具Ⅱ型Holling 功能性反应的捕食-被捕食系统

【作者】 周玲；

【导师】 刘祖汉； 林支桂；

【作者基本信息】 扬州大学 ， 基础数学， 2004， 硕士

【摘要】 种群生态学是生态学的一个重要分支，由于自然界中生态关系的复杂性，数学的方法和结果被越来越多地应用于生态学，种群生态学即是迄今数学在生态学中应用最为广泛深入，发展最为系统成熟的分支。早期的种群生态学重在研究局部的种群动力学，然而许多的生态过程都与物种的空间分布有关，仅考虑种群密度与时间的关系是不合理的，因此PDE的生态模型近年来倍受关注。 本论文讨论的是具Ⅱ型Holling功能性反应的捕食-被捕食模型。 第一章主要研究描述两物种的带齐次Neumann边界条件的弱耦合反应扩散系统，首先推出抛物问题解的先验估计，用上下解的方法研究了抛物问题解的存在唯一性，然后利用Lyapunov函数及局部稳定性给出了正常数解全局渐近稳定的充分条件，该条件说明，只要食饵的出生率足够大；或者捕食者的捕获率足够小；或者捕食者的内部竞争充分强，正常数解就是全局渐近稳定的。最后还证明了只要一个物种的扩散率足够大，稳态系统不存在非常数解。 第二章研究带齐次Dirichlet边界条件的强耦合椭圆系统，首先推出非线性椭圆系统解的先验估计，然后证明了当食饵和捕食者的扩散率足够大，或者出生率足够小时，系统不存在共存现象，并给出半平凡解存在的充分条件。最后利用Schauder不动点定理，得到强耦合的椭圆问题至少有一个正解存在的充分条件，该条件说明只要捕食者的内部竞争强，物种的交叉扩散相对弱，或者捕获率足够小，物种的交叉扩散相对弱，强耦合系统就至少有一个正解存在。更多还原

【Abstract】 Population ecology is an important branch of ecology science. Since the complexity of ecological relations, mathematical methods and results have been used in and have emerged from ecology. Now population ecology has become the branch that mathematics is most deeply applied in and which is the most systematic one. Early population studies concentrated on local population dynamics. However, it is not enough that populations of organisms are considered only in time. Many ecological processes that are distributed over some spaces should be considered in space. Therefore, ecological models of PDE have attracted considerable attention in recent years.In this paper, we discuss a predate-prey model with Holling type II functional response.First, the weakly coupled reaction-diffusion system describing two interacting species with homogeneous Neumann boundary conditions is studied. We derive the prior estimates for the solutions of parabolic system, and the global existence and uniqueness results of solution are given by upper and lower solutions. A sufficient condition for the global asymptotical stability is given by Lyapunov function and the local asymptotical stability. It is revealed that if the intrinsic growth rate of prey is slow, or if the capturing rate of predator is slow, or if the intra-specific competition of predator is strong enough, positive constant solution is globally asymptotically stable. Also, we show that the steady state has no non-constant positive solution if one of the diffusion rates is large enough.Second, we consider the strongly coupled elliptic system with homogeneous Dirichlet boundary conditions. The prior estimates for thesolution of nonlinear elliptic system are derived. It is shown that there is no coexistence state if diffusion rates are strong, or if the intrinsic growth rates are slow. Making use of the Schauder fixed point theory, we derive some sufficient conditions to have a coexistence state for the strongly coupled elliptic problem. Moreover, our results reveal that this problem possesses at least one coexistence state if the intra-specific competition of predator is strong and cross-diffusions are relatively weak, or if the capturing rate is slow and cross-diffusions are relatively weak.更多还原