Scale transition theory: Its aims, motivations and predictions

Abstract Scale transition theory is an approach to understanding population and community dynamics in the presence of spatial or temporal variation in environmental factors or population densities. It focuses on changes in the equations for population dynamics as the scale enlarges. These changes are explained in terms of interactions between nonlinearities and variation on lower scales, and they predict the emergence of new properties on larger scales that are not predicted by lower scale dynamics in the absence of variation on those lower scales. These phenomena can be understood in terms of statistical inequalities arising from the process of nonlinear averaging, which translates the rules for dynamics from lower to higher scales. Nonlinearities in population dynamics are expressions of the fundamental biology of the interactions between individual organisms. Variation that interacts with these nonlinearities also involves biology fundamentally in several different ways. First, there are the aspects of biology that are sensitive to variation in space or time. These determine which aspects of a nonlinear dynamical equation are affected by variation, and whether different individuals or different species are sensitive to different extents or to different aspects of variation. Second is the nature of the variation, for example, whether it is variation in the physical environment or variation in population densities. From the interplay between variation and nonlinearities in population dynamics, scale transition theory builds a theory of changes in dynamics with changes in scale. In this article, the focus is on spatial variation, and the theory is illustrated with examples relevant to the dynamics of insect communities. In these communities, one commonly occurring nonlinear relationship is a negative exponential relationship between survival of an organism and the densities of natural enemies or competitors. This negative exponential has a biological origin in terms of independent actions of many individuals. The subsequent effects of spatial variation can be represented naturally in terms of Laplace transforms and related statistical transforms to obtain both analytical solutions and an extra level of understanding. This process allows us to analyze the meaning and effects of aggregation of insects in space. Scale transition theory more generally, however, does not aim to have fully analytical solutions but partial analytical solutions applicable for circumstances too complex for full analytical solution. These partial solutions are intended to provide a framework for understanding of numerical solutions, simulations and field studies where key quantities can be estimated from empirical data.

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