Oscillatory dynamics and spatial scale: The role of noise and unresolved pattern

Predator-prey and other nonlinear ecological interactions often lead to osc illatory dynamics in temporal systems and in spatial systems when the rates of movement are large, so that individuals are effectively well mixed and space becomes unimportant. When individuals are not well mixed, however. pr operties of fluctuations in population densities, and in particular their a mplitudes. are known to vary with the spatial scale at which the system is observed. We investigate the relationship among dynamics at different spati al scales with an individual-based predator-prey model that is stochastic a nd nonlinear. Results elucidate the role of spatial pattern and individual variability in the dynamics of densities. We show that spatial patterns in this system reduce the per capita rates of predation and prey growth but pr eserve functional forms. The functional forms remain those one would expect in a well-mixed system in which individuals interact according to mean pop ulation densities, but with modified parameters. This similarity of the fun ctional forms allows us to approximate accurately the long-term dynamics of the spatial system at large scales with a temporal predator-prey model wit h only two variables, a simple system of ordinary differential equations of the type ecologists have been using for a long time. This approximation pr ovides an explanation for the stabilizing role of space. the decrease in th e amplitude of fluctuations from the well-mixed to the limited-movement cas e. We also provide an explanation for the previously described aperiodic dynam ics of densities at intermediate spatial scales. These irregular cycles res ult from the interplay of demographic noise with decaying oscillations, whe re the decay of the cycles is due to the spatial patterns. It is indeed pos sible to capture essential properties of these cycles. including their appa rent sensitivity to initial conditions, with a model that follows individua ls but parameterizes their spatial interactions in a simple way, using agai n the similarity of functional forms and the modified parameters. Thus, dem ographic noise appears essential at a spatial scale previously chosen for t he high degree of determinism in the dynamics. Our results illustrate a semi-empirical approach to simplify and to scale s patial ecological systems that are oscillatory from individual or local-sca le to large-scale dynamics.