Figure shows 997 feedings links (lines) among 92 taxa (nodes) [

These examples illustrate the current view of ecological systems as complex adaptive systems [

Is it possible to incorporate the effect of small-scale variability without resorting to the “brute force” approach of using higher and higher resolution? I start with examples from theoretical ecology that illustrate problems and approaches related to these scaling questions; I then present more specific examples related to global change and ecosystem dynamics, and end with a series of related problems on the dynamics of large food webs, the ultimate networks of ecological interactions.

Lotka–Volterra equations and their many descendants assume that individuals are well mixed and interact at mean population abundances. They are mean-field equations that use the mass-action law to describe the dynamics of interacting populations, and ignore both the scale of individual interactions and their spatial distribution. A key question therefore is: can the spatial variability generated at small, individual scales influence the dynamics at larger, population scales? If so, can the effect of smaller scales be represented by simply modifying mean-field models?

Stochastic models such as interacting particle systems [

On the left is a detailed model in which individual interactions in a network are described explicitly. On the right, typical “mean field” models aggregate the population into compartments (here for the three subpopulations of susceptible, infected, and recovered individuals in the dynamics of an infectious disease with permanent immunity). Computational approaches can help us understand the relationship between dynamics at these two different scales, from the individual to the population level. We can start with a stochastic individual-based model and develop approximations that simplify it (A). From this process, we can learn about the opposite direction of formulating simple models directly without sufficient knowledge to first specify the detailed interactions and components (B). These simple models represent implicitly the effect of smaller scale variability.

Recent findings on individual-based models for predator–prey dynamics in a spatial lattice indicate that simpler, low-dimensional models can still be applied at the population level [

The problem of incorporating sub-grid-scale processes into large-scale models is found in many other scientific fields in which nonlinearity allows variability to interact across spatial or organizational scales. It also applies to other ecological contexts, in particular to global change ecology and to the spatiotemporal ecosystem models used to represent feedbacks between the biota and the physical environment. At large spatial and temporal scales, the question of essential biological detail quickly becomes computationally intractable. In a recent review on ecosystem–atmosphere interactions, Moorcroft [

Similar connections are being addressed in aquatic environments for phytoplankton, the unicellular primary producers of lakes and oceans estimated to contribute 45% of global net primary production (i.e., the amount of carbon fixed by plants per unit area over time via photosynthesis; E. A. Litchman, C. A. Klausmeier, J. R. Miller, O. Schofield, P. G. Falkowski, unpublished data). The amount of net primary production depends on biological heterogeneity in the form of a taxonomically diverse group of species [

In this large simulation [

In short, the computational and conceptual challenge is to bridge not only highly disparate temporal and spatial scales, but also organizational ones, from individual physiology to ecosystem biogeochemistry, via community structure and functional diversity (

The nonlinear dynamics of large networks is a major challenge in computational biology and complex systems in general [

Stochastic assembly models are perhaps the best candidates to develop a general dynamical theory not only to address open questions on the relationship between structure and dynamics, but also to generate the macroscopic community patterns that ecologists observe in nature and characterize diversity (such as species–area curves and species-rank abundance curves). In these models, macroscopic patterns in diversity arise from the dynamic tension between extinction (as the result of ecological interactions and environmental perturbations) and innovation (as the result of evolution and the immigration of new species from outside the system) (e.g., [

The topics described here only begin to illustrate some of the many rich areas for research in computational ecology. We have moved beyond learning more details about the components of complex systems in order to reconstruct their dynamics. Instead, a more fundamental role for computation is found in exploring the relationship between dynamics across scales, in the constant dialogue between simplicity and complexity. Perhaps this is best expressed by what a famous mathematician had to say about a famous macroscopic law of physics: “If our means of investigation became more and more incisive, we would discover the simple under the complex, then the complex under the simple, then again the simple under the complex, and so on, without being able to predict which state would ultimately prevail” [

I thank Simon Levin and Andy Dobson for comments on an earlier version of this manuscript, the Pacific Ecoinformatics and Computational Ecology Lab for the food web image, and Victoria Coles, Khalid Boushaba, and Manojit Roy for their help with the other figures. I also thank the James S. McDonnell Foundation for a Centennial Fellowship on Global and Complex Systems.