種群模型的動力學分析與吸引域估計
發(fā)布時間：2018-09-16 21:40
【摘要】：種群是在一定空間范圍內(nèi)同時生活著的同種個體的集合.種群動力學是對種群生態(tài)進行定量研究的一門重要學科,通過研究種群模型的動力學性態(tài)并利用數(shù)值模擬手段,分析種群的發(fā)展過程,揭示種群生存規(guī)律,預測其變化發(fā)展趨勢,可以為種群的保護和利用提供重要的理論依據(jù)和數(shù)量依據(jù).另一方面,近年來吸引域在預防物種滅絕、控制疾病傳播和維持生態(tài)平衡等諸多領(lǐng)域的廣泛應用,引起了眾多學者的廣泛關(guān)注,成為當下的熱點研究領(lǐng)域之一.本文在分析和總結(jié)兩類種群模型研究現(xiàn)狀的基礎(chǔ)上,根據(jù)穩(wěn)定性理論、分岔理論、流形理論等分析方法,分別對兩類種群模型的動力學行為和吸引域進行研究.本文的組織如下:第一章概述浮游生物植化相克模型與腫瘤免疫模型的研究背景、研究意義與研究現(xiàn)狀,并且闡述本文的主要內(nèi)容和創(chuàng)新點.第二章給出一些預備知識.第三章研究一類浮游生物植化相克模型,并分析種群繁殖率,種間競爭率和毒素抑制率對系統(tǒng)平衡點的吸引域的影響.通過數(shù)值仿真發(fā)現(xiàn),如果某一種群的競爭力越大,則該種群的吸引域越大,從而該種群生存的可能性越大.種群的繁殖率對種群的吸引域也有一定的影響,繁殖率越高的種群生存的機會越多.此外,某一種群對另一種群的毒素抑制率越高,則該種群的吸引域越大,這說明該種群生存的機率越高.第四章討論具有時滯腫瘤免疫模型,研究系統(tǒng)的穩(wěn)定性、分岔和吸引域問題.首先,選取時滯為分岔參數(shù),通過特征根方法討論了平衡點的局部穩(wěn)定和分岔問題.其次,構(gòu)建一個合適的李雅普諾夫函數(shù),利用SOS(平方和)方法估計出穩(wěn)定平衡點的吸引域.最后,數(shù)值模擬驗證了理論分析的正確性并分別估計出邊界平衡點和正平衡點的吸引域.第五章總結(jié)全文的工作,并對今后的工作進行展望.
[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.
【學位授予單位】：南京航空航天大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O175
本文編號：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.
【學位授予單位】：南京航空航天大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O175
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