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Article Contents

# On a discrete three-dimensional Leslie-Gower competition model

• * Corresponding author

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.

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

 Citation:

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