兩類具M(jìn)ichaelis-Menten型收獲的捕食系統(tǒng)的穩(wěn)定性與分支分析
發(fā)布時(shí)間：2018-07-22 21:33
【摘要】：本文著重研究了兩類帶有Michaelis-Menten型收獲的食餌-捕食系統(tǒng)的穩(wěn)定性以及分支,其主干內(nèi)容分別是在第二章與第三章這兩章中完成的,主要工作及結(jié)論如下:第二章研究了 一類食餌種群帶有Michaelis-Menten型收獲項(xiàng)的Leslie-Gower型捕食系統(tǒng),考察了該系統(tǒng)的穩(wěn)定性以及其中各種分支的存在性等動(dòng)力學(xué)問題.該章的主要理論依據(jù)是微分方程的定性與穩(wěn)定性理論、分支理論和規(guī)范性理論等.文章首先探究了系統(tǒng)各類有效平衡點(diǎn)的存在性,通過選取適當(dāng)?shù)膮?shù)作為分析參數(shù),得到了各自平衡點(diǎn)存在時(shí)的具體對(duì)應(yīng)條件.其次,研究這些平衡點(diǎn)在其相應(yīng)存在條件下的穩(wěn)定性,針對(duì)不同情形,分別采取了特征值分析法、線性化法等方法進(jìn)行分析,得出邊界平衡點(diǎn)在其存在條件下都不穩(wěn)定,而正平衡點(diǎn)在其不同存在條件下則可能分別以匯點(diǎn)、源點(diǎn)、中心、鞍結(jié)點(diǎn)、尖點(diǎn)等類型出現(xiàn)的結(jié)論.最后,還研究了該系統(tǒng)中的一些分支,包括鞍結(jié)分支、Hopf分支和Bogdanov-Takens分支,通過選取恰當(dāng)?shù)膮?shù)作為分支參數(shù),分別計(jì)算給出了具體分支點(diǎn),并通過驗(yàn)證橫截條件的方法嚴(yán)格證明了鞍結(jié)分支、Hopf分支的存在性,利用第一李雅普諾夫方法給出了 Hopf分支的方向,還通過規(guī)范型理論的應(yīng)用論證了該系統(tǒng)中余維2的Bogdanov-Takens分支的存在性,得到了二階截?cái)嘁?guī)范型以及分支曲線表達(dá)式等.第三章探討了 一類帶有食餌種群成熟時(shí)滯和捕食者種群Michaelis-Menten型收獲的捕食系統(tǒng),研究其系統(tǒng)的穩(wěn)定性和Hopf分支等動(dòng)力學(xué)行為.這章研究的主要理論基礎(chǔ)是微分方程的穩(wěn)定性理論、Hopf分支理論和時(shí)滯泛函微分方程理論.文章給出了系統(tǒng)存在惟一正平衡點(diǎn)時(shí)的參數(shù)條件,并通過分析特征方程,利用Hurwitz判別法,得到了系統(tǒng)正平衡點(diǎn)局部漸近穩(wěn)定的充要條件.另外,文章還通過選取時(shí)滯量作為分析參數(shù),對(duì)系統(tǒng)特征方程特征根的實(shí)部進(jìn)行分析,得到了系統(tǒng)保持局部漸近穩(wěn)定的參數(shù)區(qū)間,并且給出具體的臨界滯量值,論證得出了當(dāng)該時(shí)滯參數(shù)穿越該臨界值時(shí)系統(tǒng)就會(huì)發(fā)生Hopf分支的結(jié)論.本文在主要理論結(jié)果得出之后,還借助Matlab進(jìn)行了相應(yīng)的數(shù)值模擬,驗(yàn)證了其結(jié)論的準(zhǔn)確性.
