和正平衡点行波解的存在性

Lotka—Volterra竞争扩散系统连接边界平衡点和正平衡点行波解的存在性

Lotka—Volterra竞争扩散系统连接边界平衡点和正平衡点行波解的存在性【摘要】本文探讨了Lotka—Volterra竞争扩散系统连接边界平衡点和正平衡点行波解的存在性问题。

首先介绍了竞争扩散系统的基本原理,然后分别定义和描述了边界平衡点和正平衡点的性质。

接着阐述了行波解的概念,并重点讨论了连接边界平衡点和正平衡点的行波解存在性。

探讨了存在性分析的意义,并展望了进一步的研究方向。

本文通过理论分析和数值模拟,深入探究了竞争扩散系统中行波解的存在性,对于生态学和数学建模领域具有重要的理论意义和应用价值。

【关键词】Lotka—Volterra竞争扩散系统、边界平衡点、正平衡点、行波解、存在性分析、研究展望1. 引言1.1 研究背景在生态学领域,竞争扩散系统是一种重要的研究对象,其中Lotka—Volterra模型是经典的描述种群竞争关系的数学模型。

竞争扩散系统可以模拟不同种群之间的竞争和扩散过程,揭示种群数量和空间分布之间的动态关系。

在实际生态系统中,种群之间的竞争和扩散是普遍存在的现象,对于生态系统的稳定性和可持续发展具有重要意义。

研究Lotka—Volterra竞争扩散系统的连接边界平衡点和正平衡点的行波解存在性,不仅可以加深我们对生态系统动态特性的理解,还可以为生态系统的管理和保护提供理论指导。

在实际应用中,行波解的存在性分析可以为预测种群扩散和竞争的趋势提供参考,为生态环境的健康和生物多样性的维护提供科学依据。

探究Lotka—Volterra竞争扩散系统连接边界平衡点和正平衡点行波解的存在性,具有重要的理论和应用意义。

1.2 研究目的研究目的是探讨在Lotka—Volterra竞争扩散系统中连接边界平衡点和正平衡点的行波解的存在性问题。

具体来说,我们的目的包括以下几点:1. 确定竞争扩散系统的基本原理,深入理解系统内各种影响因素之间的相互作用关系,从而为后续研究奠定基础。

2. 研究和探讨边界平衡点和正平衡点在竞争扩散系统中的定义和性质,分析它们在系统中的作用和重要性。

Lotka—Volterra竞争扩散系统连接边界平衡点和正平衡点行波解的存在性

Lotka—Volterra竞争扩散系统连接边界平衡点和正平衡点行波解的存在性

Lotka—Volterra竞争扩散系统连接边界平衡点和正平衡点行波解的存在性Lotka-Volterra竞争扩散系统是描述种群竞争和迁移的数学模型,它由Alfred J. Lotka和Vito Volterra在20世纪初提出。

这个系统描述了两个不同种群在空间中的竞争和扩散的动态过程,对于了解生态系统中物种之间的相互作用具有重要意义。

在Lotka-Volterra竞争扩散系统中,连接边界平衡点和正平衡点的行波解的存在性是一个重要问题,对于我们理解这个系统的稳定性和动态行为具有重要的意义。

让我们来了解一下Lotka-Volterra竞争扩散系统的基本形式。

该系统的基本描述是由一对常微分方程组成,考虑两个种群的竞争和扩散过程。

假设有两个物种,分别用u(x, t)和v(x, t)表示它们在空间位置x和时间t上的密度。

那么Lotka-Volterra竞争扩散系统可以用如下的方程描述:∂u/∂t = d₁∇²u + r₁u(1 - u - α₁v)∂v/∂t = d₂∇²v + r₂v(1 - v - α₂u)d₁和d₂分别表示两个种群的扩散系数,r₁和r₂分别表示两个种群的增长率,α₁和α₂表示两个种群之间的竞争系数。

这个系统描述了种群的扩散和竞争,其中扩散项描述了种群在空间中的迁移,而竞争项描述了种群之间的相互作用。

连接边界平衡点和正平衡点的行波解是指在这个系统中,当种群的密度在空间和时间上变化时,存在一种特殊的解,它以一定的速度向着某个方向传播,并且在这个速度下保持稳定。

连接边界平衡点和正平衡点的行波解的存在性意味着在这个系统中,存在着一种特殊的动态行为,种群可以在空间中形成稳定的结构,即使在竞争和扩散的作用下也能够维持一定的稳定形态。

关于连接边界平衡点和正平衡点的行波解的存在性,已经在过去的研究中得到了一些结论。

一些研究表明,在一些特定的参数范围内,Lotka-Volterra竞争扩散系统确实存在连接边界平衡点和正平衡点的行波解,而这些行波解对于了解种群的空间动态行为具有重要的意义。

Lotka—Volterra竞争扩散系统连接边界平衡点和正平衡点行波解的存在性

Lotka—Volterra竞争扩散系统连接边界平衡点和正平衡点行波解的存在性

Lotka—Volterra竞争扩散系统连接边界平衡点和正平衡点行波解的存在性Lotka-Volterra竞争扩散系统是描述两个物种之间竞争和扩散关系的模型,它在生物学和生态学领域有着重要的应用。

在该系统中,两个物种之间通过资源的竞争相互影响,并且通过空间的扩散进行传播。

本文将探讨Lotka-Volterra竞争扩散系统连接边界平衡点和正平衡点行波解的存在性。

我们来了解一下Lotka-Volterra竞争扩散系统的基本形式。

该系统描述了两个物种在时间和空间上的分布和相互作用。

假设我们有两个物种u和v,它们的分布随时间t和空间x的变化可以由以下方程描述:\[\frac{\partial u}{\partial t} = d_u \nabla^2 u + r_u u \left(1 - \frac{u + \alpha v}{K}\right)\]du和dv分别代表两个物种的扩散系数,ru和rv分别代表两个物种的增长率,K代表环境的承载能力,α和β分别表示两个物种对对方竞争的敏感度。

在上述方程中,存在两种平衡点:边界平衡点和正平衡点。

边界平衡点指的是物种在空间的边界处达到平衡状态,而正平衡点指的是物种在空间内部达到平衡状态。

连接边界平衡点和正平衡点的行波解,描述了两个物种在空间中的扩散和竞争关系。

接下来,我们将讨论连接边界平衡点和正平衡点行波解的存在性。

在实际生态系统中,很多情况下物种之间存在着空间上的扩散和竞争关系,因此连接边界平衡点和正平衡点的行波解的存在性具有重要的理论和实际意义。

通过数学分析和数值模拟可以发现,连接边界平衡点和正平衡点的行波解在Lotka-Volterra竞争扩散系统中是存在的。

具体来说,在一些特定的参数取值条件下,我们可以得到连接边界平衡点和正平衡点的行波解。

这些行波解描述了两个物种在空间中的分布和相互作用,展现了它们在空间上的动力学特性。

连接边界平衡点和正平衡点的行波解的存在性为我们理解生物群落的空间格局提供了重要的线索。

Lotka—Volterra竞争扩散系统连接边界平衡点和正平衡点行波解的存在性

Lotka—Volterra竞争扩散系统连接边界平衡点和正平衡点行波解的存在性

Lotka—Volterra竞争扩散系统连接边界平衡点和正平衡点行波解的存在性Lotka-Volterra竞争扩散系统是描述生态系统中种群竞争和扩散相互作用的数学模型,它由Alfred Lotka和Vito Volterra在20世纪初提出,并被广泛应用于生态学、生物学和数学领域。

在生态系统中,不同种群之间存在着资源的竞争和空间的扩散。

这种竞争扩散系统的动力学特性对生态系统的稳定性和多样性具有重要影响。

在过去的研究中,人们主要关注于Lotka-Volterra竞争扩散系统内部正平衡点的存在性和稳定性,但对于连接边界平衡点和正平衡点行波解的存在性研究相对较少。

本文将重点讨论Lotka-Volterra竞争扩散系统的连接边界平衡点和正平衡点行波解的存在性,探讨这一问题在生态系统稳定性和多样性中的重要意义。

我们将介绍Lotka-Volterra竞争扩散系统的基本模型和数学表达式,然后分析连接边界平衡点和正平衡点行波解的存在性,最后讨论这一研究对生态学和数学的意义和应用。

1. Lotka-Volterra竞争扩散系统的基本模型Lotka-Volterra竞争扩散系统是一种描述生态系统中种群竞争和扩散相互作用的数学模型,其基本形式可以表示为:\begin{cases}\frac{\partial u}{\partial t} = d_u\Delta u+ru(1-\frac{u}{K})-auv\\\frac{\partial v}{\partial t} = d_v\Delta v+sv(1-\frac{v}{L})-buv\end{cases}u和v分别表示两个种群的密度,t表示时间,d_u和d_v表示扩散系数,r和s分别表示种群的增长率,K和L分别表示种群的最大容纳量,a和b分别表示种群之间的竞争强度。

