具有离散尺度结构的非线性种群模型的长期行为

发布时间：2018-01-16 21:47

本文关键词：具有离散尺度结构的非线性种群模型的长期行为　出处：《杭州电子科技大学》2015年硕士论文　论文类型：学位论文

【摘要】：在研究生物种群的长期演化行为以及最优调控问题的时候，往往都会基于一定的假设，建立相应的生物种群数学模型。这样一来，就把种群问题的研究转化为数学问题分析。应用较为完善的数学工具，解决的问题就更为广泛。这种研究手段能充分发挥数学理论库的优势。另外，某些预测结构是现场分析和实验研究所得不到的结论。所以，数学建模在分析生物动力学中发挥着不可忽略的作用。相比连续结构模型，离散结构种群模型更为自然，更为切合生态系统和数据记录实际情况。自1945年H. Leslie提出一类非常重要的离散种群模型以来，很多学者把注意力转移到离散生物种群模型领域。研究离散结构的种群模型一方面可以分析种群的长期演化行为，另一方面可以根据种群的变化规律，制定科学的资源开发管理办法，比如怎样捕捞、怎样砍伐才不会影响资源的可持续发展，又能够得到最佳的经济利益。本文主要讨论三类离散尺度结构的种群模型，分别是具有线性形式、非线性形式的矩阵模型和斑块模型。前者重点研究模型的平衡态存在性、稳定性条件等，后者主要讨论了模型解的有界性和种群长期演化行为。应用矩阵理论、数值分析等工具，，得到一些新的结论，为实际应用提供了可靠的理论依据。第二、三章主要讨论线性形式、非线性形式下的矩阵模型。第二章提出一类广义Leslie模型，从不同的角度，应用不同的方法分析了平衡态及其稳定性等问题。第三章引进一类较为典型的非线性繁殖力函数，体现种群内部的个体竞争或密度制约，应用矩阵特理论等知识，得到了平衡态的存在性和稳定性条件。最后给出具体实例，用Matlab等软件进行了数值模拟，展示了种群长期的演化发展趋势。第四章考虑的是同一种群个体生活在两个有通道连接的斑块环境中，每个斑块中的种群个体按照尺度分为三个小组，其中第一小组无繁殖能力。该模型在斑块间考虑扩散情形，在同一斑块内的各组个体考虑正常生长、迟滞生长和跨组生长等，证明了种群分布的有界性，给出了种群零平衡态存在的条件。
[Abstract]:When studying the long-term evolutionary behavior and optimal regulation of biological population, the mathematical model of biological population is usually established based on certain assumptions. The research on population problem is transformed into mathematical problem analysis, and the more perfect mathematical tools are used to solve the problem more widely. This research method can give full play to the advantage of mathematical theory database. In addition, this method can give full play to the advantages of mathematical theory database. Some prediction structures are not available in field analysis and experimental research. Therefore, mathematical modeling plays an important role in the analysis of biodynamics. The discrete structure population model is more natural and more suitable for the actual situation of ecosystem and data recording. In 1945, H. Leslie proposed a very important discrete population model. Many scholars have turned their attention to the field of discrete biological population model. On the one hand, the study of discrete structure population model can analyze the long-term evolution behavior of the population, on the other hand, it can be based on the law of population change. Scientific methods of resource development and management, such as how to catch, how to cut down, will not affect the sustainable development of resources, and can obtain the best economic benefits. In this paper, we mainly discuss three kinds of population models with discrete scale structure, which are matrix model with linear form, nonlinear form and patch model. The former focuses on the existence of equilibrium state and stability conditions of the model. The latter mainly discusses the boundedness of the model solution and the long-term evolution behavior of the population. By using matrix theory and numerical analysis, some new conclusions are obtained, which provide a reliable theoretical basis for practical application. In the second and third chapters, we mainly discuss the matrix model in the linear form and the nonlinear form. In the second chapter, we propose a kind of generalized Leslie model from different angles. Different methods are used to analyze the equilibrium state and its stability. Chapter three introduces a class of typical nonlinear fecundity functions to reflect the individual competition or density constraints within the population. The existence and stability conditions of equilibrium state are obtained by using the knowledge of matrix special theory. Finally, a concrete example is given, and numerical simulation is carried out by using Matlab and other software. The long-term evolution trend of the population is shown. Chapter 4th considers that the individuals of the same species live in two patch environments with channels connected, and the individuals in each patch are divided into three groups according to the scale. The first group has no reproductive ability. The model considers diffusion among patches, and individuals in the same patch consider normal growth, hysteresis growth and cross-group growth, which proves the boundedness of population distribution. The conditions for the existence of zero equilibrium state of population are given.
【学位授予单位】：杭州电子科技大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175

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