种群模型的动力学分析与吸引域估计
发布时间：2018-09-16 21:40
【摘要】：种群是在一定空间范围内同时生活着的同种个体的集合.种群动力学是对种群生态进行定量研究的一门重要学科,通过研究种群模型的动力学性态并利用数值模拟手段,分析种群的发展过程,揭示种群生存规律,预测其变化发展趋势,可以为种群的保护和利用提供重要的理论依据和数量依据.另一方面,近年来吸引域在预防物种灭绝、控制疾病传播和维持生态平衡等诸多领域的广泛应用,引起了众多学者的广泛关注,成为当下的热点研究领域之一.本文在分析和总结两类种群模型研究现状的基础上,根据稳定性理论、分岔理论、流形理论等分析方法,分别对两类种群模型的动力学行为和吸引域进行研究.本文的组织如下:第一章概述浮游生物植化相克模型与肿瘤免疫模型的研究背景、研究意义与研究现状,并且阐述本文的主要内容和创新点.第二章给出一些预备知识.第三章研究一类浮游生物植化相克模型,并分析种群繁殖率,种间竞争率和毒素抑制率对系统平衡点的吸引域的影响.通过数值仿真发现,如果某一种群的竞争力越大,则该种群的吸引域越大,从而该种群生存的可能性越大.种群的繁殖率对种群的吸引域也有一定的影响,繁殖率越高的种群生存的机会越多.此外,某一种群对另一种群的毒素抑制率越高,则该种群的吸引域越大,这说明该种群生存的机率越高.第四章讨论具有时滞肿瘤免疫模型,研究系统的稳定性、分岔和吸引域问题.首先,选取时滞为分岔参数,通过特征根方法讨论了平衡点的局部稳定和分岔问题.其次,构建一个合适的李雅普诺夫函数,利用SOS(平方和)方法估计出稳定平衡点的吸引域.最后,数值模拟验证了理论分析的正确性并分别估计出边界平衡点和正平衡点的吸引域.第五章总结全文的工作,并对今后的工作进行展望.
[Abstract]:A population is a collection of the same kind of individuals living simultaneously in a certain space. Population dynamics is an important subject in the quantitative study of population ecology. By studying the dynamics of population model and using numerical simulation, the development process of population is analyzed, and the law of population survival is revealed. The prediction of its changing trend can provide an important theoretical and quantitative basis for the protection and utilization of the population. On the other hand, the wide application of attraction domain in many fields, such as preventing species extinction, controlling the spread of disease and maintaining ecological balance in recent years, has attracted the attention of many scholars and become one of the hot research fields. On the basis of analyzing and summarizing the present situation of the two kinds of population models, according to the stability theory, bifurcation theory, manifold theory and other analytical methods, the dynamic behavior and attraction region of the two kinds of population models are studied respectively. The organization of this paper is as follows: in the first chapter, the background, significance and research status of phytoplankton culture model and tumor immune model are summarized, and the main contents and innovations of this paper are described. The second chapter gives some preparatory knowledge. In chapter 3, a phytoplankton model is studied, and the effects of population reproduction rate, interspecific competition rate and toxin inhibition rate on the attractive region of the equilibrium point of the system are analyzed. It is found by numerical simulation that the more competitive a population is, the greater the region of attraction of the population is, and the more likely the population is to survive. The population reproduction rate also has a certain influence on the population attraction region, and the higher the population reproduction rate is, the more chances it is to survive. In addition, the higher the inhibition rate of toxin on another population is, the greater the attractive region of the population is, which means that the higher the survival probability of the population is. In chapter 4, we discuss the tumor immune model with delay, and study the stability, bifurcation and attraction domain of the system. Firstly, the local stability and bifurcation of the equilibrium point are discussed by means of the eigenvalue method with time delay as the bifurcation parameter. Secondly, a suitable Lyapunov function is constructed and the attractive region of the stable equilibrium point is estimated by using the SOS (square sum) method. Finally, the numerical simulation verifies the correctness of the theoretical analysis and estimates the attraction regions of the boundary equilibrium points and the positive equilibrium points, respectively. Chapter V summarizes the work of the full text and prospects for future work.
【学位授予单位】：南京航空航天大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175
本文编号：2244892
[Abstract]:A population is a collection of the same kind of individuals living simultaneously in a certain space. Population dynamics is an important subject in the quantitative study of population ecology. By studying the dynamics of population model and using numerical simulation, the development process of population is analyzed, and the law of population survival is revealed. The prediction of its changing trend can provide an important theoretical and quantitative basis for the protection and utilization of the population. On the other hand, the wide application of attraction domain in many fields, such as preventing species extinction, controlling the spread of disease and maintaining ecological balance in recent years, has attracted the attention of many scholars and become one of the hot research fields. On the basis of analyzing and summarizing the present situation of the two kinds of population models, according to the stability theory, bifurcation theory, manifold theory and other analytical methods, the dynamic behavior and attraction region of the two kinds of population models are studied respectively. The organization of this paper is as follows: in the first chapter, the background, significance and research status of phytoplankton culture model and tumor immune model are summarized, and the main contents and innovations of this paper are described. The second chapter gives some preparatory knowledge. In chapter 3, a phytoplankton model is studied, and the effects of population reproduction rate, interspecific competition rate and toxin inhibition rate on the attractive region of the equilibrium point of the system are analyzed. It is found by numerical simulation that the more competitive a population is, the greater the region of attraction of the population is, and the more likely the population is to survive. The population reproduction rate also has a certain influence on the population attraction region, and the higher the population reproduction rate is, the more chances it is to survive. In addition, the higher the inhibition rate of toxin on another population is, the greater the attractive region of the population is, which means that the higher the survival probability of the population is. In chapter 4, we discuss the tumor immune model with delay, and study the stability, bifurcation and attraction domain of the system. Firstly, the local stability and bifurcation of the equilibrium point are discussed by means of the eigenvalue method with time delay as the bifurcation parameter. Secondly, a suitable Lyapunov function is constructed and the attractive region of the stable equilibrium point is estimated by using the SOS (square sum) method. Finally, the numerical simulation verifies the correctness of the theoretical analysis and estimates the attraction regions of the boundary equilibrium points and the positive equilibrium points, respectively. Chapter V summarizes the work of the full text and prospects for future work.
【学位授予单位】：南京航空航天大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175
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