上式中的第一项表示扩散项,第二项表示种群的自我增长,第三项表示种群之间的竞争作用。

这个模型描述了种群在空间中的扩散和竞争,可以用来研究生态系统中种群的动态演变和空间分布。

一类含时滞和扩散的Lotka-Volterra生态模型的渐近性质的开题报告

一类含时滞和扩散的Lotka-Volterra生态模型的渐近性质的开题报告

一类含时滞和扩散的Lotka-Volterra生态模型的渐近性质的开题报告1. 研究背景和意义生态系统中的物种相互作用一直是生态学研究的重要课题。

Lotka-Volterra模型是描述物种间相互作用的经典模型之一,其追溯至上个世纪初。

在实际应用中,我们会发现很多生态系统中,信息(或种群)的传递存在滞后。

这种滞后被称为时滞。

时滞往往会带来很多复杂性质,因此,时滞的Lotka-Volterra模型成为深入研究这类生态系统的关键性工具。

同时,扩散是许多生态系统中的普遍现象。

因此,在实际中,研究时滞和扩散的Lotka-Volterra模型对于理解和预测生态系统的演变是非常重要的。

近年来,研究人员关注含时滞和扩散的Lotka-Volterra模型的渐近性质,如渐近稳定性、渐近周期性和渐近分散性等特性。

这些性质可以帮助我们更好地理解生态系统的动态过程和演变规律。

2. 研究内容和方法本研究将着重研究含时滞和扩散的Lotka-Volterra生态模型的渐近性质。

具体研究内容包括:(1)建立含时滞和扩散的Lotka-Volterra生态模型,并加入一些实际中常见的因素,如随机扰动、外部干扰等。

(2)研究模型的渐近稳定性。

通过构造Lyapunov函数和LaSalle不变集理论研究模型的渐近稳定性和渐近吸引性。

(3)研究模型的渐近周期性。

通过构造周期函和Poincaré-Bendixon定理研究模型的周期性和极限环。

(4)研究模型的渐近分散性。

通过构造分散基函数或分散集理论分析模型的扩散性和分散性。

本研究采用的方法包括数学分析方法,如微分方程理论、稳定性分析、周期分析和分散基函数分析等。

3. 预期研究结果和意义本研究预期获得含时滞和扩散的Lotka-Volterra生态模型的渐近性质,并在此基础上提出一些基于实际的生态管理和保护策略。

此外,该研究可以为理解和预测生态系统的动态演变提供理论基础和方法指导。

Lotka—Volterra竞争扩散系统连接边界平衡点和正平衡点行波解的存在性

Lotka—Volterra竞争扩散系统连接边界平衡点和正平衡点行波解的存在性

Lotka—Volterra竞争扩散系统连接边界平衡点和正平衡点行波解的存在性1. 引言1.1 背景介绍Lotka—Volterra竞争扩散系统是一种描述生态系统中物种之间相互作用的数学模型,它结合了Lotka—Volterra竞争模型和扩散方程,能够更全面地描述物种之间的竞争和扩散行为。

在生态学中,理解物种之间的竞争对于生态系统的稳定和演化具有重要意义。

研究Lotka—Volterra竞争扩散系统连接边界平衡点和正平衡点行波解的存在性,对于深入理解生态系统动态过程具有重要意义。

在过去的研究中,人们已经开始对Lotka—Volterra竞争扩散系统进行了一些探究。

对于连接边界平衡点和正平衡点行波解的存在性,仍然存在一定的研究空白。

本文旨在通过数学模型分析和数值模拟的方法,探讨Lotka—Volterra竞争扩散系统连接边界平衡点和正平衡点行波解的存在性,以期为生态系统动态过程的理解提供新的视角和研究途径。

1.2 研究目的本研究旨在通过探讨Lotka—Volterra竞争扩散系统连接边界平衡点和正平衡点行波解的存在性,深入理解这一系统在生态学领域的重要性和影响。

具体而言,我们的研究目的包括以下几个方面:2. 探究正平衡点行波解的存在性:分析在系统中是否存在正平衡点行波解,并研究其在生态学中的实际意义和应用价值。

3. 提出数学模型分析和数值模拟方法:通过建立相应的数学模型和进行数值模拟,揭示系统的特征和行为规律,从而更好地理解Lotka—Volterra竞争扩散系统的内在机制。

通过对以上研究目的的探讨和实证分析,本研究旨在为生态学领域的相关研究提供新的理论和方法支持,促进生态系统的可持续发展和管理。

1.3 文献综述在过去的几十年中,关于Lotka—Volterra竞争扩散系统连接边界平衡点和正平衡点行波解的存在性的研究取得了一系列重要进展。

许多学者对这一领域展开了深入的探讨,提出了许多重要的理论和结论。

几类具时滞的Lotka-Volterra互惠系统的定性分析

几类具时滞的Lotka-Volterra互惠系统的定性分析

几类具时滞的Lotka-Volterra互惠系统的定性分析生态系统的持久性、周期解的存在性及稳定性、全局吸引性等问题是数学生态学理论中的一个重要研究内容,对上述问题给出明确的判定准则是数学生态学的一个重要课题,历来受到生态学家与数学家的广泛重视。

研究非自治的Lotka-Volterra互惠系统解的定性性质,在实际应用方面有着广阔前景。

对于生态系统而言,我们既要考虑种群的成长规律(即时滞效应),又要考虑环境对生态系统的影响(即系数是时间t的函数),因此,我们一般研究具时滞的非自治的生态系统。

本文利用微分方程定性理论,M-矩阵理论,非线性常微分方程泛函方法等理论工具,研究了几类具时滞的非自治Lotka-Volterra 互惠系统,讨论了系统解的持久性、有界性、周期解的存在性、全局吸引性,得到了一些有用的结论。

全文安排如下:第1章简要介绍了研究的理论背景与现实意义;第2章讨论了一类二维变时滞系统的持久性、有界性、周期解的存在性及全局吸引性等问题,给出了一系列判别准则;第3章讨论了一类三维多时滞系统的持久性、有界性、周期解的存在性及全局吸引性;第4章研究一类具变时滞的三维互惠系统的动力学性质,证明了在一定条件下,系统周期正解的存在性;第5章讨论一类具分布时滞的n维互惠系统,给出了系统周期正解全局吸引的充分条件。

基于Lotka-Volterra模型的氢能市场系统演化分析

基于Lotka-Volterra模型的氢能市场系统演化分析

基于Lotka-Volterra模型的氢能市场系统演化分析马涛;陈鸣歧;杨建华;赵泽斌【期刊名称】《哈尔滨商业大学学报(自然科学版)》【年(卷),期】2009(025)005【摘要】作为良好替代性的清洁能源,氢能商业层面产业化进程相对缓慢.从氢能与非氢能的替代性入手,通过Lotka-Volterra系统模型建立氢能市场系统的演化模型分析以增加氢能供给为目的氢能市场演化性质.基于替代性和增加氢能供给量的原则,界定了一个相对的氢能市场和氢能市场结构的演化系统,在此基础上通过加入时间变量改进了基本的Lotka-Volterra模型,从而初步建立了氢能市场的系统演化分析模型.采用平均值法求解模型,分别从供给增加与非氢能消减的关系,以及氢能供给增长速度、非氢能消减及其涨落间的关系,两个层面上进行系统演化分析.在上述过程中,分别得出了在氢能供给的限度、独立性和增长率的四个关于氢能市场系统演化的推论.【总页数】4页(P528-530,539)【作者】马涛;陈鸣歧;杨建华;赵泽斌【作者单位】哈尔滨工业大学,管理学院,哈尔滨,150001;哈尔滨工业大学,市政环境学院,哈尔滨,150001;山东工商学院,工商管理学院,山东,烟台,264005;哈尔滨工业大学,管理学院,哈尔滨,150001【正文语种】中文【中图分类】F416.2【相关文献】1.基于Lotka-Volterra模型的中国股票市场非线性特征--一个生态学的视角 [J], 刘辉煌;莫宪;饶彬2.沪港通市场动态互动关系研究r——基于Lotka-Volterra金融生态模型 [J], 张本照;姚刚;宋平凡3.国内民航客运市场竞争态势及其演化分析——基于非自治Lotka-Volterra模型[J], 李爽;李帅华4.基于Lotka-Volterra模型-Logistic模型高校主导的创新生态系统动态演化研究[J], 孙金花;苟晓朦;杜姣5.基于风氢耦合的氢储能系统参与电力市场机制与风险量化模型设计 [J], 孔飘红;蒋正威;杨力强;甘倍瑜;龚昭宇因版权原因,仅展示原文概要,查看原文内容请购买。

时滞Lotka-Volterra扩散系统的分支与周期解的开题报告

时滞Lotka-Volterra扩散系统的分支与周期解的开题报告

时滞Lotka-Volterra扩散系统的分支与周期解的开题报告研究背景和意义Lotka-Volterra模型被广泛用于描述捕食者和猎物之间的相互作用。

在扩散系统中,物种的扩散会对模型的动态行为产生影响。

然而,由于生态系统内部的非线性相互作用和外部环境的影响,实际的生态系统往往存在时滞现象,这使得当时间与空间耦合时,动态行为会变得更加复杂。

因此,研究时滞Lotka-Volterra扩散系统的分支和周期解对于理解生态系统动态行为、预测物种演化和保护生物多样性都具有重要意义。

研究内容本文将研究时滞Lotka-Volterra扩散系统的分支和周期解的存在性和稳定性。

具体研究内容包括以下方面:1. 构建时滞Lotka-Volterra扩散系统的数学模型并对其进行数学描述。

2. 利用动力学系统理论和广义极值理论,研究时滞Lotka-Volterra 扩散系统的分支解。

分析分支解存在的条件和稳定性。

3. 利用周期函数理论和分支理论,研究时滞Lotka-Volterra扩散系统的周期解。

分析周期解存在的条件和稳定性。

4. 通过数值模拟,验证理论结果的正确性和可行性。

研究方法本文将主要采用以下研究方法:1. 动力学系统理论:利用微分方程理论、极限环理论和广义极值理论,对时滞Lotka-Volterra扩散系统的分支解进行分析和研究。

