Define the Future challenges of spatial processes?
Understanding the dynamics of stochastic spatial systems, systems with a multitude of spatial scales, and systems along with underlying heterogeneity present fundamental mathematical challenges. From a biological point of view, queries such as the performance of moderately-sized neuronal networks (Pinto and Ermentrout 2001a,b), coexistence in ecological communities such as forests, and where to site marine or terrestrial reserves along heterogeneous coastlines and landscapes, provide motivation for the development of new mathematical approaches. For deterministic models, much more attention needs to be paid to the development of methods for gaining an understanding of the dynamics of nonlinear spatial systems described by reaction diffusion equations, or integro-difference equations, or perhaps more common treatments (for example Weinberger et al., 2002).
Methods will need to be developed to provide analytic approximations to describe the dynamics of stochastic spatial models (building upon current work such as Bolker and Pacala, 1997) that can be applied to more complex systems. Additional understanding of the dynamics of coupled oscillators (see Pinto and Ermentrout, 2001a, b) will offer insights not just into the dynamics of neural networks, but of coupled systems at another biological scales ranging from the population to the ecosystem. Ascertaining the impact of heterogeneities at small spatial scales on the performance of ecological and other systems at larger spatial scales can depend on the use and further development of methods based on homogenization (for example Powell and Zimmerman, 2002), which offers an appropriate way to look at spatial averages of dynamics.