Scaling up predator-prey dynamics using spatial moment equations

Classical models of predatorprey dynamics, commonly used in community and evolutionary ecology to explain population cycles, species coexistence, the effects of enrichment, or predict the evolution of behavioural traits, are often based on the mass-action assumption. This means encounter rates between predators and prey are expressed as a product of predator and prey landscape densities; as if the system was well-mixed. While mass-action may occur at small spatial scales, spatial variances and covariances in prey and predator densities affect encounter rates at large spatial scales. In the context of hostparasitoid interactions, this has been incorporated into theory for some time, but for predators, well-mixed or other ad hoc models are often used despite empirical evidence for intricate spatial variation in predator and prey numbers. We review the classical models and concepts, their strengths and weaknesses, and we present two recent spatial moment approaches that scale up predatorprey population dynamics from the individual or patch level to large spatial scales. Both methods include descriptors of spatial structure as corrections to encounter rates, but differ in whether or not these descriptors have dynamics that are explicit functions of movements, births and deaths. We describe how these spatial moment techniques work, what new results they have so far produced, and provide some suggestions to improve the connection of predatorprey theoretical models to empirical studies.