Along the lines of the 'polluter pays principle', it has recently been proposed that the local long-line fishing industry should fund eradication of terrestrial predators at seabird breeding colonies, as a compensatory measure for the bycatch caused by the fishing activity. The measure is economically sound, but a quantitative and reliable test of its biological efficacy has never been conducted. Here, we investigated the demographic consequences of predator eradication for Cory's shearwater Calonectris diomedea, breeding in the Mediterranean, using a population model that integrates demographic rates estimated from individual life-history information with experimental measures of predation and habitat structure. We found that similar values of population growth rate can be obtained by different combinations of habitat characteristics, predator abundance and adult mortality, which explains the persistence of shearwater colonies in islands with introduced predators. Even so, given the empirically obtained values of survival, all combinations of predator abundance and habitat characteristics projected a decline in shearwater numbers. Perturbation analyses indicated that the value and the sensitivity of shearwater population growth rates were affected by all covariates considered and their interactions. A decrease in rat abundance delivered only a small increase in the population growth rate, whereas a change in adult survival (a parameter independent of rat abundance) had the strongest impact on population dynamics. When adult survival is low, rat eradication would allow us to "buy" years before extinction but does not reverse the process. Rat eradication can therefore be seen as an emergency measure if threats on adult survival are eliminated in the medium-term period. For species with low fecundity and long life expectancy, our results suggest that rat control campaigns are not a sufficient, self-standing measure to compensate the biological toll of long-line fisheries.