Asymptotic dynamics of a modified discrete Leslie-Gower competition system

We propose a modified discrete-time Leslie-Gower competition system of two populations to study competition outcomes. Depending on the magnitude of a particular model parameter that measures intraspecific competition between individuals within the same population, either one or both populations may be subject to Allee effects. The resulting system can have up to four coexisting steady states. Using the theory of planar competitive maps, it is shown that the model has only equilibrium dynamics. The competition outcomes then depend not only on the parameter regimes but may also depend on the initial population distributions.