**By Peter Chesson** (University of Arizona)

All natural populations and communities occupy spatially heterogeneous
landscapes. Individual fitnesses vary in space, and the contributions
to population growth from different localities can be profoundly unequal
even though directed dispersal, and retention of organisms in favorable
localities, reduce the potential inequalities. Elaborate methods have
been developed for solving models of population and community dynamics
on heterogeneous landscapes, but to be science rather than technology,
they must also impart understanding. Scale transition theory aims to
provide this understanding through concepts that explain how the rules
for population dynamics on local spatial scales interact with
heterogeneity between localities, leading to novel outcomes on larger
spatial scales. These novel outcomes include changes in population
densities, stabilization of population dynamics, and promotion of
species coexistence.
Scale transition theory is founded on the idea of nonlinear averaging:
the average of a nonlinear function is in general different from the
average of the nonlinear function. Jensenâ€™s inequality is a simple
instance of nonlinear averaging where the function is convex and the
average of the function is uniformly greater than the function of the
average. More generally, the function is multidimensional and not
necessarily convex. A product of two variables is a simple but
extremely important example of a nonlinear function of multiple
variables. The average of the product is the product of the averages
(the function of the average), plus the covariance between the two
variables in space. Two key concepts emerge from this simple
observation: (1) fitness-density covariance, which explains how fitness
at the landscape scale differs from the simple spatial average of local
scale fitness, and (2) covariance between environment and competition,
which explains how the ability of an organism to benefit from favorable
environmental conditions can be limited by competition. Both concepts
have powerful roles showing how population stabilization and species
coexistence can emerge from the interaction between nonlinearities and
spatial heterogeneity. Another concept, nonlinear competitive variance,
shows how different nonlinear reactions to the same underlying
spatially varying density-dependent processes can profoundly change
higher scale dynamics.
Scale transition theory can be applied to models using quadratic
approximations related to moment closure techniques. However, exact
approaches are being developed. In general these exact approaches
require numerical evaluation of the key quantities, but still provide
general understanding through elucidation of the role of the key ideas
in specific instances. Other approximations use spatially implicit
models and Laplace transforms to provide detailed understanding of the
interactions between different kinds of spatial variation and nonlinear
functions.
Recent developments in scale transition theory abandon the usual
stationary assumptions that the environment repeats statistically in
time and space, allowing realistic landscapes, and the consequences of
temporal nonstationarity, including long-term climate change, to be
studied rigorously with mathematical models, extending traditional
theory to richer more realistic domains.