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系統識別號 U0002-1607201317310100
中文論文名稱 三物種之食物鏈模型的動態行為研究
英文論文名稱 Dynamics of Three Species Food Chain Models with Holling type II Functional Response
校院名稱 淡江大學
系所名稱(中) 數學學系碩士班
系所名稱(英) Department of Mathematics
學年度 101
學期 2
出版年 102
研究生中文姓名 潘虹霖
研究生英文姓名 Hung-Lin Pan
學號 600190101
學位類別 碩士
語文別 英文
口試日期 2013-06-27
論文頁數 25頁
口試委員 指導教授-楊定揮
委員-許正雄
委員-楊智烜
中文關鍵字 霍林型二  食物鏈模型  三物種 
英文關鍵字 Holling-type II  Food Chain Model  Three Species 
學科別分類 學科別自然科學數學
中文摘要 本次研究的三個物種食物鏈模型是最簡單的三層食物網類型,一個變數變換後將模型轉化為三個一階常微分方程且有六個參數,起始值為正且有界。然後我們分類邊界平衡點的存在與局部穩定性。並且我們發現一個完整的參數分類,它的必要條件和充分條件存在,但不存在正重根,但正平衡點的局部穩定性數值已驗證。最後我們進行一些數值模擬我們所分類的每個區域。
英文摘要 In this work, the three species food chain models which are the simplest biological models with three trophic levels are investigated. After a non-dimensionless transformation, the model is transformed to a three first order ordinary differential equations with six parameters. The boundedness and positivity of solution with positive initial conditions are first established. Then we classify the existence and local stability of all boundary equilibria. Moreover, we obtain a complete classification of parameter space. For each region of the classification, the necessary and sufficient conditions of existence, non-existence and multiplicity of positive equilibria are obtained. However, the local stability of positive equilibrium is verified numerically. Finally, some numerical simulations are performed for each region of our classification.
論文目次 Contents
1 Introduction 2
2 Preliminary Results 3
2.1 Boundedness of Solutions 3
2.2 Subsystems and its Equilibria, Dynamics 4
3 Dynamics of Equilibria in R3 5
3.1 Local Stability of Equilibria in R3 7
3.2 Existence of Coexistence State and its Local Stability 9
4 Numerical Results 14
5 Reference 23

List of Figures
3.1 The region of parameters pace with various a1, a2, xed m1, m2, d1,
d2 under assumptions (A1) and (A2) 9
3.2 The region of parameters pace with various a1, a2 and xed m1, m2,
d1, d2 under assumptions (A1) and (A2) 12
4.1 The boundary equilibrium Ex is globally asymptotically stable if
assumption (A1) does not hold 16
4.2 The dynamics is a unique limit cycle if d2 m2 and d1 < m1
a1+1( x <
1+a1
2 ) 16
4.3 Proposition 3.2. If d2 m2 and d1 < m1
a1+1 ( x > 1+a1
2 ) 18
4.4 Boundary equilibrium Exy is globally asymptotically stable if pa-
rameters in the green region of Figure 3.2 19
4.5 The orbit is periodic on x-y plane if parameters in the light green
region of Figure 3.2 19
4.6 Boundary equilibrium Ex is globally asymptotically stable if param-
eters in the black region of Figure 3.2 20
4.7 Boundary equilibrium Exy is globally asymptotically stable if pa-
rameters in the white region of Figure 3.2 20
4.8 The orbit is periodic on x-y plane if parameters in the orange region
of Figure 3.2 21
4.9 The orbit is periodic on x-y plane if parameters in he left hand of
yellow area of Figure 3.2 21
4.10 Positive equilibrium E is globally asymptotically stable if parame-
ters in the right hand of yellow region of Figure 3.2 22
4.11 The orbit is periodic on x-y plane if parameters in the left hand of
red region of Figure 3.2 22
4.12 Positive equilibrium E is globally asymptotically stable if parame-
ters in the right hand of red region of Figure 3.2 23

List of Tables
2.1 Subsystems, equilibria and its stability for system (1.2) 6
4.1 The dynamics and parameter values with xed d1 = 0:45, m1 = 0:6,
m2 = 0:6 and various a1, a2, d2 for each numerical simulation 17
4.2 The corresponding analytical conditions for each numerical simulation 18
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