Predator-prey systems can extend over large geographical areas but empirical modelling of predator-prey dynamics has been largely limited to localised scales. This is due partly to difficulties in estimating predator and prey abundances over large areas. Collection of data at suitably large scales has been a major problem in previous studies of European rabbits (Oryctolagus cuniculus) and their predators. This applies in Western Europe, where conserving rabbits and predators such as Iberian lynx (Lynx pardinus) is important, and in other parts of the world where rabbits are an invasive species supporting populations of introduced, and sometimes native, predators. In pastoral regions of New Zealand, rabbits are the primary prey of feral cats (Felis catus) that threaten native fauna. We estimate the seasonal numerical response of cats to fluctuations in rabbit numbers in grassland-shrubland habitat across the Otago and Mackenzie regions of the South Island of New Zealand. We use spotlight counts over 1645 km of transects to estimate rabbit and cat abundances with a novel modelling approach that accounts simultaneously for environmental stochasticity, density dependence and varying detection probability. Our model suggests that cat abundance is related consistently to rabbit abundance in spring and summer, possibly through increased rabbit numbers improving the fecundity and juvenile survival of cats. Maintaining rabbits at low abundance should therefore suppress cat numbers, relieving predation pressure on native prey. Our approach provided estimates of the abundance of cats and rabbits over a large geographical area. This was made possible by repeated sampling within each season, which allows estimation of detection probabilities. A similar approach could be applied to predator-prey systems elsewhere, and could be adapted to any method of direct observation in which there is no double-counting of individuals. Reliable estimates of numerical responses are essential for managing both invasive and threatened predators and prey.