在這項工作中，我們考慮具有Holling II型功能反應的三種食物鏈模型。在重新調整的變換之後，我們有一個具有三個方程和6個參數的ODE系統。基於一些滅絕結果，假設一些溫和的假設。然後研究了每個邊界平衡點的局部穩定性，並完全分析了正平衡點的存在性。此外，By Routh-Hurwitz準則驗證了共存的局部穩定性。最後，進行了一些有趣的數值模擬來說明我們的理論結果

In this work, we consider a three species food chain model with Holling type II functional response. After the rescaled transformation, we have a system of ODE with three equations and 6 parameters. Based on some extinction results, some mild hypothesis are assumed. Then local stability of each boundary equilibria are investigated, and the existence of positive equilibrium are analyzed completely. Moreover, the local stability of coexistence are verified by By Routh-Hurwitz criterion. Finally, some interesting numerical simulation are performed to illustrate our theoretical results.

1 Introduction ..........................................................................1
2 Preliminary Results.............................................................. 2
2.1 BoundednessofSolutions....................................................3 
2.2 Dynamicsof(1.2)onInvariantSubspaces .............................5
3 Dynamics of Equilibria in R3 .................................................7 
3.1 LocalStabilityofBoundaryEquilibriainR3 ............................. 7
3.2 Existence of Coexistence State and its Local Stability .......9
4 Numerical Results and Discussions ......................................13
References...............................................................................16

[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. Chˆeng. 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 eciency. 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, S. P. Hubbell, and P. Waltman. A Contribution to the Theory of Competing Predators. Ecological Monographs, 48(3):337–349, 1978.
[18] 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.
[19] A. Klebano↵ and A. Hastings. Chaos in three-species food chains. Journal of Mathematical Biology, 32(5):427–451, 1994.
[20] 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.
[21] 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.
[22] 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.
[23] L. Markus. Asymptotically autonomous di↵erential systems, Contributions to the Theory of Nonlinear Oscillation Vol. 3. Princeton University Press, 1956, pp.17-29.
[24] 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.
[25] S. G. Ruan and H. I. Freedman. Persistence in three-species food chain models with group defense. Mathematical Biosciences, 107(1):111–125, 1991.
[26] P. T. Saunders and M. J. Bazin. On the stability of food chains. Journal of theoretical biology, 52(1):121–142, 1975.
[27] 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.