两种群L-V捕食-食饵模型分类及相关研究
毕业论文关键词：捕食-食饵；功能反应；稳定性；Hopf分支；数值模拟
Classification and Study of Two-Species L-V Predator-Prey Models
Abstract： Ecology is an edge science which involves a wide range of disciplines, such as mathematics, biology, physics, geology, chemistry and other disciplines. According to the scope and method of the study, ecology has derived more specific branches, including mathematical ecology, population ecology and so on. Since ecological relations are often extremely complex, we need a lot of mathematical methods and conclusions to conduct in-depth research, mathematical ecology will be born, its purpose is to reveal the natural relationship between mathematics and ecology. In this paper, two species of Lotka-Volterra predator-prey model, related functional response, growth function, mortality rate are introduced. The origins of the models, the meaning of the parameter are also introduced. And then the stability of two specific models and the analysis of the bifurcation are listed. Firstly, the positive equilibrium point of the model is given, and then the stability of the positive equilibrium point is obtained by the linear approximation method, and the Hopf bifurcation condition is found. Finally, the model is numerically simulated to verify the results.
KeyWords: Predator-prey; Functional response; Stability; Hopf bifurcation; Numerical simulation
目录
摘要..I
AbstractII
1. 引言 1
1.1 课题的目的和意义 . 1
2. L-V模型分类 3
2.1 初始模型 .. 3
2.1.1 功能性反应 4
2.1.2 增长函数与死亡率 8
3. 模型分析 .. 10
3.1 预备知识 10
3.1.1 非线性驻定系统的线性近似法 .. 10
3.1.2 Hopf分支简介 .. 10
3.2 具有Gause 功能反应的模型分析 .. 11
3.3 具有Beddington-DeAngelis功能反应的模型分析 .. 13
4. 数值模拟 .. 16
4.1 模型(3)的模拟 .. 16
4.2 模型(4)的模拟 .. 18
5. 总结 . 22
致谢 23
参考文献.. 24
1. 引言 1.1 课题的目的和意义 数学生态学是数学和生态学之间交叉形成的一门边缘学科，近年来得到了巨大发展，同时也吸引了很多数学家和生态学家的关注。研究种群成长、灭亡、竞争、捕食关系的种群生态学或许是在数学上研究的最成熟的生态学领域。通过数学家和生物学家们的不懈努力，对一些复杂的种群生态学模型已累积了大量研究结果。种群生态学中建立的模型可以划分为很多类型，比如按互相影响的种群数量来分，可以分为：两种群模型、三种群模型等，按影响关系来分，又可以分为：猎食-被猎食关系、角逐关系、互补关系等。同时，对于不同的功能反应函数、出生率等函数的选取，模型也会有很大的差别。随着数学生态学的发展，将会演化出更多更为细致、复杂的模型，对它们分析研究会让我们逐步接近种群间影响关系的本质。 种群生态学中有很多的模型，此中非常典型的是 Lotka-Volterra 模型。虽然自它被提出以来，历史上各位学者对该模型有许多改进和深入的研究，但缺少相关的归纳和总结。论文网