本論文關注在一封閉區域中兩個或三個競爭物種存在的生長情況。我們著眼於以下問題：考慮到物種間的某些生長參數以及不同物種所擁有不同的擴散速率，將探討持續一段持間之後，哪個物種將生存而哪些將不生存。
我們對於這個問題，再給定Neumann邊界條件下，我們利用有限元方法對此提供了數值解上的簡短Matlab實現，提出一些動態模型來進行研究，包含有一維與二維的情形；而在二維的討論中，
除了凸域(矩形)以外，我們將通過有限元方法在L形域（特殊的多邊形域）上比較這些物種，並通過數值計算驗證理論預測。

This paper concerns the growth of two or three competing species in the same bounded domain. We focus on the following issue:
given the same response growth parameters with different diffusion rate of two different species,which species will survive and which
will not after a period of time. For this problem, given Neumann boundary conditions, we provide a short Matlab implementation
of the numerical solution for this by using the finite element method, and propose some dynamic models to carry out
the numerical research, including one-dimensional and two-dimensional cases. In two-dimensional cases, except for the convex domain
(rectangular domain), we will compare the performance of these species by the finite element method on L-shape domain (a special polyhedral
domain) and verify the theoretical predictions by the numerical computation.

1. He, X. and Wei-Ming Ni, Global dynamics of the Lotka-Volterra competition-diffusion system: diffusion and spatial heterogeneity, I., Commun. Pure Appl. Math. doi:10.1002/cpa.21596.
2. He, X. and Wei-Ming Ni, Global dynamics of the Lotka-Volterra competition-diffusion system: with equal amount of total resources, II., Calc. Var. Partial Diff. Equ., 55(2016), Art.25, 20 pp.
3. Dockery J., Huston V., Mischaikow K.,Pernarowski M., The evolution of slow dispersal rates: a reaction diffusion model, J. Math. Biol. 37(1998),no. 1,61-83. doi:10.1007/s002850050120.
4. A. Hastings, Can spatial variation alone lead to selection for dispersal?, Theor. Pop. Biol., 24(1983), 244-251.
5. J. Alberty, C. Carstensen and Stefan A. Funken, Remarks around 50 lines of Matlab: short finite element implementation, Numer Algorithms 20(1999):117–137.
6. Hsu, S., Smith, H. and Waltman, P., Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Trans. Amer. Math. Soc. 348: 4083—4094 (1996)
7. Hirsch, M., Stability and convergence in strongly monotone dynamical systems, J. reine angew. Math. 383: 1—53 (1988)
8. Kishimoto, K., Instability of non-constant equilibrium solutions of a system of competition-diffusion equations, J. Math. Biol., 13(1981) 105-114.
9. Kishimoto, K. and Weinberger, H. F., The spatial homogeneity of stable equilibria of some reaction-diffusion systems on convex domains }, Math. Biol., 13(1982) 16: 103-112.
10. Hiroshi Matano and Masayasu Mimura, Pattern Formation in Competition-Diffusion Systems in Nonconvex Domains, Publ. RIMs, Kyoto Univ. 19(1983), 1049-1079
11. Cosner, C., Reaction-Diffusion Equations and Ecological Modeling, Tutorials in Mathematical Biosciences IV. Lecture Notes in Mathematics, vol 1922. Springer, Berlin, Heidelberg.
12. Vivan Hutson, Salome Martinez, Konstantin Mischaikow, and Glenn T. Vickers, The evolution of dispersal, J. Math. Biol., 47(6):483-517, 2003.
13. Lenggyel, I. and Rpstein, I. R., A Chemical approach to designing Turing patterns in reaction-diffusion systems, Proc. Natl. Acad. Sci. USA, Vol. 89, pp. 3977-3979, May 1922.
14. Kishimoto, K., The Diffusion Lotka-Volterra System with Three Species Can Have a Stable Non-Constant Equilibrium Solution, J. Math. Biology (1982) 16: 103-112.
15. Chiun-Chuan C., Li-Chang H., Masayasu M., Makoto T. and Daishin U., Semi-exact equilibrium solutions for three-species competition-diffusion systems, Hiroshima Math. J. 43(2013), 179-206.
16. P. Hess and A.C. Lazer, On an abstract competition model and applications, Nonlinear Analysis, T.M.A. 16 (1991), 917–940. MR 92f:92036.