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。

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.

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

[1] M. P. Boer, B. W. Kooi, and S. A. L. M. Kooijman. Homoclinic and hetero-
clinic orbits to a cycle in a tri-trophic food chain. Journal of Mathematical
Biology, 39(1):19{38, 1999.
[2] K. S. Ch^eng. Uniqueness of a limit cycle for a predator-prey system. SIAM
Journal on Mathematical Analysis, 12(4):541{548, 1981.
[3] C.-H. Chiu. Global qualitative analysis of coupled three-level food chains.
Journal of Mathematical Analysis and Applications, 349(1):272{279, 2009.
[4] C.-H. Chiu and S.-B. Hsu. Extinction of top-predator in a three-level food-
chain model. Journal of Mathematical Biology, 37(4):372{380, 1998.
[5] B. Deng. Food chain chaos due to junction-fold point. Chaos. An Interdisci-
plinary Journal of Nonlinear Science, 11(3):514{525, 2001.
[6] B. Deng. Food chain chaos with canard explosion. Chaos. An Interdisciplinary
Journal of Nonlinear Science, 14(4):1083{1092, 2004.
[7] B. Deng. Equilibriumizing all food chain chaos through reproductive
e ciency. Chaos. An Interdisciplinary Journal of Nonlinear Science,
16(4):043125, 7, 2006.
[8] B. Deng and G. Hines. Food chain chaos due to Shilnikov's orbit. Chaos. An
Interdisciplinary Journal of Nonlinear Science, 12(3):533{538, 2002.
[9] B. Deng and G. Hines. Food chain chaos due to transcritical point. Chaos.
An Interdisciplinary Journal of Nonlinear Science, 13(2):578{585, 2003.
[10] H. I. Freedman and J. W. H. So. Global stability and persistence of simple
food chains. Mathematical Biosciences, 76(1):69{86, 1985.
[11] H. I. Freedman and P. Waltman. Mathematical analysis of some three-species
food-chain models. Mathematical Biosciences, 33(3-4):257{276, 1977.
[12] T. C. Gard. Persistence in food chains with general interactions. Mathematical
Biosciences, 51(1-2):165{174, 1980.
[13] T. C. Gard. Persistence in food webs: Holling-type food chains. Mathematical
Biosciences, 49(1-2):61{67, 1980.
[14] T. C. Gard and T. G. Hallam. Persistence in food webs. I. Lotka-Volterra
food chains. Bulletin of Mathematical Biology, 41(6):877{891, 1979.
[15] A. Hastings and T. Powell. Chaos in a three-species food chain. Ecology,
72(3):896{903, June 1991.
[16] S. B. Hsu. On global stability of a predator-prey system. Mathematical
Biosciences, 39(1-2):1{10, May 1978.
[17] S.-B. Hsu, C. A. Klausmeier, and C.-J. Lin. Analysis of a model of two
parallel food chains. Discrete and Continuous Dynamical Systems. Series B.
A Journal Bridging Mathematics and Sciences, 12(2):337{359, 2009.
[18] A. Klebano  and A. Hastings. Chaos in three-species food chains. Journal of
Mathematical Biology, 32(5):427{451, 1994.
[19] Y. A. Kuznetsov, O. De Feo, and S. Rinaldi. Belyakov homoclinic bifurcations
in a tritrophic food chain model. SIAM Journal on Applied Mathematics,
62(2):462{487 (electronic), 2001.
[20] C. Letellier, L. A. Aguirre, J. Maquet, and M. A. Aziz-Alaoui. Should all the
species of a food chain be counted to investigate the global dynamics? Chaos,
Solitons and Fractals, 13(5):1099{1113, 2002.
[21] S. Mandal, M. M. Panja, S. Ray, and S. K. Roy. Qualitative behavior of
three species food chain around inner equilibrium point: spectral analysis.
Nonlinear Analysis-Modelling And Control, 15(4):459{472, 2010.
[22] R. K. Naji and A. T. Balasim. Dynamical behavior of a three species food
chain model with Beddington-DeAngelis functional response. Chaos, Solitons
and Fractals, 32(5):1853{1866, 2007.
[23] S. G. Ruan and H. I. Freedman. Persistence in three-species food chain models
with group defense. Mathematical Biosciences, 107(1):111{125, 1991.
[24] P. T. Saunders and M. J. Bazin. On the stability of food chains. Journal of
theoretical biology, 52(1):121{142, 1975.
[25] J. W.-H. So. A note on the global stability and bifurcation phenomenon of a
Lotka-Volterra food chain. Journal of theoretical biology, 80(2):185{187, 1979.