2. 周期函数理论和分支理论:利用微分方程周期函数理论和分支理论,对时滞Lotka-Volterra扩散系统的周期解进行分析和研究。

3. 数值模拟:利用Matlab等数值计算软件,对分支解和周期解的数值解进行模拟和求解,验证理论结果的正确性和可行性。

预期成果1. 建立了时滞Lotka-Volterra扩散系统的数学模型,并对其进行了数学描述。

2. 研究了时滞Lotka-Volterra扩散系统的分支解,并分析了其存在的条件和稳定性。

3. 研究了时滞Lotka-Volterra扩散系统的周期解,并分析了其存在的条件和稳定性。

具有捕获的三种群Lotka-Volterra系统的多个周期解

具有捕获的三种群Lotka-Volterra系统的多个周期解

具有捕获的三种群Lotka-Volterra系统的多个周期解刘娟风;魏凤英【摘要】本文讨论具有捕获的三种群Lotka-Volterra系统的正周期解,利用重合度定理.得到该系统至少存在八个正周期解.【期刊名称】《福建师大福清分校学报》【年(卷),期】2010(000)005【总页数】8页(P1-8)【关键词】正周期解;捕食-食饵;重合度定理;捕获【作者】刘娟风;魏凤英【作者单位】福州大学数学与计算机科学学院,福建福州,350108;福州大学数学与计算机科学学院,福建福州,350108【正文语种】中文【中图分类】O175.12近些年,关于生物模型的正周期解的存在性被广泛研究[1-4].本文考虑具有人为捕获的三种群Lotka-Volterra捕食-食饵系统:其中xi(t),yi(t)表示种群密度,ri(t)表示第i个种群的内禀增长率;aii(t)表示第i个种群的密度制约率;a12(t),a23(t)分别表示第2个种群对第1个种群、第3个种群对第2个种群的捕食率;a21(t),a32(t)分别表示第1个种群对第2个种群、第2个种群对第3个种群的贡献率;hi(t)表示第i个种群的人为捕获率,且ri(t),aij(t),hi(t)在[0,∞)上是连续有界正值函数(i=1,2,3).为了方便,我们记,f(t)是连续ω-周期函数.为了便于理解,我们首先引入Gains and Mawhin中的延拓定理.设X和Z是赋范向量空间,L∶Dom L⊂X→Z为线性映射.N∶X×[0,1]→Z为连续映射.L是指标为零的Fredholm算子,若dim Ker L=co dim Im L<∞且Im L为Z 中的闭集.若L是指标为零的Fredholm算子,又存在连续投影P∶X→X和Q∶Z→Z使得X=Ker L⊕Im Q,Z=Im L⊕Im Q,Im P=Ker L,Im L=KerQ=Im (I-Q),则L|DomL∩Ker P∶(I-P)X→Im L是可逆的,并记其逆为Kp.假设Ω是X的有界开集,若QN(Ω¯×[0,1])有界,且Kp(I-Q)N∶Ω¯×[0,1]→X是紧的,则N在Ω¯×[0,1]是L-紧的.由于ImQ与Ker L同构,因此存在同构映射J∶ImQ→Ker L.引理2.1[5]令L为指标为零的Fredholm算子且N在Ω¯是L-紧的.假设(a)对任意的λ∈(0,1),Lx=λN(x,λ)的每个解x都满足x∉əΩ∩Dom L;(b)对任意的x∈∂Ω∩Ker L,都有QN(x,0)≠0;(c)deg{JQN(x,0),Ω∩Ker L,0}≠0;则方程Lx=Nx在Dom L∩至少存在一个解.引理2.2[2]设则函数满足:(1)f(x,y,z)和g(x,y,z)在x∈(0,∞)分别单调递增和单调递减,(2)f(x,y,z)和g(x,y,z)在y∈(0,∞)分别单调递减和单调递增,(3)f(x,y,z)和g(x,y,z)在z∈(0,∞)分别单调递减和单调递增.定理2.3若(A1),(A2),(A3)成立,则系统(1.1)至少有八个正ω-周期解. 证明.对系统(1.1)作变换,令xi(t)=eu1(t)(i=1,2,3),则有令,定义范数或X或Z,则X和Z是Banach空间.设因此}是Z的闭集,dim Ker L=3=co dim Im L,且P,Q连续映射,则Im Q=KerL,Im L=Ker Q=Im(I-Q).故而,L是指标为零的Fredholm算子.定义L的广义逆Kp∶Im L→Ker P∩Dom L如下:则有其中显然,QN和Kp(I-Q)N是连续的,根据Arzela-Ascoli定理,Kp(I-Q)N对任何有界开集Ω⊂X是紧的,QN是有界的,N是关于任何有界开集Ω⊂X在是L-紧的.对于算子方程Lu=λ(u,λ),λ∈(0,1),我们得到假设u∈X是系统(2.2)的ω-周期解,λ∈(0,1),因此存在ξi,ζi∈[0,ω]满足显然u′i(ξi)=0,u′i(ζi)=0,i=1,2,3.由(2.2)可得和由(2.3)的第一个方程,有类似的,由方程(2.4)第一个方程得到其中根据(2.3)第二个方程,有从而有类似的,由(2.4)的第二个方程,可得其中由(2.3)的第三个方程,可得,所以由(2.4)的第三个方程,有其中另一方面,由(2.3)的第一个方程,有,因此类似的,由(2.4)第一个方程,可得其中事实上,应用引理2.2可得由(2.5),(2.6),(2.11)和(2.12),可得因此,对所有t∈R根据(2.4)的第二个方程,有可得类似的,由(2.5)的第二个方程,可得其中事实上,由引理2.2,可得由(2.7),(2.8),(2.14)和(2.15),可得或则对所有t∈R,有由(2.4)的第三个方程,可得,从而类似的,由(2.5)的第三个方程,有其中事实上,根据引理2.2,可得由(2.9),(2.10),(2.17)和(2.18),可得故对t∈R,有或显然,lnl±1,lnl±2,lnl±3,lnH±1,lnH±2和lnH±3与λ无关.设则Ωi(i=1,2,3,4,5,6,7,8)是X的有界开子集,Ωi∩Ωj=Φ(i≠j,i,j=1,2,3,4,5,6,7,8).因此Ωi,i=1,2,3,4,5,6,7,8满足引理2.1条件(a).现在证当u∈əΩi∩Κer L=əΩi∩R3时,QN(u,0)≠(0,0,0)T(i=1,2,3,4,5,6,7,8).假设它不成立,则当u∈əΩi∩Κer L=əΩi∩R3时,常向量u=(u1,u2,u3)T∈əΩi满足由微分中值定理可知,存在三个点tj∈[0,ω],j=1,2,3满足根据(2.13),(2.16)和(2.19),得到则u∈Ωi∩R3(i=1,2,3,4,5,6,7,8),这与已知条件u∈∂Ωi∩R3(i=1,2,3,4,5,6,7,8)相矛盾.因此引理2.1条件(b)成立.最后,证明引理2.1条件(c)成立.因为(A1),(A2),(A3)成立,所以代数方程有八个不同的解:其中由引理2.1,很容易证明因此由Ker L=ImQ,令J=I,则因为所以因此至此,引理2.1的条件都成立,于是,系统(1.1)至少有八个ω-周期解.定理2.3得证. 考虑下面具有捕获的三种群捕食-食饵系统因此则因此,由定理2.3,系统(3.1)至少存在八个正周期解.【相关文献】[1]D.Hu,Z.Zhang.Four positive periodic solutions to a Lotka-Volterra cooperative system with harvesting terms[J].Nonlinear Analysis:Real World Applications,2010(11):1115-1121.[2]K.Zhao,Y Ye.Four positive periodic solutions to a periodic Lotka-Volterra predator-prey system with harvesting terms[J].Nonlinear Analysis:Real World Applications,2010(11):2448-2455.[3]Li Z,K.Zhao,Y.Li.Multiple positive periodic solutions for a non-autonomous stage-structured predator-prey system with harvesting terms[J].Communications in Nonlinear Science and Numerical Simulation,2010(15):2140-2148.[4]Z.Zhang,Z.Hou.Existence of four positive periodic solutions for a ratio-dependent predator-prey system with harvesting terms[J].Nonlinear Analysis:Real World Applications,2010(11):1560-1571.[5]R.Gaines,J.Mawhin.Coincidence Degree and Nonlinear DifferentialEquations[M].Springer Verlag,Berlin,1997.[6]Z.Ma.Mathematical Modelling and Study on Species Ecology[M].Hefei:Education Press,1996(in Chinese).[7]Horst R.Thieme.Mathematics in population biology.In:Princeton syries in theoretical and computation biology[M].Princeton,NJ:Princeton University Press,2003.。

Lotka—Volterra竞争扩散系统连接边界平衡点和正平衡点行波解的存在性

Lotka—Volterra竞争扩散系统连接边界平衡点和正平衡点行波解的存在性

Lotka—Volterra竞争扩散系统连接边界平衡点和正平衡点行波解的存在性1. 引言1.1 研究背景研究如何将Lotka-Volterra竞争扩散系统与边界条件结合起来,探讨边界平衡点和正平衡点的性质以及它们之间的连接行波解的存在性是一个具有挑战性和重要意义的课题。

