Predator–prey and other nonlinear ecological interactions often lead to oscillatory 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, properties of fluctuations in population densities, and in particular their amplitudes, are known to vary with the spatial scale at which the system is observed. We investigate the relationship among dynamics at different spatial scales with an individual-based predator–prey model that is stochastic and 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 preserve functional forms. The functional forms remain those one would expect in a well-mixed system in which individuals interact according to mean population densities, but with modified parameters. This similarity of the functional forms allows us to approximate accurately the long-term dynamics of the spatial system at large scales with a temporal predator–prey model with only two variables, a simple system of ordinary differential equations of the type ecologists have been using for a long time. This approximation provides an explanation for the stabilizing role of space, the decrease in the amplitude of fluctuations from the well-mixed to the limited-movement case.
We also provide an explanation for the previously described aperiodic dynamics of densities at intermediate spatial scales. These irregular cycles result from the interplay of demographic noise with decaying oscillations, where the decay of the cycles is due to the spatial patterns. It is indeed possible to capture essential properties of these cycles, including their apparent sensitivity to initial conditions, with a model that follows individuals but parameterizes their spatial interactions in a simple way, using again the similarity of functional forms and the modified parameters. Thus, demographic noise appears essential at a spatial scale previously chosen for the high degree of determinism in the dynamics.
Our results illustrate a semi-empirical approach to simplify and to scale spatial ecological systems that are oscillatory from individual or local-scale to large-scale dynamics.
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