# American Institute of Mathematical Sciences

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

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