To eradicate or effectively contain a biological invasion, all or most reproductive individuals of the invasion must be found and destroyed. To help find individual invading organisms, predictions of probable locations can be made with statistical models. We estimated spread dynamics based on time-series data and then used model-derived predictions of probable locations of individuals. We considered one of the largest data sets available for an eradication program: the campaign to eradicate the red imported fire ant (Solenopsis invicta) from around Brisbane, Australia. After estimating within-site growth (local growth) and inter site dispersal (saltatory spread) of fire ant nests, we modeled probabilities of fire ant presence for >600000 1-ha sites, including uncertainties about fire ant population and spatial dynamics. Such a high level of spatial detail is required to assist surveillance efforts but is difficult to incorporate into common modeling methods because of high computational costs. More than twice as many fire ant nests would have been found in 2008 using predictions made with our method rather than those made with the method currently used in the study region. Our ethod is suited to considering invasions in which a large area is occupied by the invader at low density. Improved predictions of such invasions can dramatically reduce the area that needs to be searched to find the majority of individuals, assisting containment efforts and potentially making eradication a realistic goal for many invasions previously thought to be ineradicable.