Mathematics of Planet Earth

Creating usable models for the sustainability of ecosystems has many mathematical challenges. Ecosystems are complex because they involve multiple interactions among organisms and between organisms and the physical environment, at multiple spatial and temporal scales, and with multiple feedback loops making connections between and across scales. The issue of scaling and deriving models at one scale from another is well known to lead to substantial mathematical issues, as in going from descriptions of stochastic spatial movement at the population scale from the individual scale and as in getting diffusion limits. Here, for example, recent work has focussed on alternatives to the diffusion limit. The mathematical challenges in the analysis of full ecosystems are truly great.

Many modeling approaches have been used in studying ecosystems, ranging from simple dynamical systems to highly detailed computational models. Relatively simple models are essential to gain insight into fundamental features of complex systems and the mechanisms behind them, whereas highly detailed models are essential for making predictions about the specific effects that changes may have on ecosystem functioning. The complex ones include agent-based models and models that place biological models into realistic and detailed models for physical processes such as ocean dynamics. There is a need to develop new mathematical tools for making connections among different processes at different scales and thus provide a robust framework for assessing the sustainability of ecosystem processes.

Understanding models at multiple scales also requires case studies of particular systems. Plankton dynamics provide a good case study. At one extreme, low dimensional Nutrient-Phytoplankton-Zooplankton (NPZ) models give insight into the balance between light penetration and nutrient upwelling that underlie patterns of plankton blooms. At the other extreme, computational models of a myriad competing plankton species, rapidly evolving in the face of changing ocean temperature and salinity, are numerically incorporated into global climate models. As a second example, forests and savanna are complex systems where organisms interact with physical processes, specifically fire and hydrology, and they have been studied from the viewpoint of individual-based modeling but also with simple models.

This workshop aims to engage computational and mathematical modelers, empiricists, and mathematicians in a dialogue about how to best address the problems raised by the pressing need to understand complex ecological interactions at many scales. Its ultimate goal is to initiate transformative research that will provide new approaches and techniques, and perhaps new paradigms, for modeling complex systems and for connecting different types of models operating at different levels of detail. An important feature of the workshop will be afternoon sessions devoted to case studies rather than lectures with the goal of starting new collaborations and new research directions.