我們選了一個 Holling Type II 的兩個掠食者一個被掠食的動態模型，
討論這模型在二維的平衡點穩定性、邊界的穩定性、局部的穩定性和唯一性，
接著是在三維的情況下，邊界平衡點的局部分析。另外，還探討了這模型在
Global 的動態情況是否會有出現共存的可能。最後經過數據的分析，我們可
以知道我們的模型是不會出現共存的情形，只會有一個死亡，另一個存活的狀
況。

In the work, we consider the two-predator-one prey models with Holling type II functional response. First, we show that the model is dissipative and the existence, local stability of all boundary equilibrium are clarifi ed in R3 with some suitable conditions. Then it is verifi ed that the positive equilibrium does not exist generically which is the so-called Competitive Exclusion Principle. Moreover, we obtain a classifi cation of parameter space to clarify all local dynamics of the model and two global extinction results are showed analytically. Finally, numerical simulations are presented for all regions of parameter space of our classifi cation.

Contents
1 Introduction                                       2
2 Steady States, Local Stability, Boundedness, Uniform Persistence, and Global Stability                    3
2.1 Boundedness of Solutions . . . . . . . . . . . . 3
2.2 Subsystems and Boundary Equilibria . . . . . . . 4
3 Local Analysis of Boundary Equilibria in R3        7
3.1 Local Analysis in R3 . . . . . . . . . . . . . . 8
4 Some Global Dynamics                               12
5 Numerical Results                                  17
6 Conclusion and Remarks                             24
7 References                                         25

List of Figures
3.1 Two generic typical pictures of parameter space of two possibilities with varied a1, a2 and  xed d1 = d2 = 1, m1 = 1:7, and m2 = 1:8. . . . . . . . . . . . . . . . . . 11
5.1 Ex is globally asymptotically stable. .. . . . . 19
5.2 Exy2 is globally asymptotically stable. .  . . . 19
5.3 A periodic solution on x-y2 plane. . . . . . . . 20
5.4 Exy1 is globally asymptotically stable. .. . . . 20
5.5 A periodic solution on x-y1 plane. . . . . . . . 21
5.6 Exy1 is globally asymptotically stable.  . . . . 21
5.7 Exy1 is globally asymptotically stable.. . . . . 22
5.8 A periodic solution on x-y1 plane.. . . .. . . . 22
5.9 A periodic solution on x-y1 plane. . . . . . . . 23
5.10 A periodic solution on x-y2 plane. .  . . . . . 23
5.11 A periodic solution on x-y1 plane. . .. . . . . 24
5.12 A periodic solution in R3. . . . . .. . . . . . 24

List of Tables
5.1 Equilibria and its stability for system (1.2)  . 17
5.2 d1 = d2 = 1, m1 = 1:7, m2 = 1:8 . . . .  . . . . 18

References
[1] K. S. Cheng. Uniqueness of a limit cycle for a predator-prey system. SIAM Journal on Mathematical Analysis, 12(4):541-548, 1981.
[2] G. Hardin. The Competitive Exclusion Principle. Science, 131:1292-1297, 1960.
[3] S. B. Hsu, S. P. Hubbell, and P. Waltman. A Contribution to the Theory of Competing Predators. Ecological Monographs, 48(3):337-349, 1978.
[4] S. B. Hsu, S. P. Hubbell, and P.Waltman. Competing predators. SIAM Journal on Applied Mathematics, 35(4):617-625, 1978.
[5] A. L. Koch. Competitive coexistence of two predators utilizing the same prey under constant environmental conditions. Journal of Theoretical Biol-ogy, 44(2):387-395, 1974.
[6] W. Liu, D. Xiao, and Y. Yi. Relaxation oscillations in a class of predator-prey systems. Journal of Di fferential Equations, 188(1):306-331, 2003.
[7] H. L. Smith. The interaction of steady state and Hopf bifurcations in a two-predator-one-prey competition model. SIAM Journal on Applied Mathematics, 42(1):27-43, 1982.