摘要
通过建立生物模型,利用丰富的数学理论和方法来研究生物学中的问题是现代科技发展的重要方向之一.大量的生物模型可归纳为反应扩散方程(组).利用反应扩散方程(组)来研究这些模型也是偏微分方程研究领域中的一个重要研究方向.
本文基于反应扩散系统理论的研究及应用现状,在前人研究成果的基础上利用非线性分析、非线性偏微分方程,特别是反应扩散方程和对应的椭圆型方程的理论和方法,分别对两类生物模型：恒化器(chemostat)模型和衰减(depletion)模型,进行了深入系统的研究(包括恒化器问题正平衡态解的存在性,惟一性,稳定性及长时行为,衰减模型常数解的稳定性及非常数解的存在性),得到了一些有益的结果.所涉及的理论有上下解方法、比较原理、局部和全局分歧理论、线性算子的稳定性理论、不动点指标理论、正则化理论、Lyapunov函数、摄动理论以及数值模拟等.
下面是本文的结构和主要内容：
第一章介绍模型背景和文中将要用到的反应扩散系统研究领域的一些基本理论及经典结果,包括特征值问题、不动点指标理论、分歧理论等,这些理论及结果是以后各章内容能够得以进行的基础.
第二章研究了带有B-D型功能反应函数的非均匀恒化器竞争模型.首先,利用全局分歧理论得到由半平凡解产生分歧的全局结构.结果表明,在一定条件下由半平凡解产生的分歧解支在某点会与另一半平凡解相连.然后,利用抛物型方程的比较原理、正则化理论及Lyapunov函数,研究了该模型解的渐近行为,得到其极限系统全局吸引子存在的一个充分条件.最后,利用不动点指标理论、摄动理论,重点分析了物种υ的种内竞争参数β1对模型正平衡态解的影响.结果表明当β1很大时,如果物种υ的生长率满足一定条件,则此模型的所有正解由一个极限问题所决定.特别地,当υ.υ的生长率适当大时,模型存在惟一正解,且该正解非退化线性稳定.
第三章讨论了一类带有齐次Neumann边界条件的活化基质系统——生物衰减模型(未补充活化剂).主要对其平衡态问题进行定性分析和数值模拟.首先,运用最大值原理、能量积分的方法建立了解的比较精确的先验估计,并分析非常数正解的不存在性.结果表明当活化剂的扩散率d较大时,平衡态问题不存在非常数正解.其次,利用线性算子的稳定性理论详细讨论了系统常数解的稳定性.结果表明当d较小时,系统会产生Turing不稳定现象.然后,在一维情形以d为分歧参数,利用局部和全局分歧理论详细分析了非常数解集的全局分歧结构.指出在空间为一维时,系统发自正常数平衡解处的分歧解支一定是关于υ延伸至无穷远的,同时也说明了,当d适当小时,系统存在非常数正解.这一结果进一步表明在一定条件下扩散导致模式生成.最后,通过大量的数值模拟来验证和补充之前的理论结果.
第四章继续考虑在反应中活化剂以常数率被补充的衰减模型.这时,常数平衡态的复杂性导致了理论分析具有一定的难度.为此,本章首先讨论常数平衡态与系统中参数之间的关系,然后通过对常数平衡态的稳定性及其产生的分歧进行总的分析,得到每种具体的常数解的稳定性和分歧结构.最后通过数值模拟来验证理论部分.
Researching biological phenomena by establishing biological models and using rich mathematics theories and methods now becomes an important aspect of modern science and technology development. A large number of biological models can be summarized as reaction diffusion equations, so it is a crucial research aspect in the region of partial differential equations for us to investigate these models by reaction diffusion equations.
In the light of the current researches and applications of the theoretics of re-action diffusion systems, based on sonic pioneering works, using the theories of nonlinear analysis and nonlinear partial differential equations, especially those of reaction diffusion equations and corresponding elliptic equations, we have system-atically studied two biological models:chemostat model and depletion model. By studying the coexistence, uniqueness, stability of positive steady states and the longtime behavior of species for the chemostat model, stability of positive constant steady states and existence of non-constant steady states for the depletion model, some valuable results are obtained. The tools used here include super-sub solutions method, comparison principle, local and global bifurcation theories, stability the-ory, fixed point index theory. regularity theorem, Lyapunov function, perturbation technique and numerical simulation.
The structure and contains of this paper are as follows.
In chapter 1, we introduce the background of models and some classical results of reaction diffusion systems, such as eigenvalue problems, fixed point index theory, bifurcation theories and so on. These are the basic parts that will be very useful in the forthcoming chapters.
In chapter 2. an unstirred chemostat model with B-D functional response is studied. Firstly, we obtain the global structure of this system by global bifurcation theory. It turns out that the bifurcation curve bifurcating from one of the semi-trivial equilibria can finally meet the other at some point in certain condition. Secondly, by the means of comparison principle, regularity theorem and Lyapunov function. the asymptotic behavior of solutions of the system is investigated, and we obtain a sufficient condition of a global attractor for the limit of the system. Finally, the effect of the parameterβ1 in the B-D functional response which models mutual interference between speciesμis considered carefully by making use of the fixed point index theory and perturbation technique. The result shows that ifβ1 is sufficiently large, the solution of this model is determined by a limiting equation when the growth rate of the specialμlies in certain range. Especially, when the growth rates ofμ,τare suitable large, this model has a unique positive steady solution which is non-degenerate and linear stable.
In chapter 3, an activator-substrate system with homogeneous Neumann bound-ary condition—a biological depletion model is discussed. For the case that no activator is supplied to the system, we mainly analyze the steady-state problem qualitatively and numerically. Firstly, we establish the fine apriori estimate for positive solutions and the non-existence of non-constant positive solution by the maximum principle and energy integral method. The result shows that the system has no any non-constant positive solution when the diffusion rate d of activator is large. Secondly, the stability of positive constant solutions is discussed in detail by means of stability theory. It turns out that there is Turing instable occurring when d is small. Thirdly, in the one dimensional case, we regard d as the bifurcation parameter to make a detailed description for the global bifurcation structure of the set of the non-constant solutions using bifurcation theory. The results indicate that if d is properly small, the bifurcation curve of the system from positive constant solutions finally reach infinity with respect toμ, and the system has at least one non-constant positive solution, which say that diffusion can create pattern forma-tion. Some results on extensive numerical studies are reported in the last confirming and complementing the previous results.
In chapter 4, we continue to consider the depletion model, in which the activator is supplied at a constant rate. In this case, the constant steady states are complicated which leads to the difficulty of theoretical analysis. So we first establish the relation between constant steady states and parameters of the system in detail. Then, by a general discussion for the stability of constant steady states and their bifurcation, we get the stability and the bifurcation structure for the concrete constant steady state. Finally, the predictions from linear theory are confirmed through extensive numerical simulations.
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