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On a discrete three-dimensional Leslie-Gower competition model

 1 Institute of Mathematics, Academia Sinica, Taipei, Taiwan 106 2 Department of Mathematics, National Taiwan University, Taipei, Taiwan 106

* Corresponding author

Received  April 2018 Revised  January 2019 Published  June 2019

Fund Project: 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

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.

Citation: Yunshyong Chow, Kenneth Palmer. On a discrete three-dimensional Leslie-Gower competition model. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019123
References:
 [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. Ackleh, R. 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. Balreira, S. 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. Cushing, S. Levarge, N. 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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References:
 [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. Ackleh, R. 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. Balreira, S. 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. Cushing, S. Levarge, N. 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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