通过对这些问题进行深入研究,不仅可以丰富我们对竞争扩散系统的理解,还可以为生态学和地理学领域提供新的理论基础和实际应用价值。

本文旨在探讨Lotka-Volterra竞争扩散系统连接边界平衡点和正平衡点行波解的存在性问题,为相关领域的研究提供新的思路和方法。

1.2 研究目的本文旨在探讨Lotka—Volterra竞争扩散系统连接边界平衡点和正平衡点行波解的存在性。

具体地,我们将通过对系统中边界平衡点和正平衡点的定义和性质进行分析,揭示它们之间的联系与特点。

我们希望能够证明在这样的竞争扩散系统中,存在连接边界平衡点和正平衡点的行波解。

通过对这一问题的研究,我们可以更深入地理解竞争扩散系统的演化过程,揭示其中蕴含的规律和机理。

这对于生态学和数学建模领域具有重要的理论意义,并且具有一定的应用价值。

通过揭示行波解的存在性,我们可以为解释生态系统中物种竞争与扩散的相互作用提供新的视角和方法,为未来研究提供新的思路和方向。

1.3 相关工作相关工作部分主要回顾了之前在Lotka—Volterra竞争扩散系统连接边界平衡点和正平衡点行波解存在性方面的研究成果。

早期的相关工作主要集中在竞争扩散系统的理论分析和数值模拟方面,较少涉及到连接边界平衡点和正平衡点行波解的存在性证明。

Jin等人(2010)通过数值模拟研究了Lotka—Volterra竞争扩散系统中行波解的形成机制,但并未深入探讨存在性证明。

Li和Wang (2015)对竞争扩散系统中的边界平衡点和正平衡点行波解进行了理论分析,提出了一些重要的结论,但仍存在一些待解决的问题。

近年来,越来越多的研究者开始关注连接边界平衡点和正平衡点行波解的存在性证明问题。

高维lotka-volterra系统的周期解

高维lotka-volterra系统的周期解

高维lotka-volterra系统的周期解
Lotka-Volterra系统是生态学中常用的模型之一,用来描述生态系统中生物种群数量的变化。

高维Lotka-Volterra系统是指将Lotka-Volterra系统扩展到描述多种生物种群数量变化的情况。

在高维Lotka-Volterra系统中,周期解是指系统中所有生物种群数量在一个固定周期内互相循环变化的解。

高维Lotka-Volterra系统的周期解可能需要使用复杂的数学方法来求解,并且其周期解的存在性和具体形式可能受到系统参数的影响。

需要注意的是,高维Lotka-Volterra系统不一定存在周期解,在某些情况下可能存在混沌解或稳定解。

因此在研究高维Lotka-Volterra系统时需要综合考虑系统参数和初始条件等因素。

信息与计算科学论文 生物数学-Lotka-Volterra模型的数值解法

信息与计算科学论文 生物数学-Lotka-Volterra模型的数值解法

摘要 (1)Abstract (2)1. 绪论............................................................. 1 1.1 生物数学背景 (1)1.2 生物数学的发展现状 (2)1.3 微分方程数值解法的产生 (2)2.预备知识 (4)2.1数值解法 (4)2.1.1数值解法的引出(初值问题)[2] (4)2.1.2数值解法的基本实现和途径 (4)2.1.3数值解法的分类[3] (6)(1)单步法 (6)(2)多步法..................................................6 2.1.4数值解法的常用方法 (6)(1)Euler 法[4]............................................... 6 (2)Runge-Kutta 法[5].. (7)(3)数值积分梯形积分 (11)2.2生态数学 (12)2.2.1 Volterra 模型的原理 (12)2.2.2 Volterra 模型的应用 (13)2.2.3 Volterra 模型的相关定理及证明 (14)3.数值解法在生物模型中的应用.......................................15 3.1模型建立....................................................16 3.2对问题进行分析 (17)3.3求解 (17)3.3.1数值解 (17)3.1.2平衡点及相轨线 (21)3.1.3 )(t x ,)(t y 在一个周期内的平均值 (24)4.结论......................................................... 26 参 考 文 献.. (27)附录...............................................................27生物数学-Lotka-Volterra模型的数值解法摘要数值解法是研究有关微分方程的近似解的数值方法和相关理论。

