本文選題：分?jǐn)?shù)階 + 捕食者-食餌系統(tǒng)　； 參考：《北京交通大學(xué)》2015年碩士論文

【摘要】：近年來,分?jǐn)?shù)階動力學(xué)系統(tǒng)因其在生物數(shù)學(xué)、社會科學(xué)、統(tǒng)計力學(xué)等領(lǐng)域中存在巨大的應(yīng)用價值而成為當(dāng)前國內(nèi)外的熱點(diǎn)研究課題.作為非線性科學(xué)的一個新的研究方向,分?jǐn)?shù)階種群系統(tǒng)的動力學(xué)研究具有重要的理論與實際意義.目前,對于分?jǐn)?shù)階種群系統(tǒng)的動力學(xué)研究,主要是針對分?jǐn)?shù)階單種群和兩種群生物系統(tǒng),探討其平衡點(diǎn)的穩(wěn)定性和數(shù)值解.然而,對于分?jǐn)?shù)階三種群生物系統(tǒng)的研究還十分欠缺.鑒于此,本文以三維分?jǐn)?shù)階捕食者-食餌系統(tǒng)為研究對象,建立了四類分?jǐn)?shù)階捕食者-食餌系統(tǒng),并對其進(jìn)行了深入細(xì)致的研究. 一方面,建立了兩類分?jǐn)?shù)階三種群捕食者-食餌系統(tǒng),分別研究了這兩類系統(tǒng)的穩(wěn)定性和分岔性質(zhì).其一,研究了一類具有種間競爭的分?jǐn)?shù)階捕食者-食餌系統(tǒng)的穩(wěn)定性,給出了系統(tǒng)各個平衡點(diǎn)局部漸近穩(wěn)定的充分性條件,并通過數(shù)值仿真驗證了所給條件的正確性；其二,研究了另一類具有種內(nèi)競爭的分?jǐn)?shù)階捕食者-食餌系統(tǒng)的分岔行為,將分?jǐn)?shù)階階數(shù)作為分岔參數(shù),得到了系統(tǒng)發(fā)生分岔的臨界值,并在數(shù)值仿真中發(fā)現(xiàn)了Hopf分岔現(xiàn)象. 另一方面,建立了兩類分?jǐn)?shù)階具有時滯和階段結(jié)構(gòu)的捕食者-食餌系統(tǒng),研究了這兩類系統(tǒng)的穩(wěn)定性.根據(jù)分?jǐn)?shù)階時滯系統(tǒng)的穩(wěn)定性定理,得到了系統(tǒng)各個平衡點(diǎn)局部漸近穩(wěn)定的充分性條件,并通過數(shù)值仿真驗證了理論分析的有效性.此外,在數(shù)值仿真中討論了時滯參數(shù)對于系統(tǒng)收斂速率的影響.
[Abstract]:In recent years, fractional order dynamics system has become a hot research topic at home and abroad because of its great application value in the fields of biology mathematics, social science, statistical mechanics and so on. As a new research direction of nonlinear science, the dynamics of fractional population system has important theoretical and practical significance. At present, the stability and numerical solution of the equilibrium point of the fractional population system are studied mainly for the fractional order single population and two species biological systems. However, the study of fractional three species biological system is still very lacking. In view of this, four kinds of fractional predator-prey systems are established and studied in detail, taking the three-dimensional fractional predator-prey system as the research object. In this paper, two classes of fractional order three species predator-prey systems are established, and their stability and bifurcation properties are studied respectively. First, the stability of a class of fractional predator-prey systems with interspecific competition is studied. The sufficient conditions for the local asymptotic stability of each equilibrium point of the system are given, and the correctness of the conditions is verified by numerical simulation. The bifurcation behavior of another kind of fractional predator-prey system with intraspecific competition is studied. The fractional order is taken as the bifurcation parameter and the critical value of bifurcation is obtained. Hopf bifurcation is found in numerical simulation. On the other hand, two classes of fractional order predator-prey systems with time delay and stage structure are established, and the stability of these two systems is studied. According to the stability theorem of fractional delay systems, the sufficient conditions for the local asymptotic stability of each equilibrium point of the system are obtained, and the validity of the theoretical analysis is verified by numerical simulation. In addition, the effect of delay parameters on the convergence rate of the system is discussed in the numerical simulation.
【學(xué)位授予單位】：北京交通大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2015
【分類號】：O175

【引證文獻(xiàn)】

相關(guān)碩士學(xué)位論文 前1條

1 宋萍;分?jǐn)?shù)階種群模型的動態(tài)分析[D];南京航空航天大學(xué);2016年

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本文編號：1995329