A general notion of the Allee effect for higher dimensional triangular maps is proposed. A global dynamics theory is established. The theory is applied to multi-species hierarchical models. Then we provide a detailed study of the global dynamics of three-species Ricker competition models with the Allee effect. Regions of extinction, exclusion and coexistence are identified.
Keywords: 39A11; 92D25; Allee effect; competition models; global dynamics; invariant manifolds; omega limit set; triangular maps.