# American Institute of Mathematical Sciences

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2014, 11(4): 947-970. doi: 10.3934/mbe.2014.11.947

## Global dynamics for two-species competition in patchy environment

 1 Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan, Taiwan 2 Department of Mathematics, Mathematical Bioscience Institute, Ohio State University, Columbus, Ohio 43210

Received  January 2013 Revised  July 2013 Published  March 2014

An ODE system modeling the competition between two species in a two-patch environment is studied. Both species move between the patches with the same dispersal rate. It is shown that the species with larger birth rates in both patches drives the other species to extinction, regardless of the dispersal rate. The more interesting case is when both species have the same average birth rate but each species has larger birth rate in one patch. It has previously been conjectured by Gourley and Kuang that the species that can concentrate its birth in a single patch wins if the diffusion rate is large enough, and two species will coexist if the diffusion rate is small. We solve these two conjectures by applying the monotone dynamics theory, incorporated with a complete characterization of the positive equilibrium and a thorough analysis on the stability of the semi-trivial equilibria with respect to the dispersal rate. Our result on the winning strategy for sufficiently large dispersal rate might explain the group breeding behavior that is observed in some animals under certain ecological conditions.
Citation: Kuang-Hui Lin, Yuan Lou, Chih-Wen Shih, Tze-Hung Tsai. Global dynamics for two-species competition in patchy environment. Mathematical Biosciences & Engineering, 2014, 11 (4) : 947-970. doi: 10.3934/mbe.2014.11.947
##### References:
 [1] R. S. Cantrell, C. Cosner and Y. Lou, Evolutionary stability of ideal free dispersal strategies in patchy environments, J. Math. Biol., 65 (2012), 943-965. doi: 10.1007/s00285-011-0486-5. [2] B. S. Goh, Global stability in many-species systems, American Naturalist, 111 (1977), 135-143. doi: 10.1086/283144. [3] S. A. Gourley and Y. Kuang, Two-species competition with high dispersal: The winning strategy, Math. Biosci. Eng., 2 (2005), 345-362. doi: 10.3934/mbe.2005.2.345. [4] S. B. Hsu, H. Smith and P. Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Trans. Amer. Math. Soc., 348 (1996), 4083-4094. doi: 10.1090/S0002-9947-96-01724-2. [5] M. Y. Li and Z. Shuai, Global-stability problem for coupled system of differential equations on networks, J. Differential Eqations., 248 (2010), 1-20. doi: 10.1016/j.jde.2009.09.003. [6] H. L. Smith, Competing subcommunities of mutualists and a generalized Kamke theorem, SIAM J. Appl. Math., 46 (1986), 856-874. doi: 10.1137/0146052. [7] H. L. Smith, Monotone Dynamical Systems: An Introduction To The Theory of Competitive and Cooperative Systems, Math. Surveys and Monographs, Amer. Math. Soc., 1995. [8] Y. Takeuchi and Z. Lu, Permanence and global stability for competitive Lotka-Volterra diffusion systems, Nonlinear Anal., 24 (1995), 91-104. doi: 10.1016/0362-546X(94)E0024-B.

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##### References:
 [1] R. S. Cantrell, C. Cosner and Y. Lou, Evolutionary stability of ideal free dispersal strategies in patchy environments, J. Math. Biol., 65 (2012), 943-965. doi: 10.1007/s00285-011-0486-5. [2] B. S. Goh, Global stability in many-species systems, American Naturalist, 111 (1977), 135-143. doi: 10.1086/283144. [3] S. A. Gourley and Y. Kuang, Two-species competition with high dispersal: The winning strategy, Math. Biosci. Eng., 2 (2005), 345-362. doi: 10.3934/mbe.2005.2.345. [4] S. B. Hsu, H. Smith and P. Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Trans. Amer. Math. Soc., 348 (1996), 4083-4094. doi: 10.1090/S0002-9947-96-01724-2. [5] M. Y. Li and Z. Shuai, Global-stability problem for coupled system of differential equations on networks, J. Differential Eqations., 248 (2010), 1-20. doi: 10.1016/j.jde.2009.09.003. [6] H. L. Smith, Competing subcommunities of mutualists and a generalized Kamke theorem, SIAM J. Appl. Math., 46 (1986), 856-874. doi: 10.1137/0146052. [7] H. L. Smith, Monotone Dynamical Systems: An Introduction To The Theory of Competitive and Cooperative Systems, Math. Surveys and Monographs, Amer. Math. Soc., 1995. [8] Y. Takeuchi and Z. Lu, Permanence and global stability for competitive Lotka-Volterra diffusion systems, Nonlinear Anal., 24 (1995), 91-104. doi: 10.1016/0362-546X(94)E0024-B.
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