二类反应扩散问题解的性质

【作者】 甘文珍；

【导师】 林支桂；

【作者基本信息】 扬州大学 ， 基础数学， 2006， 硕士

【摘要】 在数学生态学领域,用数学的方法研究传染病动力学和种群生态学很受重视。自Kermack和Mckendrick于1927年构造了经典的SIR仓室模型以来,传染病模型一直被广泛研究,是现在研究的热点之一。与此同时,种群生态学也倍受关注,它是迄今数学在生态学中应用最为广泛和深入、发展得最为系统和成熟的分支。但传染病动力学和种群生态学中的大多数模型是用常微分方程表示的,它们只考虑了时间对种群密度的影响。实际上,在某个区域内,为了获得充足的食物,种群本身有从密度高的地方向密度低的地方迁移的自扩散性质,为了抵御疾病和天敌的侵袭,种群还有交叉扩散的性质。因此,我们用反应扩散方程组来表示这些模型就更为合理。本论文主要讨论两类问题:一、具时滞和扩散的传染病模型;二、带Dirichlet边界条件的强耦合椭圆问题。第一部分,研究了一类具时滞和扩散的传染病模型,这一模型数学上可表示为一类非线性弱耦合反应扩散方程组。我们重点研究了该方程组解的定性性质。首先推出抛物问题解的正性及小初值条件下的有界性,然后利用线性化和特征值方法讨论了无病平衡点和染病平衡点局部稳定性,最后利用Liapunov函数和局部稳定性给出了无病平衡点在小初值条件下渐近稳定的充分条件。我们的结果表明,在小初值条件下,当接触率小的时候,该问题的无病平衡点是渐近稳定的。第二部分,研究了带Dirichlet边界条件的竞争共栖模型的强耦合椭圆问题。首先介绍一般的强耦合椭圆问题解的情况,然后利用上下解和单调迭代的方法结合Schauder不动点定理给出竞争共栖模型的强耦合问题有共存解的充分条件。该条件说明,当交错扩散和种间作用相对弱时,强耦合系统就至少有一个正解存在。更多还原

【Abstract】 In the field of mathematical ecology, it is important to use mathematics to study epidemic dynamics and population ecology. Since the pioneering work of Kermack-Mekendrick (1927) on SIR, epidemiological models have received much attention from scientist and have been an important aspect in the field of applied mathematics. At the same time, population ecology has attracted conside- rable attention. It has become the branch that mathematic is the most deeply and extensively applied in ecology and which is the most systematic and matured one. However, most models of epidemic dynamics and population ecology are ordina- ry models where time is only considered as the effect on population density. In fact, to gain abundant food in some spatial domain, population move from high density region to low density region. In order to defend the disease and rivals, population have the crossing diffusion. So, it is reasonable for us to make use of reaction-diffusion equations to show these models.In this presentation, we discuss two problems. one is an epidemic model with delay and diffusion, the other is a strongly coupled elliptic system with Dirichlet boundary conditions.The first part is devoted to an epidemic model with time delay and diffusion. It can be described mathematically by nonlinear reaction-diffusion equations. Its qualitative properties are studied. Firstly, we derive the positive and prior estima- tes for the solutions of the parabolic system if the initial values are in some range. And then the local asymptotical stabilities are given by linearization and eigenva- lue. Finally, the asymptotic stabilities are investigated using the method of Liapunov functions. Our results show that the disease-free equilibrium is asympt- otically stable if the contact rate is small and the initial values are in some range.The second part is devoted to a strongly coupled elliptic system of competitor -competitor-mutualist model with Dirichlet boundary conditions. On one hand,更多还原