We present a model that predicts the level of gene flow mediated by animal pollinators from a source population to a sink population.  The model requires specification of three elements: (1) the paternity that originates from a single flower, the paternity shadow; (2) the mean number of flowers that pollinators visit during stays in the sink population, the residence; (3) the proportion of pollinators arriving at the sink that carry pollen from the source population.  Provided that pollinators visit enough flowers in the sink to exhaust the paternity shadows from the source, the general results are that gene flow is inversely proportional to the mean pollinator residence in the sink population, and is proportional to the fraction of pollinators arriving with pollen from the source.  These results are used to propose explanations for two of the widely observed patterns in gene flow among plant populations.  Numerical solutions to the model are derived using experimentally determined values of elements (1) and (2) that represent bumblebees, Bombus spp., visiting agricultural fields of oilseed rape, Brassica napus L.   In B. napus, the paternity shadow attenuates rapidly over approximately 20 recipient flowers.  Mean bumblebee residences in the fields studied varied between 490 and 720 flowers.  In the absence of a direct measurement of element (3), we calculated the maximum level of bumblebee-mediated gene flow by assuming that all bees arrived at the sink saturated with pollen from extrinsic sources.  In this case, the model predicts that bumblebee-mediated gene flow accounted for between 0.1% and 0.5% of the progeny in the agricultural fields studied.  A likelihood analysis of our observations is unable to reveal convincingly the proportion of bees arriving at the sink via a source population, but the literature suggests that bumblebees have high site fidelity, which implies that bee-mediated gene flow may be substantially less than our estimated maximum.  We consider the role of various factors, including wind pollination, in accounting for the differences between the model's predictions and the generally higher levels of gene flow observed in previous studies of oilseed rape.