1. The fitness of individual species depends on their ability to persist and establish at low densities, just as the diversity of ecological communities depends on the establishment and persistence of low-density, ‘invader’ species. Theory predicts that abiotic conditions and the competitive make-up of resident communities jointly shape invader fitness, limiting the phenotypic identity of successful invaders. 2. We use an invasion experiment to ask how competitive traits of 20 introduced plant species alter their absolute fitness in fragments that differ in size, abiotic conditions and traits of the resident community. 3. We show that abiotic conditions interact with both invader traits and resident functional diversity to determine invader survival. Optimal invader traits depended on the soil characteristics, while greater resident trait diversity lowered invader fitness and had especially strong effects in low resource environments. Unlike other abiotic conditions, fragment size had consistent effects irrespective of invader identity, decreasing survival in larger fragments. 4. Synthesis. Our results illustrate how the abiotic environment mediates the effects of resident and invader traits on establishment, creating fitness landscapes that structure local diversity and the functional identities of successful species.