Elimination, permanence, and exclusion in a competition model under Allee effects

In this paper, we study a system of nonlinear partial differential equations that models the population dynamics of two competitive species both under Allee effects. The consideration of the model includes Logistic growth with Allee effects, Lotka-Volterra competition, diffusion, initial density and boundary conditions on the habitat. In the reaction-diffusion system, we employ the method of upper and lower solutions to address questions on self-elimination or persistence, as well as permanence or competitive exclusion. Specific conditions on biological parameters are explicitly given for extinction, coexistence and competitive exclusion of the species under various boundary conditions. Numerical simulations for the model are demonstrated to illustrate our results from mathematical analysis.