本文關(guān)鍵詞：具有離散尺度結(jié)構(gòu)的非線性種群模型的長期行為　出處：《杭州電子科技大學(xué)》2015年碩士論文　論文類型：學(xué)位論文

  更多相關(guān)文章： 生物種群 尺度結(jié)構(gòu) 差分方程組 平衡態(tài) 穩(wěn)定性 本原矩陣 圓盤定理 極限集

【摘要】：在研究生物種群的長期演化行為以及最優(yōu)調(diào)控問題的時候，往往都會基于一定的假設(shè)，建立相應(yīng)的生物種群數(shù)學(xué)模型。這樣一來，就把種群問題的研究轉(zhuǎn)化為數(shù)學(xué)問題分析。應(yīng)用較為完善的數(shù)學(xué)工具，解決的問題就更為廣泛。這種研究手段能充分發(fā)揮數(shù)學(xué)理論庫的優(yōu)勢。另外，某些預(yù)測結(jié)構(gòu)是現(xiàn)場分析和實(shí)驗(yàn)研究所得不到的結(jié)論。所以，數(shù)學(xué)建模在分析生物動力學(xué)中發(fā)揮著不可忽略的作用。相比連續(xù)結(jié)構(gòu)模型，離散結(jié)構(gòu)種群模型更為自然，更為切合生態(tài)系統(tǒng)和數(shù)據(jù)記錄實(shí)際情況。自1945年H. Leslie提出一類非常重要的離散種群模型以來，很多學(xué)者把注意力轉(zhuǎn)移到離散生物種群模型領(lǐng)域。研究離散結(jié)構(gòu)的種群模型一方面可以分析種群的長期演化行為，另一方面可以根據(jù)種群的變化規(guī)律，制定科學(xué)的資源開發(fā)管理辦法，比如怎樣捕撈、怎樣砍伐才不會影響資源的可持續(xù)發(fā)展，又能夠得到最佳的經(jīng)濟(jì)利益。 本文主要討論三類離散尺度結(jié)構(gòu)的種群模型，分別是具有線性形式、非線性形式的矩陣模型和斑塊模型。前者重點(diǎn)研究模型的平衡態(tài)存在性、穩(wěn)定性條件等，后者主要討論了模型解的有界性和種群長期演化行為。應(yīng)用矩陣?yán)碚�、�?shù)值分析等工具，，得到一些新的結(jié)論，為實(shí)際應(yīng)用提供了可靠的理論依據(jù)。 第二、三章主要討論線性形式、非線性形式下的矩陣模型。第二章提出一類廣義Leslie模型，從不同的角度，應(yīng)用不同的方法分析了平衡態(tài)及其穩(wěn)定性等問題。第三章引進(jìn)一類較為典型的非線性繁殖力函數(shù)，體現(xiàn)種群內(nèi)部的個體競爭或密度制約，應(yīng)用矩陣特理論等知識，得到了平衡態(tài)的存在性和穩(wěn)定性條件。最后給出具體實(shí)例，用Matlab等軟件進(jìn)行了數(shù)值模擬，展示了種群長期的演化發(fā)展趨勢。 第四章考慮的是同一種群個體生活在兩個有通道連接的斑塊環(huán)境中，每個斑塊中的種群個體按照尺度分為三個小組，其中第一小組無繁殖能力。該模型在斑塊間考慮擴(kuò)散情形，在同一斑塊內(nèi)的各組個體考慮正常生長、遲滯生長和跨組生長等，證明了種群分布的有界性，給出了種群零平衡態(tài)存在的條件。
[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.
【學(xué)位授予單位】：杭州電子科技大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2015
【分類號】：O175

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