The scale transition: scaling up population dynamics with field data.

Applying the recent developments of scale transition theory, we demonstrate a systematic approach to the problem of scaling up local scale interactions to regional scale dynamics with field data. Dynamics on larger spatial scales differ from the predictions of local dynamics alone because of an interaction between nonlinearity in population dynamics at the local scale and spatial variation in density and environmental factors over the regional population. Our systematic approach to scaling up involves the following five steps. First, define a model for dynamics on the local spatial scale. Second, apply scale transition theory to identify key interactions between nonlinearity and spatial variation that translate local dynamics to the regional scale. Third, measure local-scale model parameters to determine nonlinearities at local scales. Fourth, measure spatial variation. Finally, combine nonlinearity and variation measures to obtain the scale transition. Using field data for the dynamics of grazers and periphyton in a freshwater stream, we show that scale transition terms greatly reduce the growth and equilibrium density of the periphyton population at the stream scale compared to rock scale populations, confirming the importance of spatial mechanisms to stream-scale dynamics.