变时滞随机Lotka-Volterra_生物模型的渐近性质

变时滞随机Lotka-Volterra_生物模型的渐近性质

第 42 卷第 3 期2023年 5 月Vol.42 No.3May 2023中南民族大学学报(自然科学版)Journal of South-Central Minzu University(Natural Science Edition)变时滞随机Lotka-Volterra生物模型的渐近性质胡军浩,王朝航*(中南民族大学数学与统计学学院,武汉430074)摘要研究了变时滞随机Lotka-Volterra (LV)生物模型,其中变时滞函数是不可微的,放宽了现有文献变时滞是可微且导数小于1的假设. 使用Ito公式和线性矩阵不等式(LMI)研究了这类生物模型全局正解的存在性和唯一性,并进一步给出了其正解渐近有界、时间均值意义下矩有界和多项式增长的充分条件. 最后给出实例验证了结论的有效性.关键词随机生物模型;Lotka-Volterra生物模型;不可微时滞函数;LMI不等式;渐近有界性中图分类号O241.8 文献标志码 A 文章编号1672-4321(2023)03-0402-06doi:10.20056/ki.ZNMDZK.20230316Asymptotic properties of the stochastic Lotka-Volterra system withvariable time delayHU Junhao,WANG Zhaohang*(College of Mathematics and Statistics, South-Central Minzu University, Wuhan 430074, China)Abstract This paper is concerned with stochastic Lotka-Volterra (LV) system with variable time delay. Comparing with most existing papers,the time delay functions in the LV system are no longer required to be differentiable,their derivatives are less than 1 is not to be mentioned. The existence and uniqueness of the global positive solutions of this system are investigated by using Ito formula and linear matrix inequality (LMI). Further,sufficient conditions are also obtained for the asymptotic boundedness,time average moment boundedness and the polynomial pathwise growth of the positive solution. Finally, an example is given to illustrate the effectiveness of the results.Keywords stochastic biological model; Lotka-Volterra system; non-differentiable time delay function; LMI inequality;asymptotic boundedness1 模型介绍本文考虑如下形式的变时滞随机Lotka-Volterra 系统:d x(t)=diag(x(t)){[a+Ax(t)+By(t)]d t+[b+Dx(t)+Ey(t)]dω(t)},(1)其中x(t),y(t)=x(t-δ(t))∈R n分别表示种群和变时滞种群,δ(t)表示变时滞函数.diag(x)= diag(x1,⋯,x n)表示n×n阶对角矩阵. a,b∈R n,矩阵A=[a ij],B=[b ij],D=[d ij],E=[e ij]∈R n×n. 记f(x,y)=a+Ax+By,g(x,y)=b+Dx+Ey.过去几十年,Lotka-Volterra (LV)生物模型受到了越来越多学者的关注[1-2]. 生物模型经常受到噪音因素的影响,用随机微分方程(SDE)来描述这类生物模型更具现实意义. NIE和MEI[3]研究了白噪声与时滞对LV生物模型的影响,证明了白噪声和时滞完全抑制了LV生物模型种群的爆破. LI和MAO[4]进一步研究了非自治的LV生物模型在随机扰动下的持久性和非持久性. 文献[5-8]研究了常收稿日期2022-11-07 * 通信作者王朝航,研究方向:随机生物数学,E-mail:****************作者简介胡军浩(1974-),男,教授,博士,研究方向:随机系统理论及应用,E-mail:******************基金项目国家自然科学基金资助项目(61876192);中南民族大学研究生学术创新基金项目(3212023sycxjj003)第 3 期胡军浩,等:变时滞随机Lotka-Volterra生物模型的渐近性质时滞随机LV生物模型正解的存在唯一性及相关性质. HU等[9]研究了变时滞随机LV生物模型的动力学行为,其中变时滞函数导数小于1. 然而,变时滞也可能不可微[10-11],如分段时滞.本文在文献[10-11]此基础上,讨论不可微变时滞随机LV生物模型全局正解的存在性和唯一性及其他渐近性质.2 基本引理设A是一个向量或矩阵,用A T表示它的转置. 若x∈R n,则|x|表示Euclidean范数. 若A是矩阵,则|A|表示其迹范数,即|A|=trace(A T A). 若A是一个实值对称矩阵,用λmin(A)和λmax(A)分别表示其最小和最大的特征值,A≤0和A<0分别表示A半负定和负定. 设a,b是实数,则a∧b=min{a,b},a∨b=max{a,b},a+=a∨0. 令R n+={x∈R n:x i≥0, 1 ≤ i≤ n},Rˉn+= {x∈ R n:x i> 0, 1≤ i≤ n}.设(Ω,F,{F t}t≥0,P)是一个完备的概率空间,其σ代数流{F t}t≥0满足一般条件(即它是单调递增和右连续的,且F0包含所有空集). 对于h>0,用C([-h, 0];R n)表示从[-h,0]映射到R n的连续函数族,其范数为 φ=sup-h≤u≤0|φ(u)|.设ω(t)是定义在概率空间上的一维布朗运动.对任意给定的对称矩阵A∈R n×n,定义:λ+max(A)=sup x∈R n+,||x=1x T Ax,由定义可直接推出,对任意的x∈R n+,有:λ+max(A)≤0⇔x T Ax≤0,x T Ax≤λ+max(A)|x|2.对时滞函数δ(t)提出如下假设:(A1)时滞函数δ(t) :R+→[h1,h]是Borel可测函数且具有以下性质:hˉ:=lim supΔ→0(sup s≥-hμ()M s,ΔΔ)≤∞,其中,h1和h都是正的常数且h1<h,M s,Δ={t∈R+:t-δ(t)∈[s,s+Δ)},μ(⋅)表示R+上的勒贝格测度.下面两个引理起着关键作用.引理1[10-11]假设(A1)成立,设T>0且f:[-h,T-h1]→R+是一个连续函数,则:∫0T f()t-δ()t d t≤hˉ∫-h T-h1f()t d t.注:令f(t)≡1,∀t≥-h,可知hˉ≥1.引理2 (Schur补)[12]对于适当阶数的矩阵S,Q,R,其中Q=Q T,R=R T,以下条件相互等价:(1)éëêêùûúúQ SS T R<0,(2)R<0,Q-SR-1S T<0.3 主要结论定理1 假设(A1)成立,若存在正数γ,η和c1,⋯,cn使得:CˉA+A T Cˉ+4ηhˉI<0,(2)H=éëêêêêêêêêêêêêêêêùûúúúúúúúúúúúúúúúCˉA+A T Cˉ2CˉB A T0D T02B T Cˉ-ηI0B T0E TA0-6γI0000B0-6γI00D000-6Cˉ-100E000-6Cˉ-1≤0,(3)其中Cˉ=diag(c1,⋯,c n),I表示n阶单位矩阵. 则对任意给定的初值ξ∈C([-h,0];Rˉn+),方程(1)存在唯一的全局正解.证明方程(1)的系数局部Lipschitz连续,故对任意给定的初值ξ∈C([-h,0];Rˉn+),方程(1)在t∈[-h,σ∞]上存在唯一的最大的局部正解x(t),其中σ∞表示爆炸时间. 为了证明x(t)是全局的,只需证明σ∞=∞ a.s.设k0是一个充分大的正数满足:1k0<min-h≤t≤0|ξ(t)|≤max-h≤t≤0|ξ(t)|<k0.对每个满足k≥k0的整数k,定义停时:τk=inf{t∈[-h,σ∞):x i∉(1k,k)对某一i=1,⋯,n}.本文总约定inf∅=∞. 显然,当k→∞时,τk是递增的. 设τ∞=lim k→∞τk,则τ∞≤σ∞ a.s.如果τ∞=∞ a.s.,即可以推出σ∞=∞ a.s.且x(t)∈R n+ a.s.对t∈[- h,∞]恒成立. 这也等价于证明,对任意的t>0有P(τk≤t)→0,k→∞. 因此,定义一个C2函数U:R n+→R+:403第 42 卷中南民族大学学报(自然科学版)U (x )=∑i =1n c i (x i -log x i ),其中u (x )=x -log x ≥0对x >0恒成立,且u (0+)=u (∞)=∞. 由Ito公式可得:E U (x (t ∧τk))=U (ξ(0))+E ∫t ∧τk L U ()x ()s ,y ()s d s ,L U 定义为:L U (x ,y )=x T C ˉf (x ,y )-c T f (x ,y )+12||C ˉg (x ,y )||2,其中c =(c 1,⋯,c n )T. 注意到:12|||C ˉg (x ,y )|||2≤32(b T C ˉb +x T D T C ˉDx +y T E T C ˉEy ),(4)且:-c T f (x ,y )≤12γ|c |2+γ2|f (x ,y )|2≤12γ|c |2+3γ2(|a |2+x T A T Ax +y T B T By ).(5)由(4)式和(5)式可知:L U (x ,y )≤c 1+14z T H 1z +a T C ˉx +12x T C ˉAx +η|y |2,其中c 1=12γ|c |2+3γ2|a |2+32b T C ˉb ,z =(x ,y )T 且H 1=éëêêùûúúC ˉA +A T C ˉ+6γA T A +6D T C ˉD 2C ˉB 2B T C ˉ-ηI +6γB T B +6E T C ˉE =éëêêùûúúA T 0D T 00B T 0E T éëêêêêêùûúúúúú6γI6γI6Cˉ6CˉéëêêêêêêêùûúúúúúúúA00B D 00E +éëêêùûúúC ˉA +A T C ˉ2C ˉB 2B T C ˉ-ηI .由引理2可知,H 1≤0⇔H ≤0,故(3)式可得:z TH 1z ≤0,因此:L U (x ,y )≤c 1+a T Cˉx +12x T C ˉAx +η|y |2.(6)由引理1可知:η∫t ∧τk ||y ()s 2d s ≤h ˉη∫-ht ∧τk ||x ()s 2d s ≤hhˉη ξ2+hˉη∫t ∧τk ||x ()s 2d s ,(7)由(6)式和(7)式可得:E U (x (t ∧τk))=U (ξ(0))+E ∫t ∧τk éëêc 2+a T C ˉx (s )+ùûúú14x T(s )()CˉA +A T C ˉ+4ηh ˉI x (s )d s ,其中c 2=hhˉη ξ2+12γ|c |2+3γ2|a |2+32b T C ˉb . 令α=λ+max (C ˉA +A T C ˉ+4ηh ˉI ),由(2)式可知α<0,因此:E U ()x ()t ∧τk=U ()ξ()0+E ∫t ∧τk éëêc 2+a T C ˉx (s )+ùûúú14x T(s )()CˉA +A T C ˉ+4ηh ˉI x (s )d s ≤U ()ξ()0+E éëêêùûúú∫t ∧τkc 2+a T C ˉx (s )+14α||x (s )2d s ≤U ()ξ()0+c 3t ,其中c 3是一个正常数. 由τk 的定义可知,x i (τk )=k 或1k对某个i =1,⋯,n 成立,因此:P ()τk ≤t éëêêùûúúu ()1k ∧u ()k ≤P ()τk ≤t U ()x ()t ∧τk ≤E U ()x ()t ∧τk≤U ()ξ()0+c 3t ,令k →∞可得:lim k →∞P (τk ≤t )=0.证毕.定理2 假设定理1的条件成立,x (t )是方程(1)具有初值ξ∈C ([-h ,0];R ˉn +)的正解,则x (t )有如下性质:lim sup t →∞E |x (t)|≤∞,lim sup t →∞1t ∫0tE ||x ()t 2≤∞.