American Institute of Mathematical Sciences

Advanced
August  2019, 24(8): 4367-4377. doi: 10.3934/dcdsb.2019123

On a discrete three-dimensional Leslie-Gower competition model

Yunshyong Chow 1, and  Kenneth Palmer 2,,

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.

Keywords: Leslie-Gower competition model, discrete, equilibrium, stability, asymptotic behaviour.
Mathematics Subject Classification: Primary: 39A30; Secondary: 37N25.
Citation: Yunshyong Chow, Kenneth Palmer. On a discrete three-dimensional Leslie-Gower competition model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4367-4377. 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.  Google Scholar

[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.  Google Scholar

[3]

L. J. S. Allen, An Introduction to Mathematical Biology, Pearson, Upper Saddle River, 2007. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

Y. Chow, Asymptotic behavior of a special Leslie-Gower competition model for n species, preprint. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

L. J. S. Allen, An Introduction to Mathematical Biology, Pearson, Upper Saddle River, 2007. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

Y. Chow, Asymptotic behavior of a special Leslie-Gower competition model for n species, preprint. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Andrei Korobeinikov, William T. Lee. Global asymptotic properties for a Leslie-Gower food chain model. Mathematical Biosciences & Engineering, 2009, 6 (3) : 585-590. doi: 10.3934/mbe.2009.6.585
[2]

Mingxin Wang, Qianying Zhang. Dynamics for the diffusive Leslie-Gower model with double free boundaries. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2591-2607. doi: 10.3934/dcds.2018109
[3]

Yunfeng Liu, Zhiming Guo, Mohammad El Smaily, Lin Wang. A Leslie-Gower predator-prey model with a free boundary. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2063-2084. doi: 10.3934/dcdss.2019133
[4]

Hongmei Cheng, Rong Yuan. Existence and stability of traveling waves for Leslie-Gower predator-prey system with nonlocal diffusion. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5433-5454. doi: 10.3934/dcds.2017236
[5]

Wen-Bin Yang, Yan-Ling Li, Jianhua Wu, Hai-Xia Li. Dynamics of a food chain model with ratio-dependent and modified Leslie-Gower functional responses. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2269-2290. doi: 10.3934/dcdsb.2015.20.2269
[6]

Wenjie Ni, Mingxin Wang. Dynamical properties of a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3409-3420. doi: 10.3934/dcdsb.2017172
[7]

Xiaofeng Xu, Junjie Wei. Turing-Hopf bifurcation of a class of modified Leslie-Gower model with diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 765-783. doi: 10.3934/dcdsb.2018042
[8]

Jun Zhou. Qualitative analysis of a modified Leslie-Gower predator-prey model with Crowley-Martin functional responses. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1127-1145. doi: 10.3934/cpaa.2015.14.1127
[9]

Hongwei Yin, Xiaoyong Xiao, Xiaoqing Wen. Analysis of a Lévy-diffusion Leslie-Gower predator-prey model with nonmonotonic functional response. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2121-2151. doi: 10.3934/dcdsb.2018228
[10]

Jun Zhou, Chan-Gyun Kim, Junping Shi. Positive steady state solutions of a diffusive Leslie-Gower predator-prey model with Holling type II functional response and cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3875-3899. doi: 10.3934/dcds.2014.34.3875
[11]

Walid Abid, Radouane Yafia, M.A. Aziz-Alaoui, Habib Bouhafa, Azgal Abichou. Global dynamics on a circular domain of a diffusion predator-prey model with modified Leslie-Gower and Beddington-DeAngelis functional type. Evolution Equations & Control Theory, 2015, 4 (2) : 115-129. doi: 10.3934/eect.2015.4.115
[12]

Zengji Du, Xiao Chen, Zhaosheng Feng. Multiple positive periodic solutions to a predator-prey model with Leslie-Gower Holling-type II functional response and harvesting terms. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1203-1214. doi: 10.3934/dcdss.2014.7.1203
[13]

Na Min, Mingxin Wang. Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1071-1099. doi: 10.3934/dcds.2019045
[14]

Changrong Zhu, Lei Kong. Bifurcations analysis of Leslie-Gower predator-prey models with nonlinear predator-harvesting. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1187-1206. doi: 10.3934/dcdss.2017065
[15]

C. R. Zhu, K. Q. Lan. Phase portraits, Hopf bifurcations and limit cycles of Leslie-Gower predator-prey systems with harvesting rates. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 289-306. doi: 10.3934/dcdsb.2010.14.289
[16]

Safia Slimani, Paul Raynaud de Fitte, Islam Boussaada. Dynamics of a prey-predator system with modified Leslie-Gower and Holling type Ⅱ schemes incorporating a prey refuge. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 5003-5039. doi: 10.3934/dcdsb.2019042
[17]

Aicha Balhag, Zaki Chbani, Hassan Riahi. Existence and continuous-discrete asymptotic behaviour for Tikhonov-like dynamical equilibrium systems. Evolution Equations & Control Theory, 2018, 7 (3) : 373-401. doi: 10.3934/eect.2018019
[18]

Zaki Chbani, Hassan Riahi. Existence and asymptotic behaviour for solutions of dynamical equilibrium systems. Evolution Equations & Control Theory, 2014, 3 (1) : 1-14. doi: 10.3934/eect.2014.3.1
[19]

Masaaki Mizukami. Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2301-2319. doi: 10.3934/dcdsb.2017097
[20]

Masaaki Mizukami. Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 269-278. doi: 10.3934/dcdss.2020015

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (84)
  • HTML views (138)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences