Biological invasions are a growing aspect of global biodiversity change. In many regions, introduced species richness increases supralinearly over time. This does not, however, necessarily indicate increasing introduction rates or invasion success. We develop a simple null model to identify the expected trend in invasion records over time. For constant introduction rates and success, the expected trend is exponentially increasing. Model extensions with varying introduction rate and success can also generate exponential distributions. We then analyse temporal trends in aquatic, marine and terrestrial invasion records. Most data sets support an exponential distribution (15/16) and the null invasion model (12/16). Thus, our model shows that no change in introduction rate or success need be invoked to explain the majority of observed trends. Further, an exponential trend does not necessarily indicate increasing invasion success or ‘invasional meltdown’, and a saturating trend does not necessarily indicate decreasing success or biotic resistance.