几类生化模型的共存态和稳定性分析
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摘要
中国有句谚语：“人往高处走,水往低处流”,它揭示了事物运动具有某些共同的趋势.在自然界中,很多生物运动和化学变化也都具有同一规律.例如,从小到分子,离子,细胞,细菌运动,大到动植物生长,疾病感染,肿瘤扩散等,在微观上,它们表现为分子作无规则布朗运动,宏观上表现为物质从浓度高的地方向浓度低的地方运动,我们把这一现象称为扩散.当然,生物运动和化学变化过程中伴随着生老病死,弱肉强食,聚合分解等,我们称之为反应.为了揭示生化反应扩散过程,人们提出了大量的数学模型.应用微分方程研究生化动力系统的思想可以追溯到20世纪10-20年代Lotka-Volterra的论著或者更早,20世纪30年代Fisher将扩散引入到种群遗传动力系统中,20世纪50年代初Skellam, Turing等人又将扩散引入到种群动力系统和化学反应系统之中,20世纪60年代,Belusov等人开始深入研究化学反应中的振荡现象,20世纪70年代以后,反应扩散系统越来越受到了人们广泛地关注.
本文基于两类生化数学模型的研究现状,主要运用非线性分析和非线性偏微分方程工具,特别是反应扩散方程(组)和对应椭圆方程(组)的理论和方法,深入系统地研究了自催化反应扩散模型和具有非单调转换率的Lotka-Volterra模型的动力学行为,包括正平衡态解的存在性、多解性、稳定性以及长时行为.所涉及的数学理论包括：上下解方法、比较原理、单调动力系统理论、全局分歧理论、拓扑不动点理论、Lyapunov函数等.本文的主要内容包括以下几个方面：
第一章建立了一般形式的自催化反应扩散数学模型,详尽列举了基元化学反应模型和Lotka-Volterra模型的研究现状,介绍了以后章节所需的最大值原理、拓扑不动点理论,分歧理论等等.
第二章讨论了一类多级自催化模型,利用锥映射上的不动点指标理论给出系统存在正稳定态的条件.在齐次Dirichlet边界条件下,把转化率c作为参数,证明了当c适当小时系统没有正平衡态,当c适当大时系统至少有两个正平衡态,当c充分大系统至少有一个正平衡态.我们还决定了分歧方向以及全局分歧的性质等.
第三章考察了一类二级基元化学反应模型,在齐次Dirichlet或Robin边界条件下,利用锥映射上的不动点指标理论给出系统存在稳定态的条件,利用局部分歧讨论了分歧点附近解的性质,利用线性化理论讨论了分歧解的稳定性.利用全局分歧理论讨论解与分歧参数的依赖关系,计算了分歧的方向,讨论了参数在无穷远附近解的极限行为以及唯一性,证明了系统在一维空间非常数稳态解是唯一的.在齐次Neumann边界条件下,利用构造Lyapunov函数方法证明了系统常数平衡态解的全局稳定性条件.本章的难点在于对共存解的唯一性证明.
第四章研究了一类三级基元化学反应模型—Schnakenberg模型,在一维空间和齐次Neumann边界下,利用Hopf分歧理论给出系统存在周期解的条件,利用局部分歧讨论了系统存在’Turing分歧,利用数值模拟验证了理论结果,也进一步说明了系统是一个富动力系统.本章的突出工作在于给出了计算了分歧方向一般方法.
第五章分析了一类带有阶段结构的捕食-食饵模型,利用线性稳定性的方法分析了半平凡解,正常数解的稳定性以及长时行为,利用构造Lyapunov函数方法给出系统常数平衡态解的全局稳定性条件.利用全局稳定和能量模方法给出了不存在正稳定态的条件,在先验估计的基础之上,仔细研究了系统在正常数平衡态解附近的线性化算子的性质,利用锥映射上的不动点指标理论给出系统存在正稳定态的条件.本章难点在于正解的有界估计以及拓扑度理论的应用.
