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On a discrete three-dimensional Leslie-Gower competition model

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The first author is partially supported by a research grant from MOST, ROC; the second author was partially supported by Academia Sinica during a visit to the Mathematics Institute

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  • We consider a special discrete time Leslie-Gower competition models for three species: $ x_i(t+1) = \frac{a_ix_i(t)}{1+x_i(t) +c \sum_{j\not = i} x_j(t)} $   for $ 1\leq i \leq 3 $ and $ t \geq 0 $. Here $ c $ is the interspecific coefficient among different species. Assume $ a_1>a_2>a_3>1 $. It is shown that when $ 0<c< c_0: = (a_3-1)/(a_1+a_2-a_3-1) $, a unique interior equilibrium $ E^* $ exists and is locally stable. Then from a general theorem in Balreira, Elaydi and Luis (2017), it follows that $ E^* $ is globally asymptotically stable. Using a result of Ruiz-Herrera [11], it is shown that the unique positive equilibrium in the $ x_1x_2 $-plane is globally asymptotically stable for $ c_0<c<\beta_{21} = (a_2-1)/(a_1-1) $. Then it is shown that $ (a_1-1, 0, 0) $ is globally asymptotically stable for $ \beta_{21} <c<\beta_{12} = (a_1-1)/(a_2-1) $. This partially generalizes a result in Chow and Hsieh (2013) and Ackleh, Sacker and Salceanu (2014). For $ c>\beta_{12} $, it is shown that there are multiple asymptotically stable equilibria.

    Mathematics Subject Classification: Primary: 39A30; Secondary: 37N25.

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  • [1] A. S. Ackleh, Y. M. Dib and S. Jang, A discrete-time Beverton–Holt competition model, Proc. 9th International Conference on Difference Equations and Discrete Dynamical Systems (eds. L. Allen, B. Aulbach, S. Elaydi, and R. Sacker), World Scientific, (2005), 1–9. doi: 10.1142/9789812701572_0001.
    [2] A. S. AcklehR. J. Sacker and P. Salceanu, On a discrete selection-mutation model, J. Difference Eqn. Appl., 20 (2014), 1383-1403.  doi: 10.1080/10236198.2014.933819.
    [3] L. J. S. Allen, An Introduction to Mathematical Biology, Pearson, Upper Saddle River, 2007.
    [4] E. C. BalreiraS. Elaydi and R. Luis, Global stability of higher dimensional monotone maps, J. Difference Eqn. Appl., 23 (2017), 2037-2071.  doi: 10.1080/10236198.2017.1388375.
    [5] Y. Chow and J. Hsieh, On multi-dimensional discrete-time Beverton-Holt competition models, J. Difference Eqn. Appl., 19 (2013), 491-506.  doi: 10.1080/10236198.2012.656618.
    [6] Y. Chow, Asymptotic behavior of a special Leslie-Gower competition model for n species, preprint.
    [7] J. M. CushingS. LevargeN. Chitnis and S. M. Henson, Some discrete competition models and the competitive exclusion principle, J. Difference Eqn. Appl., 10 (2004), 1139-1151.  doi: 10.1080/10236190410001652739.
    [8] M. R. S. Kulenovic and O. Merino, Competitive-exclusion versus competitive-coexistence for systems in the plane, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1141-1156.  doi: 10.3934/dcdsb.2006.6.1141.
    [9] M. R. S. Kulenovic and O. Merino, Invariant manifolds for competitive discrete systems in the plane, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2471-2486.  doi: 10.1142/S0218127410027118.
    [10] P. Liu and S. Elaydi, Discrete competitive and cooperative models of Lotka-Volterra type, J. Comp. Anal. Appl., 3 (2001), 53-73.  doi: 10.1023/A:1011539901001.
    [11] A. Ruiz-Herrera, Exclusion and dominance in discrete population models via the carrying simplex, J. Difference Eqn. Appl., 19 (2013), 96-113.  doi: 10.1080/10236198.2011.628663.
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