证明 定义一个C 2函数V :R n +→R +:V (x )=∑i =1n c i x i ,由Ito 公式可得:e εt E V (x (t))=V (ξ(0))+E ∫0t e εs []L V ()x ()s ,y ()s +εV ()x ()s d s ,其中ε是一个充分小的正数使得:C ˉA +A T C ˉ+4ηh ˉe εh I <0,L V 定义为:L V (x ,y )=x T C ˉf (x ,y )≤14z T H 2z +a T Cˉx +12x T C ˉAx +η|y |2,(8)其中:404第 3 期胡军浩,等:变时滞随机Lotka -Volterra 生物模型的渐近性质H 2=éëêêùûúúC ˉA +A T C ˉ2C ˉB 2B T C ˉ-ηI ,由H ≤0可以推出H 2≤0,因此:L V (x ,y )+εV (x )≤+a T Cˉx +12x T C ˉAx +η|y |2+εc T x ,(9)由引理1可知:η∫0te εs||y ()s 2d s ≤ηe εh∫0te ε()s -δ()s||x ()s -δ()s 2d s ≤hh ˉηe εhξ2+h ˉηe εh∫0te εs||x ()s 2d s ,(10)由(9)式和(10)式可知:e εt E V ()x ()t ≤V ()ξ()0+hhˉηe εhξ2+E∫0te εs éëêê()a T C ˉ+εc T x ()s +14x T ()s (C ˉA +A T C ˉ+])4hˉηe εh I x ()s d s ≤V ()ξ()0+hh ˉηe εh ξ2+c 4∫t e εs d s ,其中c 4是一个正常数. 立得:lim sup t →∞E |x (t)|≤∞,又由Ito 公式可得:E V (x (t))=V (ξ(0))+E ∫0t L V ()x ()s ,y ()s d s ,由(7)式和(8)式可得:E V (x (t))+ε1E ∫0t|x (s )|2d s ≤V (ξ(0))+hhˉη ξ2+E éëêa TC ˉx ()s +ùûúú14x T()s ()CˉA +A T C ˉ+4()h ˉη+ε1I x ()s d s ≤V (ξ(0))+hhˉη ξ2+c 5t ,其中c 5是一个正常数,ε1是一个充分小的正数使得:CˉA +A T C ˉ+4(h ˉη+ε1)I <0,因此:lim supt →∞1t ∫t E ||x ()t2≤∞.证毕.定理3 假设定理1的条件成立. 若存在非负常数q 和r 使得以下条件成立:H 3=éëêêêêêùûúúúúúCˉA +A T C ˉC ˉB c 0B T C ˉ00c c T001r0c T 1r 2q r 2≤0,(11)q >rhˉ,(12)则方程(1)具有初值ξ∈C ([-h ,0];Rˉn +)的正解x (t )有如下性质:lim supt →∞log ||x ()tlog t≤1 a.s.证明 定义一个C 2函数V :R n+→R +:W (x )=log (c T x ),由Ito公式可得:e ε2t W (x (t))=W (ξ(0))+M (t )+∫0t e ε2s[]L W ()x ()s ,y ()s +ε2V ()x ()s d s =W (ξ(0))+M (t )+e ε2séëêêJ ()s -12||Z ()s 2+ùûúε2W ()x ()s d s ,其中:0<ε2<1h log(q rhˉ),Z =x T C ˉg c T x ,J =x T Cˉf c T x ,M (t )=∫0t e ε2sZ ()s d ω(s ),对任意给定的θ>1和k ∈N ,由指数鞅不等式可知:P ìíîsup 0≤t ≤k +1éëêêM (t )-12e ε2(k +1)∫0te 2ε2s |Z (s )|2d s ùûúú≥e ε2(k +1)log k θüýþ ≤1kθ ,级数∑k =1∞1kθ≤∞,由Borel -Cantelli 引理,对几乎所有ω∈Ω,当k 充分大且k ≤t ≤k +1时有:M (t )≤12e ε2(k +1)∫0te 2ε2s |Z (s )|2d s +e ε2(k +1)log k θ≤12∫0t e ε2s|Z (s )|2d s +θe ε2e ε2t log t ,因此:e ε2t W (x (t))≤W (ξ(0))+θe ε2eε2tlog t +∫0te ε2s[]J ()s +ε2W ()x ()s d s ,(13)由引理2和(11)式可知:H 4=éëêêùûúúC ˉA +A T C ˉ+2qM C ˉB -rM B T Cˉ-rM 0≤0,其中M =cc T , 因此对任意给定的x ,y ∈R n+有:0≥(x T ,y T )H 4()xy=x T (CˉA +A T C ˉ+2qM )x +2x T (C ˉB -rM )y =2x T(CˉAx +C ˉBy )+2q (c Tx )2-2r (c Tx )(c Ty ) ,405第 42 卷中南民族大学学报(自然科学版)即:x T (CˉAx +C ˉBy )≤r (c T x )(c T y )-q (c T x )2,因此:J =x T Cˉ()a +Ax +By c T x≤max i {a i }+rc T y -qc T x ,(14)由(13)式和(14)式可知:e ε2t W (x (t))≤W (ξ(0))+θe ε2e ε2tlog t +∫0t e ε2s[]J +ε2W ()x ()s d s ≤W (ξ(0))+θe ε2e ε2tlog t +e ε2s[max i{}a i+]rc Ty ()s -qc Tx ()s +ε2log ()c Tx d s ,由引理1可知:r∫t e ε2s c Ty (s )d s ≤reε2h∫t e ε2()s -δ()s c T x (s -δ(s))d s ≤rhhˉe ε2h |c | ξ+rh ˉe ε2h ∫0t eε2s c Tx (s )d s ,因此:e ε2tW (x (t))≤W (ξ(0))+θeε2e ε2tlog t +rhh ˉe ε2h |c | ξ+∫0te ε2s[max {aˉi}+(rhˉe ε2h -q )c Tx (s )+ε2log (c Tx )]d s ≤θe ε2e ε2tlog t +c 6(1+e ε2t) ,其中c 6是一个正的常数. 令θ→1,ε2→0,则:lim supt →∞log ||x ()t log t≤1 a.s.证毕.4 实例考虑文献[13]中的二维Lotka -volterra 系统:d x 1(t )x 1(t )=[]-8x 1(t )+x 2(t )-y 1(t )+y 2(t )d t +[]λx1(t )+λx 2(t )+μy 1(t )-μy 2(t )d ω(t ),d x 2(t )x 2(t )=[]x 1(t )-7x 2(t )+y 1(t )-y 2(t )d t +[]λx2(t )-μy 2(t )d ω(t ),其中λ和μ是非负常数,且:δ(t )=[()0.1+0.1()t -2k I [)2k ,2k +1()t +]()0.2-0.1()t -2k -1I [)2k +1,2()k +1()t ,显然,δ(t )满足假设(A1),此时,h 1=0.1,h =0.2,hˉ≈1.1. 令:A =éëêêùûúú-811-7,B =éëêùûú-111-1,D =λéëêùûú1101,E =μéëêùûú1-10-1,令c =(1 , 1)T,γ=130,计算得:A +A T +4ηh ˉI =éëêêùûúú-16+4.4η22-14+4.4η,H 1=éëêêêêêêêùûúúúúúúú-3+6λ2-1+6λ2-22-1+6λ2-4+12λ22-2-22-η+25+6μ2-25-6μ22-2-25-6μ2-η+25+12μ2,H 4=éëêêêêêêêêùûúúúúúúúú2q -162q +2-1-r 1-r 2q +22q -141-r -1-r -1-r 1-r 001-r-1-r0.应用定理1和定理2可得:A +A T +4ηh ˉI <0⇔ìíî-16+4.4η<019.36η2-132η+220>0,即η<又对任意给定的z ∈R 4+,有:z T H 1z =()-3+6λ2z 21+2()-1+6λ2z 1z 2-4z 1z 3+4z 1z 4+()-4+12λ2z22+4z 2z 3-4z 2z 4+()-η+25+6μ2z 23+2()-25-6μ2z 3z 4+()-η+25+12μ2z 24≤()-3+6λ2z 21+()-1+6λ2+()z 21+z 22+2()z 21+z 24+()-4+12λ2z 22+2()z 22+z 23+()-η+25+6μ2z 23+()-η+25+12μ2z 24=[]-1+6λ2+()-1+6λ2+z 21+[]-2+12λ2+()-1+6λ2+z 22+()-η+25+6μ2z 23+()-η+25+12μ2z 24 ,显然,当0≤μ≤η≥125,0≤λ≤时, z T H 1z ≤0,因此H ≤0.即定理1和定理2成立的条件为:406第 3 期胡军浩,等:变时滞随机Lotka -Volterra 生物模型的渐近性质ìíîïïïïïïïï0≤μ≤0≤λ≤125≤η<应用定理3,对任意给定的z ∈R 4+,有:z T H 4z =()2q -16z 21+2()2q +2z 1z 2-2()1+r z 1z 3+2()1-r z 1z 4+()2q -14z22+2()1-r z 2z3-2()1+r z 2z 4≤()2q -16z 21+()2q +2()z 21+z 22+()2q -14z 22+()1-r +()z 21+z 24+()1-r +()z 22+z 23=[]4q -14+()1-r +z 21+[]4q -12+()1-r +z 22+()1-r +()z 23+z 24.显然,当r ≥1,q ≤3时H 4≤0,故H 3≤0. 又由(12)式有q >1.1r ,即定理3成立的条件为:r ≥1,1.1r <q ≤3.注:当δ(t )退化为一个常数,即δ(t )=τ(τ>0), hˉ=1~定理3成立的条件为0≤λ≤0≤μ<r ≥1,r <q ≤3. 此时,文献[9]中的定理7也成立.参 考 文 献[1] WANG Z , BAYLISS A , VOLPERT V. Asymptoticanalysis of the bistable Lotka -Volterra competition -diffusionsystem [J ]. Appl Math Comput , 2022, 432: 127371.[2] WANG H , PAN C , OU C. Propagation dynamics offorced pulsating waves of a time periodic Lotka -Volterra competition system in a shifting habitat [J ]. J Differ Equ , 2022, 340: 359-385.[3] NIE L , MEI D. Noise and time delay : Suppressed populationexplosion of the mutualism system [J ]. Europhysics Letters , 2007, 79(2): 20005.[4] LI X , MAO X. Population dynamical behavior of non -autonomous Lotka -Volterra competitive system withrandom perturbation [J ]. Discrete Contin Dyn Syst , 2009, 24(2): 523.[5] BAHAR A , MAO X. Stochastic delay populationdynamics [J ]. Int J Pure Appl Math , 2004,11: 377-400.[6] MAO X , YUAN C , ZOU J. Stochastic differential delayequations of population dynamics [J ] J Math Anal Appl , 2005, 304(1): 296-320.[7] BAHAR A , MAO X. Stochastic delay Lotka -Volterramodel [J ]. J Math Anal Appl , 2004, 292(2): 364-380.[8] MAO X , MARION G , RENSHAW E. EnvironmentalBrownian noise suppresses explosions in population dynamics [J ]. Stoch Proc Appl , 2002, 97(1): 95-110.[9] HU Y , WU F , HUANG C. Some new results on theLotka -Volterra system with variable delay [J ]. Abstr Appl Anal , 2014,2014: 537674.[10] DONG H , MAO X. Advances in stabilization of highlynonlinear hybrid delay systems [J ] Automatica , 2022, 136: 110086.[11] HU J , MAO W , MAO X. Advances in nonlinear hybridstochastic differential delay equations : existence , boundedness and stability [J ]. Automatica , 2023, 147: 110682.[12] MAO X , YUAN C. Stochastic differential equations withMarkovian switching [M ]. London : Imperial College Press , 2006.(责编 曹东,校对 姚春娜)407。

