Interspecific competition influences the composition of ecological communities. Species may differ in their needs for different resources, therefore resource availability may determine the outcome of interspecific interactions. Species often compete over food, shelter or both. When more than one resource is limited, different species may prioritize different resources. To determine the impact of resource availability on the competitive relationship between an invasive and a native species, we examined interactions between groups of the invasive Argentine ant (Linepithema humile) and the native odorous ant (Tapinoma sessile) over (1) food, (2) shelter or (3) both simultaneously. We further examined the mechanisms underlying the competitive relationship, asking whether aggressive interactions, exploratory behaviour or the order of arrival at a resource explained resource use. Shelter was preferred by both species when no competitors were present. In a competitive setting, L. humile groups controlled shelter through aggressive displacement but lost control over food due to investment of workers in the control of shelter. Thus, there are tradeoffs when competing over multiple resources and aggressive interactions allow invasive species to displace native species from a preferred resource. This article is part of the theme issue 'Intergroup conflict across taxa'.