a A typology of factors, represented by intersections in the Venn diagram that explain invasions and differentiated along the introduction-naturalisation-invasion continuum, where alien macroecology refers to the richness, distribution, abundance, spatial and trait relationships of alien biota at large spatial scales. b A community trait profile, represented by the trait positions of resident species (black dots) in the two-dimensional trait space. The double-headed arrow within the green circle indicates the trait centroid and the trait periphery of the community trait profile. c The surface of invasiveness, calculated as the invasion growth rate for any given trait position. Blue to white colours indicate invasion growth rate from negative to positive values. d Invasibility of the invaded community, represented by the size of grey areas that experience positive invasiveness. e Each resident species fluctuates around a particular abundance, indicated by the size of a green circle. Abundance gradients among these resident species, represented by blue arrows, can be identified locally in the trait space. Abundance gradients do not necessarily conform to the gradient of invasiveness (c) or the centrality of trait position (b); rather, all three jointly emerge in the open community transition and turnover as a result of persistent biological invasions.