兰州大学机构库 >数学与统计学院
微分方程理论在生物学中的应用浅谈
Alternative TitleAPPLICATION OF DIFFERENTIAL EQUATION IN BIOLOGY
郑春玮
Thesis Advisor马智慧
2018-05-12
Degree Grantor兰州大学
Place of Conferral兰州
Degree Name学士
Keyword捕食者—食饵系统 平衡点 全局渐进稳定性 最优收获
Abstract

生态系统的合理开发不仅与资源的可持续发展息息相关,而且与社会的经济发展紧密关联。在生态系统中,对生物种群的研究就是利用数学模型来描述种群的生长特性,而作为生物学重要分支的种群生态学,它的主要研究对象是捕食者—食饵系统。研究方法主要是通过建立相应的数学模型,进行定性分析,预测种群的发展变化规律以及研究人类的捕获行为对种群密度的影响,进一步生态系统的合理开发不仅与资源的可持续发展息息相关,而且与社会的经济发展紧密关联。在生态系统中,对生物种群的研究就是利用数学模型来描述种群的生长特性,而作为生物学重要分支的种群生态学,它的主要研究对象是
捕食者—食饵系统。研究方法主要是通过建立相应的数学模型,进行定性分析,预测种群的发展变化规律以及研究人类的捕获行为对种群密度的影响,进一步确立正确的捕获策略,使生态系统保持平稳发展的同时收获最大的经济效益。
本论文主要根据生态系统中两种功能反应,建立了相应的捕食者—食饵系统,并分四个章节讨论了在功能反应函数分别为Holling II 型Holling III 型时,捕获函数分别为比例依赖型和非线性型时相应的捕食者—食饵系统的稳定性。
首先本论文基于各个捕食—食饵系统的所有可能的平衡点,重点讨论了正平衡点的存在性。然后通过计算相应捕食者—食饵系统的雅可比矩阵,根据其特征方程的特征根的情况,分别判断了平衡点的类型及其局部渐进稳定性。进一步通过构造适当的Lyapunov 函数,分析了正平衡点的全局渐进稳定性。最后根据最优控制理论,运用Hamilton 函数讨论了最优捕获策略,并解释了生物学意义。

Other Abstract

The rational development of the ecosystem is not only related to the sustainable development of resources, but also to the economic development of society. In the ecosystem, the study of biological population is to use mathematical model to describe the growth characteristics of the population, and as an important branch of biology, its main research object is the predator-prey model. The method of research is mainly based on the establishment of corresponding mathematical models to conduct qualitative analysis, predict the development of the population and study the influence of
human’s capture behavior on population density, further establish the correct capture strategy, so as to maintain the balanced and stable development of the ecosystem and reap the greatest economic benefits.

Based on the two functional responses in the ecosystem, this paper establishes the corresponding predator-prey system, and discusses the stability of the predator-prey system when the function response function is Holling II and Holling III, respectively, and the capture function is proportional and non-linear.
The paper first discusses the existence of a positive equilibrium point based on all possible balance points of each predator – prey system. Then by calculating the Jacobian matrix of the corresponding predator-prey system, the type of equilibrium and its local asymptotic stability are determined according to the characteristic root of the
characteristic equation. Further, by constructing suitable Lyapunov function, the global asymptotic stability of positive equilibrium are analyzed. Finally, according to the optimal control theory, the optimal acquisition strategy is discussed by using the Hamilton function, and the biological significance is explained.

URL查看原文
Language中文
Document Type学位论文
Identifierhttp://ir.lzu.edu.cn/handle/262010/224602
Collection数学与统计学院
Recommended Citation
GB/T 7714
郑春玮. 微分方程理论在生物学中的应用浅谈[D]. 兰州. 兰州大学,2018.
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