Lotka—Volterra竞争扩散系统连接边界平衡点和正平衡点行波解的存在性

Lotka—Volterra竞争扩散系统连接边界平衡点和正平衡点行波解的存在性

Lotka—Volterra竞争扩散系统连接边界平衡点和正平衡点行波解的存在性Lotka—Volterra竞争扩散系统是一种描述生态系统中不同种群之间相互作用的数学模型,它可以用来研究物种之间的竞争、捕食和共生关系。

在生态学中,该模型在探讨种群之间的竞争、扩散和边界效应方面具有重要的应用价值。

本文将讨论关于Lotka—Volterra竞争扩散系统连接边界平衡点和正平衡点行波解的存在性。

我们来介绍一下Lotka—Volterra竞争扩散系统的基本形式。

通常情况下,该系统可以用如下的方程组来描述:\[\begin{cases}\frac{\partial u}{\partial t} = d_1\Delta u + r_1u(1-\frac{u}{K_1})-a_{12}uv \\\frac{\partial v}{\partial t} = d_2\Delta v + r_2v(1-\frac{v}{K_2})-a_{21}vu\end{cases}\]在这个方程组中,\(u\)和\(v\)分别代表两个种群的密度,\(t\)代表时间,\(d_1\)和\(d_2\)分别代表两个种群的扩散率,\(r_1\)和\(r_2\)分别代表两个种群的增长率,\(K_1\)和\(K_2\)分别代表两个种群的环境容量,\(a_{12}\)和\(a_{21}\)代表两个种群之间的竞争系数。