第六章分析了一类两物种竞争一种资源的竞争-竞争-捕食模型,作为一个例子,我们仔细讨论了功能函数为HollingⅡ的情形.利用线性稳定性的方法分析了半平凡解,正常数解的稳定性以及长时行为,利用构造Lyapunov函数方法给出系统常数平衡态解的全局稳定性条件.利用全局稳定,能量模方法以及隐函数的方法给出了不存在正稳定态的条件,通过巧妙构造同伦函数,利用锥映射上的不动点指标理论给出系统存在正稳定态的条件.本章难点在于隐函数定理的应用以及同伦函数的构造.
A Chiness proverb is "People goes to high place, water flows to low place", which try to discover the laws of convergence. In a assemblage of particles, for example, the individuals can be very small such as basic particles in physics, bacteria, molecules, cell and so on, and very large objects such that as animals, plants, epidemics, rumors and so on, which have a common law:In micro presentation, the particles spread out as a result of this irregular individual particle's motion; In macro presentation, it is a nature phenomenon that a substance goes from high density regions to low density region, we can think of it as a diffusion process. On the other hand, the number of particles may change because of other reasons like birth, death, hunting, or chemical reactions, e.ct. To unravel the underlying mechanisms involved in the biological or chemical process, A large class of reaction diffusion equation and system are arose. The idea of modelling the dynamics of interactions with a system of nonlinear differential equations dates bake at least to the pioneering work of Lotka and Volterra in the 1920s. The idea of using diffusion to model the spatial dispersal of population genetics was introduced by Fisher the 1930s and applied to population dynamics by Skellam and others in the early 1950s. Such a mechanism was proposed as a model for chemical basis of morphogenesis by Turing in 1952, and such systems have been widely studied since about 1970.
In the light of the recent work in these several kinds of biological models and chemical models, mainly using the theories of nonlinear analysis and nonlinear par-tial differential equations, especially those of reaction-diffusion equations and corre-sponding elliptic equations, we have systematically studied the dynamical behavior of the autocatalytic reaction diffusion model and Lotka-Volterra model with non-monotonic conversion rate, such as coexistence, multiplicity, stability of positive steady states and the longtime behavior of species. The tools used here include super-sub solutions method, comparison principle, monotone system theory, global bifurcation theory, fixed-point theory of topology, Lyapunov function technique. The main contents and results in this dissertation are as follows:
In chapter 1, the general autocatalytic reaction models are deduced, both the foreign and domestic status of research on the autocatalytic reaction diffusion model and Lotka-Volterra model are introduced and analyzed in detail, we list some ba- sic theory and classic results of reaction diffusion systems, such as the maximum principle, the fixed points index theory, bifurcation theory and so on.
In chapter 2, an autocatalytic reaction-diffusion system is investigated. We consider the coexistence states of the system by the degree and bifurcation methods, under the boundary conditions of Dirichlet type, it turns out that if the parameter c (the reaction rate) is properly small, then the system has no coexistence state, if the parameter c is suitably large, then the system has at least two coexistence states, and if the parameter c is sufficiently large, then the system has at least one coexistence state. Moreover, the bifurcation direction and global bifurcation are determined.
In chapter 3, a tri-molecular autocatalytic reaction-diffusion system with differ-ent boundary conditions is investigated. Under the boundary conditions of Dirichlet type or Robin type, the positive steady-state solutions is established by the fixed points index theory and local bifurcation theory, the stability of bifurcation solution is determined by the linearlization thoery, the bifurcation direction and global bifur-cation are studied, the uniqueness of the positive steady-state solutions in interval and parameter large enough, under the boundary conditions of Neumann type, the global stability of the positive constant equilibrium is obtained by means of Lya-punov function. The major difficulties of this section are the proof of uniqueness.
In chapter 4, a reaction diffusion system arising from Schnakenberg chemical reaction is studied. In the one dimensional and Neumann boundary conditions case, Hopf and Turing bifurcation analyses are carried out in details, examples of numerical simulation are also shown to support our approach, which implies that the reaction is a rich dynamical system. The trait of the technique is that we obtained the algorithm to determine the bifurcation director and bifurcating from a double eigenvalue.
In chapter 5, a reaction diffusion system arising from a ratio-dependent predator-prey model with stage structure is investigated. The local and global asymptotic stability of the constant equilibrium is obtained under suitable conditions by by the method of linearization and Lyapunov functions, base on priori estimate and lin-earization at positive equilibrium, existence and non-existence results of non-trivial solution are derived by means of energy method and the fixed points index theory. Emphasis is the boundedness of positive solution and the topological theory.
