非齐性环境下捕食猎物模型半平凡解的稳定性及正解的存在性

发布时间：2018-04-19 23:23

本文选题：捕食-猎物模型 + 分歧理论　；参考：《华东师范大学》2017年硕士论文

【摘要】：本文主要考虑在空间非齐性环境下,研究带有Holling-Tanner功能反应函数的捕食猎物模型的动力行为.即在平衡态下研究该模型半平凡解的稳定性和正解的存在性.本文研究中涉及到上下解方法、极值原理、Fredholm二择一定理、分歧理论以及不动点指标理论等.本文共有四章内容:第一章简单介绍捕食猎物模型的国内外研究情况、本文所要研究的内容以及本文的结构安排.第二章主要介绍研究本文所需要的相关概念、基本定理及结论.第三章利用特征值理论研究该模型在半平凡解处的稳定性.第四章将从两个方面对该模型的正解存在性展开研究,首先,获取该模型正解的先验估计与存在正解的必要条件,此处主要利用了极值原理和构造上下解方法两方面的理论;然后,利用局部分歧理论推导出该模型存在正解的充分条件;最后,通过全局分歧理论与不动点指标理论对该模型进行分析,进一步将该模型在半平凡解处产生的局部分歧延拓到全局分歧.
[Abstract]:In this paper, the dynamic behavior of predator-prey model with Holling-Tanner functional response function is studied in a spatially inhomogeneous environment. The stability and the existence of positive solutions of the semi-trivial solutions of the model are studied in equilibrium state. In this paper, the method of upper and lower solutions, the principle of extreme value and Fredholm's alternative theorem, the theory of bifurcation and the theory of fixed point index are discussed. There are four chapters in this paper: the first chapter briefly introduces the research situation of prey model at home and abroad, the content of this paper and the structure of this paper. The second chapter mainly introduces the related concepts, basic theorems and conclusions needed to study this paper. In chapter 3, the stability of the model at the semi-trivial solution is studied by using eigenvalue theory. In chapter 4, the existence of positive solution of the model is studied from two aspects. Firstly, the priori estimate and necessary condition of the positive solution of the model are obtained. In this chapter, the theory of extreme value principle and the method of constructing upper and lower solutions are mainly used. Then, the sufficient conditions for the existence of positive solutions of the model are derived by using the local bifurcation theory. Finally, the global bifurcation theory and the fixed point index theory are used to analyze the model. Furthermore, the local bifurcation generated by the model at the semi-trivial solution is extended to the global bifurcation.
【学位授予单位】：华东师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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