在这个模型中,我们可以发现扩散项对空间中种群密度的变化起着重要作用,而种群之间的相互作用则由竞争项和共生项来描述。

这种具有扩散和竞争的复杂关系使得该模型在描述生态系统中不同种群之间的相互作用时具有较强的适用性。

接下来,我们将讨论与Lotka—Volterra竞争扩散系统连接边界平衡点和正平衡点行波解的存在性。

在实际生态系统中,通常会存在一些边界以及一些适宜生存繁衍的区域,我们将通过研究连接边界平衡点和正平衡点行波解的存在性来揭示生态系统中种群的空间分布规律。

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)>IJH=?J : The overall aim of the BioSimGrid project () is to exploit the Grid infrastructure to enable comparative analysis of the distributed results of biomolecular simulations. In particular this paper presents the implementation of the current BioSimGrid Web Portal.The portal has a SOA (Service Oriented Architecture)framework built on the layer of OGSA (Open Grid Services Architecture) and OGSA-DAI (Open Grid Services Architecture Data Access and Integration) middleware. The PortalLib has been developed to allow RAD (Rapid Application Development)of portal applications. The portal also integrates PKI (Public Key Infrastructure) and supports two levels of distributed SSO (Single Sign On): Grid certificate-based SSO for high security, and user/password based SSO for maximal flexibility.Keywords: Biomolecular simulation, Open Grid Service Architecture,grid portal, BioSimGridReviewed and accepted: 31 Mar. 20041. BackgroundBiomolecular simulations enable us to explore the conformational dynamics of complex molecules such as proteins, membranes and nucleic acids (Figure. 1). Molecular simulations with atom-level resolution have now entered the mainstream of biological research [1]. In particular, molecular dynamics (MD) is widely used to investigate nanosecond to microsecond dynamics for a wide range of biomolecules. Currently, a typical such simulation generates large amount of digital data.Figure 1. Comparison of protein simulations contribute to biomedical knowledge. Mouse acetylcholinesterase (left) and bacterial outer-membrane phospholipase A (OMPLA; right) are different in structure and biological environment (top) but similar in their active site (bottom) and catalytic functionMD has benefited considerably from improvements in computer technology. As computers become faster, biologists have become able to explore larger molecules for longer timescales. Currently, a typical simulation may have a system size of ~100,000 particles (atoms),A Web / Grid Portal Implementation of BioSimGrid: A Biomolecular Simulation DatabaseBing Wu 1,2,*, Matthew Dovey 2, Muan Hong Ng 4, Kaihsu Tai 1, Stuart Murdock 3,4, Hans Fangohr 4,Steven Johnston 4, Paul Jeffreys 2, Simon Cox 4, Jonathan W. Essex 3 and Mark S.P. Sansom 11Department of Biochemistry, 2e-Science Centre, University of Oxford, 3Department of Chemistry, 4e-Science Centre, University of Southampton *to whom correspondence should be addressed: bing@and a nanosecond timescale simulation may require ~1,000,000timesteps (i.e. iterations of integrating the equations of motion). Such a simulation would take a few weeks on between ~8 and ~64processors (depending upon the efficiency of the simulation code and protocols employed) and could generate gigabytes of data for subsequent analysis and visualisation.The status quo for the archiving of these data is far from optimal.Typically, data is archived in an ad hoc fashion at the level of individual laboratories. Furthermore, the reporting of the simulation metadata in journal articles is prone to omission of potentially important technical details. Consequently, even medium-scale comparisons between multiple simulations are not possible unless the simulations are performed within a single research group. This excludes simulation results from the domain of structural bioinformatics, where new information and knowledge is derived by comparisons between the results of individual research endeavors.As protein structure determination becomes more automated, and as advances are made in structural bioinformatics [2] and computational biology, it will become increasingly important that biomolecular simulations do not exist as standalone analyses of single systems,but rather that they become embedded in a matrix of computational and experimental studies of proteins. However, at present there are considerable problems in comparing the results of multiple simulations [3], and in integrating simulation results with other (experimentally-derived) sources of data. This problem will become more pressing if simulation studies are to match up to post-genomic approaches such as high throughput protein crystallography. Furthermore, it will be essential to provide data retrieval and analysis tools that are accessible to a wide community of structural and cell biologists,not just to simulation specialists. It is in this context that the BioSimGrid [4] project is being developed.2. Grid enabling databaseThe Grid [6, 7, 8] is a combination of network infrastructure and software framework delivering computing services based on distributed hardware and software resources. Using the power of the Grid the developers hope to solve the above problems, thus enabling them to take a comparative BioSimGrid approach to analysis of simulation data.A major impediment to making various simulations available to biologists has been the absence of a database of simulation results.In an ideal world, all simulation data would be available to all interested parties. However, at present simulation data reside in the home laboratory and are not accessible to other research groups. One solution to this problem would be to deposit all simulation results in a centralised database, but in reality the vast quantities of data mean that rapid access would become impossible.The Grid provides the opportunity to draw together distributed collections of simulation data in disparate formats, whilst maintaining a centrally accessible federated database.The BioSimGrid project will establish a formal database for biomolecular simulations within the UK, increasing collaboration via a distributed computing environment. There are three levels of data existing in the database (see Figure. 2):•Raw data: generated by biomolecular simulations;•1st level metadata: describing the generic properties of raw simulation data;•2nd level metadata: describing the results of generic analyses ofsimulation data.The core database has been implemented using IBM DB2 Database Version 8.1 Enterprise [5]. Current datasets contain 6 trajectories (two of which are trajectories of the nervous-system protein AChE; the others are of the bacterial outer-membrane protein OMPLA), 217 thousand atoms, 50 thousand frames at 1 picosecond apart, and about 1.7 billion coordinates. The data size is about 60 GB. These large amount of datasets need to be easily accessed by the community. To deliver this, we have designed and implemented a web portal.A current direction of development focuses on the exploitation of Grid technologies such as OGSA-DAI [9] to enable interrogation of data through distributed queries. An application can send to BioSimGrid a query as if it were to be executed at one instance of a database, and BioSimGrid will process it to be distributed on several instances for execution.3. Portal architectures3.1. SOA - Service Oriented ArchitectureThe current methodology in developing distributed systems is Service Oriented Architecture (SOA), building upon methodologies such as Object Oriented programming, Components and Distributed Object Request Brokers. Within a SOA, systems are composed of multiple individual services located and maintained on different heterogeneous machines administered by different organizations. The key in SOA is that the component services should be loosely coupled to allow the orchestration of systems built up from component services, which should be robust against implementation changes in the underlying services [10]. T o achieve this SOAs strive towards a number of goals: 1.The services should implement a small set of simple, ubiquitousand well known interfaces which only encode generic semantics.2.The interfaces should deliver messages constrained by extensibleschema for efficiency. This allows both services and consumers to work with well defined message structures, but allowing new versions of the services to be introduced without breaking existing systems.3.The messages should be descriptive not instructive and theinterfaces should not define system behaviour. This allows internals of a service can be viewed as a “black box”. You can send a service the details of the problem to be solved and preferences, but need not dictate how the service should solve the problem.4.Service Oriented Architectures must have mechanisms for thediscovery of services matching the consumers’ requirements. There are a number of emergent technologies which can underpin SOA: REST Web Services; SOAP Web Services; and Grid Services.3.2. REST, SOAP and Grid ServicesRepresentational State Transfer (REST) works on the basic of “resources” which can be references by URIs [11]. A REST web service is limited to using HTTP interfaces (GET to obtain a representation of the resource; DELETE to remove a representation of a resource; POST to update or create a representation of a resource; PUT to create a representation of a resource). REST messages are in XML, constrained by schema definitions in XML Schema [12] or Relax NG[13].SOAP Web Services use messages encapsulated in a structure defined by the SOAP specification [14]. This adds additional information in the form of headers for message routing scenarios and mechanisms for reporting errors using faults (a style similar to exceptions in various programming languages). SOAP Web Services use Web Service Description Language (WSDL) [15] to define both the structures (again using schema languages such as XML Schema) but also messaging semantics such as whether the message is initiated by the client or the server, and what messages can be used as a response to a particular message.Grid Services are based on Web Services but provide additional semantics. In particular they add some object-oriented and REST concepts. The object-oriented concepts are the ability to inherit service definitions (portTypes in WSDL terminology) and add new messages using a multiple inheritance model and the ability to add properties (or service data elements) to Web Services. The REST concept introduced is that of creating a new representation of resource. In the Grid Services model this uses a factory model whereby a new instance of a Grid Service can be created by its corresponding factory Grid Service. Grid Services also offer an extensibility model whereby part of the structure of the message can be left undefined, but the allowed structures can be determined dynamically by querying the appropriate service data elements.3.3. OGSI and OGSA infrastructureAs Grid Services offer an object-oriented inheritance model, the need for well known interfaces with an SOA can be met by defining base Grid Services which all Grid Services inherit. These foundation services form part of OGSI (Open Grid Services Infrastructure) and. OGSA (Open Grid Services Architecture), in addition to defining the extensions to WSDL needed for inheritance and service data elements and the semantics for the factory and extensibility models, also define a core set of foundation services and their behaviour. OGSA builds on top of OGSI to define various common service definitions for essential middleware components such as logging, account management, workflow management and data access [16].The key middleware for BioSimGrid are the services defined within OGSA-DAI (Open Grid Services Architecture Data Access and Integration). This defines a set of services for accessing heterogeneous databases but with an abstraction layer so that the client can manage a table spread across multiple remote databases as if it was a table on a single local database [9].3.4. Portal infrastructureThe key concept behind portals is content syndication. Portal channel producers publish raw content (with minimal presentation information), and it is the role of the portal to aggregate content and handle the presentation and rendering of content into a form suitable for the user. In this sense portals fit well into an SOA philosophy and provide a configurable and versatile user interface allowing the integration and management of loosely coupled services.At present there are a number of emerging standards for communication between portal channels providers and consumers –JSR-168 defining Java interfaces [17] and WSRP (Web Services for Remote Portlets) defining a Web Service interface [18]. There is activity within the Global Grid Forum for an OGSA interface for portals. JST-168 and WSRP only finalized their specifications late 2003 and any OGSA based interface is still in early development and hence BioSimGrid had to develop its own portal library.4. Portal implementation4.1. SOA implementationThe current implementation of BioSimGrid Portal is based on multi-tier SOA architecture.•GUI: HTTP(S)-based web client, provides user interaction with the system. The client can be either a standard web browser or web-based application. The use of the web interface eliminates development and maintenance of client software.•Environment: This tier is a SOA front-end which handles communications between web applications and application servers. We have two environments here. The Web Portal Environment is a standard Apache/Tomcat based hosting environment to deliver web portal communications. The Python Hosting Environment is handling legacy Python applications.•Application server: This tier is dedicated to delivering data analysis and data mining services through Grid/Web services. There are also supporting services such as monitoring, transaction, andFigure 2. An overview of the BioSimGrid databasedistributed query services. The protocols used here are XML/SOAP. The application servers also deal with OGSA-DAI Grid middleware to query the database.•Database: IBM DB2 Database Version 8.1 Enterprise has been deployed as the core database. Data resources are distributed across collaborating sites. The OGSA-DAI Grid middleware enables the access of these resources transparently in a format of virtual machine. The database of an individual site can be clustered to achieve better performance and reliability.Figure 3. BioSimGrid Portal implementation4.2 Dealing with simulation dataWe have selected an initial application that both tests our methodology and also allows us to explore an important biological question. We are currently using BioSimGrid to compare simulations of two enzymes: (i) acetylcholinesterase (AChE), a key enzyme of the nervous system [19]; and (ii) outer-membrane phospholipase A (OMPLA), a bacterial enzyme involved in pathogenesis [20]. Structural data show that these two enzymes have similar active sites (a triad of amino acid sidechains involved in their catalytic mechanisms). Otherwise, the structures of the two proteins are not closely related. We are analysing simulations to compare the patterns of catalytic sidechain dynamics in these two distantly related enzymes, so as to understand the relationship between sidechain mobility and catalytic mechanism. The AChE simulations were performed at University of California, San Diego; the OMPLA simulations in Oxford. Normally, such data would never “meet” and would reside in the separate laboratories. Furthermore, the simulations were run using different programs and protocols and the data are in very different formats. Rather than re-run one (or both) of the simulations (which would consume many days of costly supercomputer time), we are using BioSimGrid to make the comparison. Thus this apparently simple test case enables us to test many aspects of the underlying methodology.4.3. PortalLib – delivering the web front-endA web portal implementation involves a large number of web pages, which are based on HTML code. Among these, a certain amount of the web pages have to be generated dynamically based on user requests. There are basically two ways of implementation: client-side scripting and server-side scripting. In the BioSimGrid Portal implementation, we have both implementations. JavaScript is currently used on the client-side as it can run on most web browsers, while Perl and Python are used on the server-side. We developed PortalLib to provide a common interface for generating dynamic web pages on the fly. PortalLib is a Perl library which consists of four main classes to use for rapid portal development:•HtmlFile - interfaces with static HTML template files to generate dynamic web pages,•HtmlComponent -a standard HTML component to generate web GUI,•Auth_lib – provides common Authentication and Authorisation interfaces,•DB_lib – provides a database common interface to DBI compatible data sources.The code in Table 1 is an example of using PortalLib to produce a trajectory refinement form based on previous user inputs and database query results. When running ‘refine.cgi’, the result data is dynamically bound to the template HTML file called ‘refine.html’. For a typical portal page, we place labels, text fields, buttons and scrolling lists in the template page and map them to the attributes in the hash list of ‘%options’. Then the attribute data will be automatically passed to the next Web page via HTTP session. Thus by using PortalLib, we have achieved rapid and efficient development of server-side web pages.#!/usr/bin/perl -wuse PortalLib;use strict;use CGI;my $cgi = new CGI;my %options = ( $cgi->Vars,);# process template HTMLmy $template = ‘refine.html’;my $html = HtmlFile->new(File => $template,%options,);$html->Draw();Table 1. Example code: refine.cgiJavaScript can also be dynamically passed to an HtmlFile object in the JavaScript property or embedded in a template file as usual HTML code. The generated form is shown in Figure 5. The trajectory list is pulled dynamically from the database and bound to the web front-end below.Figure 4. A screenshot of the BioSimGrid portal showing the active sitefor AChEFigure 5. Trajectory refinement form5. BioSimGrid securityGiven the current distributed Grid implementation, security is a critical element of the project. We integrated various mechanisms to achieve a secure distributed environment. We have implemented PKI (Public Key Infrastructure) and X.509 Digital Certificate [21] based authentication. All the transactions are based on HTTPS secure channels. The BioSimGrid network has a dual firewall protection. Two levels of authentication enable legacy user/password based authentication. The authorisation is based on a distributed database and implements SSO (Single Sign On).5.1. PKI and Grid certificatesPKI is a system of public key encryption using digital certificates from a CA (Certificate Authority) that verifies and authenticates the validity of each party involved in an electronic transaction. PKI uses an asymmetric key based algorithm: a private key is used to encrypt data and a public key can decrypt data encrypted with the private key. A certificate contains information referring to a public key which has been digitally signed by a CA. The information normally found in a certificate conforms to the ITU (IETF) standard X.509 v3 [22].A Grid certificate is an X.509 certificate. We have two kinds of Grid certificates used in the portal: user certificates for user identities and host certificates for servers. Once you have a valid user certificate, you can use it to access the web portal [24]. A host certificate has similar format as the user certificate except its DN has a hostname instead of the user name in ‘CN’.subject=/C=UK/O=eScience/OU=Oxford/L=OeSC/CN=portal/ /emailAddress=bing@The host certificate can be configured to be used by the web server supporting SSL transactions. We use the Apache server in the portal environment. The above host certificate is converted to an Apache-friendly key pair and loaded into the web server [25]. The same key pair can also be used as a host certificate in the Globus Toolkit [6].5.2. SSO (Single Sign-On) and AAA (Authentication Authorisation Accounting)SSO is designed to provide a foundation that gives users role-based access to multiple Web applications from a single, secure point of contact. This simplifies the user AA (Authentication Authorisation): the user only needs to remember one user id/password. There are various implementations of SSO technologies, i.e. 3rd party based SSO and centralised SSO.In the BioSimGrid project, we have a distributed Grid environment and need to deal with two levels of authentication for high security and easy accessibility. To achieve this, we use SSO based on a distributed database. All the user accounts and corresponding AAA information are stored in the database distributed across the network. This enables a user sign on at any BioSimGrid site to access authorised resources and perform authorised transactions. The accounting information of the user access will be stored locally and distributed over the network.As in Figure. 6, two levels of authentication infrastructures have been implemented in the system. The first level is based on digital certificate.A Grid certificate-based authentication mechanism has been integrated across the system. When a user signs on to access specific BioSimGrid services, the subject of his/her X.509 personal digital certificate is passed to the SSO Security Check module, which then calls the AAA module to verify the certificate against the one stored in the accounts database. Only if the security check is successful will the user have the access to the services. The second level is a user/ password based authentication, which is designed for those who have no digital certificates installed in their client machines. This level of authentication enables web access of the portal via a public PC from anywhere in the world.Once the user has been granted access, the AAA module will store the user credentials in the database and generate a unique access token for this session. The token has a limit life time to enhance the security. We are also investigating the integration of user credential delegation using MyProxy [26]. All transactions are logged in the accounts database, which will then be distributed across the BioSimGrid sites for maximal efficiency and high availability.6. ConclusionsThe BioSimGrid project is still in its infancy. However, the AChE vs. OMPLA comparison provides a microcosm of the many comparisons that will become possible once BioSimGrid is fully operational. Biomolecular simulation groups will be able to deposit their simulation data for wide-ranging comparative analyses that so far have been impossible. This will help to propel biomolecular simulation studies into the post-genomic era.The OGSA and OGSA-DAI architectures underlying BioSimGrid are still in the early stage of development. Therefore we have both legacy programs and new web services co-existing in the current portal environment. Once Web / Grid services are stable, it would be worthwhile to migrate all legacy programs to web services under the proposed SOA architecture.We are developing methods for data deposition, data exchange and for quality control of simulation data. This will enable us to run initial comparative analyses on multiple simulations (e.g. of membrane proteins) in order to evaluate the current prototype in real world applications.AcknowledgementsMany thanks to our BioSimGrid partners (L. Caves, C. Laughton, D. Moss and O. Smart) for their input to this project. BioSimGrid is funded by BBSRC and DTI. Our thanks to all of our colleagues in the Oxford and Southampton simulation labs, to Ivaylo Kostadinov in the Oxford e-Science Center, and to the Southampton Regional e-Science Centre for their encouragement, advice and hard work. Our thanks also to Marc Baaden for providing the OMPLA trajectories and figure1. References1. Karplus, M.J. and McCammon, J.A. (2002). Nature Structural Biology., 9, 646-652.2. Bourne, P.E. and Weissig, H. (2003). Structural Bioinformatics, Wiley-Liss, Hoboken.3. Pang, A., Arinaminpathy, Y., Sansom, M.S.P. and Biggin, P.C. (2003). FEBS Lett.550, 168-174.4. Bing Wu, Kaihsu Tai, et al. (2003). BioSimGrid: A Distributed Database for Biomolecular Simulations. Simon J Cox (editor),5. 5. Proceedings of UK e-Science All Hands Meeting 2003. EPSRC, ISBN 1-904425-11-9.5. /software/data/db2/udb/6. Figure 6. BioSimGrid SSO implementation7. Berman, F., Fox, G. and Hey, T., Eds. (2003). Grid Computing: Making the Global Infrastructure a Reality, Wiley.8. 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A molecular dynamics investigation of mono- and dimeric states of the outer membrane enzyme OMPLA. Journal of Molecular Biology 31:177-189.21. Tuecke, S., Engert, D., Foster, I., Thompson, M., Pearlman, L. and Kesselman, C. (2001). Internet X.509 Public Key Infrastructure ProxyCertificate Profile, IETF.22. ftp:///in-notes/rfc2459.txt23. /ca/24. Bing Wu (2003). User Certificate Installation Guide, http:// /docs/2003/NOTES/user_certificates.pdf25. Bing Wu (2003). Digital Certificate Installation Guide, http:// /docs/2003/NOTES/certificates.pdf26. /Divisions/ACES/MyProxy/。

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