The stability and existence of positive solutions of an elliptic system for a modified Leslie-Gower model with Allee effect in predator are considered, subject to homogeneous Neumann boundary condition. Firstly, the priori estimates of positive solutions are obtained by using the maximum principle and the Harnack inequality,then the asymptotic stability of positive constant solution are acquired by means of stability theory. Secondly, the non-existence of the non-constant positive solutions is verified through the integral property and Poincare inequality. Finally, based on the methods of Leray-Schauder degree theory, sufficient conditions for the existence of non-constant positive solutions are derived. The fact shows that the two species can be co-existed in some conditions.

Leslie-Gower model; Allee effect; stability; existence