The rate of biological invasions is growing unprecedentedly, threatening ecological and socioeconomic systems worldwide. Quantitative understandings of invasion temporal trajectories are essential to discern current and future economic impacts of invaders, and then to inform future management strategies. Here, we examine the temporal trends of cumulative invasion costs by developing and testing a novel mathematical model with a population dynamical approach based on logistic growth. This model characterises temporal cost developments into four curve types (I - IV), each with distinct mathematical and qualitative properties, allowing for the parameterization of maximum cumulative costs, carrying capacities and growth rates. We test our model using damage cost data for eight genera (Rattus, Aedes, Canis, Oryctolagus, Sturnus, Ceratitis, Sus and Lymantria) extracted from the InvaCost database – which is the most up-to-date and comprehensive global compilation of economic cost estimates associated with invasive alien species. We find fundamental differences in the temporal dynamics of damage costs among genera, indicating they depend on invasion duration, species ecology and impacted sectors of economic activity. The fitted cost curves indicate a lack of broadscale support for saturation between invader density and impact, including for Canis, Oryctolagus and Lymantria, whereby costs continue to increase with no sign of saturation. For other taxa, predicted saturations may arise from data availability issues resulting from an underreporting of costs in many invaded regions. Overall, this population dynamical approach can produce cost trajectories for additional existing and emerging species, and can estimate the ecological parameters governing the linkage between population dynamics and cost dynamics.