[Abstract]:In this paper, the stability and bifurcation of two kinds of predator-prey systems with Michaelis-Menten harvesting are studied. The main work and conclusions are as follows: in chapter 2, we study the Leslie-Gower type predator-prey system with Michaelis-Menten harvesting term, and investigate the stability of the system and the existence of various branches of the system. The main theoretical basis of this chapter is qualitative and stability theory of differential equation, branch theory and normative theory. In this paper, the existence of all kinds of effective equilibrium points in the system is discussed. By selecting appropriate parameters as the analysis parameters, the corresponding conditions for the existence of each equilibrium point are obtained. Secondly, the stability of these equilibrium points under the corresponding conditions is studied. The eigenvalue analysis method and linearization method are used to analyze the stability of these equilibrium points under the conditions of their existence, and the results show that the boundary equilibrium points are unstable under the conditions of their existence. Under different conditions, the positive equilibrium point may appear as meeting point, source point, center, saddle node, tip point and so on. Finally, some branches of the system, including saddle node Hopf bifurcation and Bogdanov-Takens bifurcation, are studied. The existence of Hopf bifurcation of saddle node bifurcation is strictly proved by the method of verifying the transverse condition. The direction of Hopf bifurcation is given by the first Lyapunov method. The existence of Bogdanov-Takens bifurcation of codimension 2 in the system is also proved by the application of normal form theory. The second order truncated canonical form and branch curve expression are obtained. In chapter 3, we study a kind of predator system with prey population maturity delay and predator population Michaelis-Menten harvest, and study its stability and Hopf bifurcation. The main theoretical basis of this chapter is the stability theory of differential equations, Hopf bifurcation theory and delay functional differential equation theory. In this paper, the parameter conditions for the existence of a unique positive equilibrium point are given. By analyzing the characteristic equation and using Hurwitz's criterion, the sufficient and necessary conditions for the local asymptotic stability of the positive equilibrium point of the system are obtained. In addition, by selecting the time-delay as the analysis parameter, the real part of the characteristic root of the system characteristic equation is analyzed, the parameter interval of the system maintaining local asymptotic stability is obtained, and the concrete critical hysteresis value is given. It is proved that Hopf bifurcation will occur when the time-delay parameter crosses the critical value. After the main theoretical results are obtained, the corresponding numerical simulation is carried out with Matlab to verify the accuracy of the conclusions.
【學(xué)位授予單位】：蘭州交通大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O175
本文編號(hào)：2138641
[Abstract]:In this paper, the stability and bifurcation of two kinds of predator-prey systems with Michaelis-Menten harvesting are studied. The main work and conclusions are as follows: in chapter 2, we study the Leslie-Gower type predator-prey system with Michaelis-Menten harvesting term, and investigate the stability of the system and the existence of various branches of the system. The main theoretical basis of this chapter is qualitative and stability theory of differential equation, branch theory and normative theory. In this paper, the existence of all kinds of effective equilibrium points in the system is discussed. By selecting appropriate parameters as the analysis parameters, the corresponding conditions for the existence of each equilibrium point are obtained. Secondly, the stability of these equilibrium points under the corresponding conditions is studied. The eigenvalue analysis method and linearization method are used to analyze the stability of these equilibrium points under the conditions of their existence, and the results show that the boundary equilibrium points are unstable under the conditions of their existence. Under different conditions, the positive equilibrium point may appear as meeting point, source point, center, saddle node, tip point and so on. Finally, some branches of the system, including saddle node Hopf bifurcation and Bogdanov-Takens bifurcation, are studied. The existence of Hopf bifurcation of saddle node bifurcation is strictly proved by the method of verifying the transverse condition. The direction of Hopf bifurcation is given by the first Lyapunov method. The existence of Bogdanov-Takens bifurcation of codimension 2 in the system is also proved by the application of normal form theory. The second order truncated canonical form and branch curve expression are obtained. In chapter 3, we study a kind of predator system with prey population maturity delay and predator population Michaelis-Menten harvest, and study its stability and Hopf bifurcation. The main theoretical basis of this chapter is the stability theory of differential equations, Hopf bifurcation theory and delay functional differential equation theory. In this paper, the parameter conditions for the existence of a unique positive equilibrium point are given. By analyzing the characteristic equation and using Hurwitz's criterion, the sufficient and necessary conditions for the local asymptotic stability of the positive equilibrium point of the system are obtained. In addition, by selecting the time-delay as the analysis parameter, the real part of the characteristic root of the system characteristic equation is analyzed, the parameter interval of the system maintaining local asymptotic stability is obtained, and the concrete critical hysteresis value is given. It is proved that Hopf bifurcation will occur when the time-delay parameter crosses the critical value. After the main theoretical results are obtained, the corresponding numerical simulation is carried out with Matlab to verify the accuracy of the conclusions.
【學(xué)位授予單位】：蘭州交通大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O175
【參考文獻(xiàn)】
相關(guān)博士學(xué)位論文 前1條
1 袁銳;具時(shí)滯和食餌收獲的捕食—食餌系統(tǒng)的分支動(dòng)力學(xué)研究[D];哈爾濱工業(yè)大學(xué);2015年
，本文編號(hào)：2138641
本文鏈接：http://sikaile.net/kejilunwen/yysx/2138641.html
最近更新
教材專著