In chapter 6, a two-competitor/one-prey reaction diffusion system with gen-cral functional response is investigated. As an example, we consider a model with a Holling II functional response. The local and global asymptotic stability of the unique positive constant equilibrium is obtained under suitable conditions by the method of linearization and Lyapunov functions, non-existence results of non-trivial solution are derived by the energy method and implicit function theory, and exis-tence positive steady state is derived by the fixed points index theory. The major difficulties are the application of the implicit function theory and the construction of the homogeneous function.
本文基于两类生化数学模型的研究现状,主要运用非线性分析和非线性偏微分方程工具,特别是反应扩散方程(组)和对应椭圆方程(组)的理论和方法,深入系统地研究了自催化反应扩散模型和具有非单调转换率的Lotka-Volterra模型的动力学行为,包括正平衡态解的存在性、多解性、稳定性以及长时行为.所涉及的数学理论包括：上下解方法、比较原理、单调动力系统理论、全局分歧理论、拓扑不动点理论、Lyapunov函数等.本文的主要内容包括以下几个方面：
第一章建立了一般形式的自催化反应扩散数学模型,详尽列举了基元化学反应模型和Lotka-Volterra模型的研究现状,介绍了以后章节所需的最大值原理、拓扑不动点理论,分歧理论等等.
第二章讨论了一类多级自催化模型,利用锥映射上的不动点指标理论给出系统存在正稳定态的条件.在齐次Dirichlet边界条件下,把转化率c作为参数,证明了当c适当小时系统没有正平衡态,当c适当大时系统至少有两个正平衡态,当c充分大系统至少有一个正平衡态.我们还决定了分歧方向以及全局分歧的性质等.
第三章考察了一类二级基元化学反应模型,在齐次Dirichlet或Robin边界条件下,利用锥映射上的不动点指标理论给出系统存在稳定态的条件,利用局部分歧讨论了分歧点附近解的性质,利用线性化理论讨论了分歧解的稳定性.利用全局分歧理论讨论解与分歧参数的依赖关系,计算了分歧的方向,讨论了参数在无穷远附近解的极限行为以及唯一性,证明了系统在一维空间非常数稳态解是唯一的.在齐次Neumann边界条件下,利用构造Lyapunov函数方法证明了系统常数平衡态解的全局稳定性条件.本章的难点在于对共存解的唯一性证明.
第四章研究了一类三级基元化学反应模型—Schnakenberg模型,在一维空间和齐次Neumann边界下,利用Hopf分歧理论给出系统存在周期解的条件,利用局部分歧讨论了系统存在’Turing分歧,利用数值模拟验证了理论结果,也进一步说明了系统是一个富动力系统.本章的突出工作在于给出了计算了分歧方向一般方法.
第五章分析了一类带有阶段结构的捕食-食饵模型,利用线性稳定性的方法分析了半平凡解,正常数解的稳定性以及长时行为,利用构造Lyapunov函数方法给出系统常数平衡态解的全局稳定性条件.利用全局稳定和能量模方法给出了不存在正稳定态的条件,在先验估计的基础之上,仔细研究了系统在正常数平衡态解附近的线性化算子的性质,利用锥映射上的不动点指标理论给出系统存在正稳定态的条件.本章难点在于正解的有界估计以及拓扑度理论的应用.
第六章分析了一类两物种竞争一种资源的竞争-竞争-捕食模型,作为一个例子,我们仔细讨论了功能函数为HollingⅡ的情形.利用线性稳定性的方法分析了半平凡解,正常数解的稳定性以及长时行为,利用构造Lyapunov函数方法给出系统常数平衡态解的全局稳定性条件.利用全局稳定,能量模方法以及隐函数的方法给出了不存在正稳定态的条件,通过巧妙构造同伦函数,利用锥映射上的不动点指标理论给出系统存在正稳定态的条件.本章难点在于隐函数定理的应用以及同伦函数的构造.
