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Stochastic dynamics Ⅱ: Finite random dynamical systems, linear representation, and entropy production
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 |
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 [
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. |
show all references
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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