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研究生： 段嘯晨
Duan, Xiao-Chen
論文名稱： 微生物在恒化器中並伴有器壁粘著與交換之效應分析
The variable-yield model with the wall growth under the exchange rate in a chemostat
指導教授： 許世壁
Hsu, Sze-Bi
口試委員: 蔡志強
Tsai, Je-Chiang
王埄彬
Wang, Feng-Bin
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系
Department of Mathematics
論文出版年： 2018
畢業學年度： 106
語文別： 英文
論文頁數： 33
中文關鍵詞： 恆化器 、變產量參數模型 、器壁粘著 、周期重置機制 、合作系統 、極限系統 、離散系統穩定性分析
外文關鍵詞： chemostat, Variable-yield model, Wall effect, Periodic resetting, Cooperative System, Limiting System, Stability analysis of Discrete System
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    • 本文首先討論了簡易恆化器中伴有器壁粘著與交換之變產量常數模型，之後進一步研究了納米恆化器中的周期重置機制之變產量常數模型。在簡易恆化器模型中，我們證明了解的有界性，分析了平衡點的局部穩定性，構建了滅絕平衡點的全局穩定性，並給出了系統一致持續生存的條件。在納米恆化器模型中，我們證明了解的有界性，分析了系統最終滅絕或持續生存的條件，並證明了滅絕不動點的全局穩定性。

    In thesis, we first discuss simple chemostat with wall growth [1] for the variable-yield model. Then we study the variable-yield model in the nano-chemostat with the periodic setting [2]. In the first model we prove the boundedness of solutions, analyze the local stability of equilibria, establish the global stability for the extinction equilibrium and give the condition such that the system is uniform persistence. For the second model, we prove the the boundedness of solutions, analyze the condition of extinction or persistence of solutions and prove the global stability of the extinction equilibrium.

    1 Introduction . . . . .. . . . . . . . . . . . . . . . . . . . . . . .1 2 Boundedness of solutions of (1.1) . . . . . . . . . .. . . .. . . . .1 3 Stability analysis of the System (1.1) . . . . . . . . . . . . . . .4 3.1 The Local Stability Analysis of extinction equilibrium . . . . . 5 3.2 The Global Stability Analysis of extinction equilibrium . . . . 6 3.3 Uniform persistence of the system (1.1) . .. . . . . . . . . . . 7 3.4 The Existence and Uniqueness of coexistence equilibrium . . . . 8 4 Periodic Resetting System . . . . . . . . . . . . . . . . . . . . . 11 4.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.2 Boundedness of solution of (4.1) . . . . . . . . . . . .. . . . 11 4.3 Stability Analysis of the System (4.1) . . . . . . . . . . . . 13 5 Numerical Result. . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.1 Initial value situation . . . . . . . . . . . . . . . . . . . . 19 5.1.1 Operation Diagram . . . . . . . . . . . . . . . . . . . . . 19 5.1.2 Extinction situation . . . . . . . . . . . . . . . . . . . .21 5.1.3 Coexistence situation . . . . . . . . . . . . . . . . . . . 22 5.2 Periodic setting. . . . . . . . . . . . . . . . . . . . . . . . 24 5.2.1 Operation Diagram. . . . . . . . . . . . . . . . . . . . . 24 5.2.2 Extinction situation. . . . . . . . . . . . . . . . . . . . 29 5.2.3 Coexistence situation . . . . . . . . . . . . . . . . . . . 31

    [1] Sergei S. Pilyugin, Paul Waltman, The simple chemostat with
    wall growth, SIAM. J. Appl. Math, Vol. 59(1999) No. 5, p1552-1572.

    [2] Sze-Bi Hsu, Ya-Tang Yang, Theory of a microfluidic serial dilution bioreactor for growth of planktonic and biofilm populations, J. Math. Biology 72 (2016), 1401-1427.

    [3] Hal L.Smith, Paul Waltman, The Theory of the Chemostat Dynamics of Microbial Competition, Cambridge University Press, 1994.

    [4] Ting-Hao Hsu, Sze-Bi Hsu, Competitive Exclusion of Microbial Species for a Single-Nutrient with Internal Storage, SIAM. J. Appl. Math, Vol 68(2008) No.6, p1600-1617.

    [5] Xiaoqiang Zhao, Dynamical systems in population biology, Springer, 2003.

    [6] Tom. M.Apostol, Calculus, 2nd edition, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, Waltham, MA: Blaisdell, 1967.

    [7] Sze-Bi Hsu, Ordinary Differential Equations, 2nd Edition, World Scietific, 2013.

    [8] Allen Linda J.S., Introduction to Mathematical Bioligy, Practice Hall, Upper Saddle River, 2007.

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