On a discrete three-dimensional Leslie-Gower competition model

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.