How localized consumption stabilizes predator-prey systems with finite frequency of mixing

Predator-prey theory began with aspatial models that assumed organisms interacted as if they were well-mixed particles that obey the laws of mass action, but it has become clear that both the spatial and individual nature of many organisms can change how the dynamics of such systems function. Here I examine how localized consumption of prey by predators changes the dynamics of predator-prey systems; I use an individual-based simulation of the Rosenzweig-MacArthur model in implicit space and its mean-field approximation. In combination with limited movement, localized consumption makes the predator-prey dynamics more stable than the comparable well-mixed Rosenzweig-MacArthur model. Using a spatial correlation, one can directly compare a simplified version of the individual-based model with the Rosenzweig-MacArthur model. While this comparison allows the changes in the dynamics to be captured by the well-mixed Rosenzweig-MacArthur model, the parameters of the functional response are now dependent on the movement parameters, and so the functional response must be estimated statistically from the dynamics of the individual-based model. Yet this implies that aspatial models may work in a scale-specific fashion for spatial systems. Unlike many recent spatial models, the localized consumption and limited movement in the model presented here cannot produce coherent spatial patterns and do not depend on a patchy structure, as found in metapopulation models. Instead, the individual nature of the interactions creates a diffusion-limited reaction, which appears closer to a form of ephemeral refuge.