A Chiness proverb is "People goes to high place, water flows to low place", which try to discover the laws of convergence. In a assemblage of particles, for example, the individuals can be very small such as basic particles in physics, bacteria, molecules, cell and so on, and very large objects such that as animals, plants, epidemics, rumors and so on, which have a common law:In micro presentation, the particles spread out as a result of this irregular individual particle's motion; In macro presentation, it is a nature phenomenon that a substance goes from high density regions to low density region, we can think of it as a diffusion process. On the other hand, the number of particles may change because of other reasons like birth, death, hunting, or chemical reactions, e.ct. To unravel the underlying mechanisms involved in the biological or chemical process, A large class of reaction diffusion equation and system are arose. The idea of modelling the dynamics of interactions with a system of nonlinear differential equations dates bake at least to the pioneering work of Lotka and Volterra in the 1920s. The idea of using diffusion to model the spatial dispersal of population genetics was introduced by Fisher the 1930s and applied to population dynamics by Skellam and others in the early 1950s. Such a mechanism was proposed as a model for chemical basis of morphogenesis by Turing in 1952, and such systems have been widely studied since about 1970.
In the light of the recent work in these several kinds of biological models and chemical models, mainly using the theories of nonlinear analysis and nonlinear par-tial differential equations, especially those of reaction-diffusion equations and corre-sponding elliptic equations, we have systematically studied the dynamical behavior of the autocatalytic reaction diffusion model and Lotka-Volterra model with non-monotonic conversion rate, such as coexistence, multiplicity, stability of positive steady states and the longtime behavior of species. The tools used here include super-sub solutions method, comparison principle, monotone system theory, global bifurcation theory, fixed-point theory of topology, Lyapunov function technique. The main contents and results in this dissertation are as follows:
In chapter 1, the general autocatalytic reaction models are deduced, both the foreign and domestic status of research on the autocatalytic reaction diffusion model and Lotka-Volterra model are introduced and analyzed in detail, we list some ba- sic theory and classic results of reaction diffusion systems, such as the maximum principle, the fixed points index theory, bifurcation theory and so on.
In chapter 2, an autocatalytic reaction-diffusion system is investigated. We consider the coexistence states of the system by the degree and bifurcation methods, under the boundary conditions of Dirichlet type, it turns out that if the parameter c (the reaction rate) is properly small, then the system has no coexistence state, if the parameter c is suitably large, then the system has at least two coexistence states, and if the parameter c is sufficiently large, then the system has at least one coexistence state. Moreover, the bifurcation direction and global bifurcation are determined.
In chapter 3, a tri-molecular autocatalytic reaction-diffusion system with differ-ent boundary conditions is investigated. Under the boundary conditions of Dirichlet type or Robin type, the positive steady-state solutions is established by the fixed points index theory and local bifurcation theory, the stability of bifurcation solution is determined by the linearlization thoery, the bifurcation direction and global bifur-cation are studied, the uniqueness of the positive steady-state solutions in interval and parameter large enough, under the boundary conditions of Neumann type, the global stability of the positive constant equilibrium is obtained by means of Lya-punov function. The major difficulties of this section are the proof of uniqueness.
In chapter 4, a reaction diffusion system arising from Schnakenberg chemical reaction is studied. In the one dimensional and Neumann boundary conditions case, Hopf and Turing bifurcation analyses are carried out in details, examples of numerical simulation are also shown to support our approach, which implies that the reaction is a rich dynamical system. The trait of the technique is that we obtained the algorithm to determine the bifurcation director and bifurcating from a double eigenvalue.
In chapter 5, a reaction diffusion system arising from a ratio-dependent predator-prey model with stage structure is investigated. The local and global asymptotic stability of the constant equilibrium is obtained under suitable conditions by by the method of linearization and Lyapunov functions, base on priori estimate and lin-earization at positive equilibrium, existence and non-existence results of non-trivial solution are derived by means of energy method and the fixed points index theory. Emphasis is the boundedness of positive solution and the topological theory.
In chapter 6, a two-competitor/one-prey reaction diffusion system with gen-cral functional response is investigated. As an example, we consider a model with a Holling II functional response. The local and global asymptotic stability of the unique positive constant equilibrium is obtained under suitable conditions by the method of linearization and Lyapunov functions, non-existence results of non-trivial solution are derived by the energy method and implicit function theory, and exis-tence positive steady state is derived by the fixed points index theory. The major difficulties are the application of the implicit function theory and the construction of the homogeneous function.
引文
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