framework to assess compliance, and summarize available knowledge needed for regulatory decisions.

EARLY ENVIRONMENTAL MODELS

The earliest uses of mathematics to explain the physical world, an important element of environmental models, came in response to the desire to explain and predict the movement of the night sky, the relationship of notes in a musical scale, and other scientific observations (Mahoney 1998; Eagleton 1999; O’Connor and Robertson 2003; Schichl 2004). Later developments of basic conceptual models that helped further the connections of mathematics and modeling to science include the thirteenth century Fibonacci sequences of rabbit population, Paracelsus’s connection of dose to disease in the fifteenth century, and the Copernican model of planetary motions in the sixteenth century. The role that mathematics would play in explaining the physical world is evident in the seventeenth century roots of differential calculus, where physical observations of moving objects led to conceptual models of motion, mathematical representations of motions, and finally predictions of locations (Herrmann 1997).

A large expansion in the use of computational models for understanding environmental science and management came in the nineteenth and early twentieth centuries.1 Mathematical formulations of basic models were developed for many problems, including atmospheric plume motion (Taylor 1915), human dose-response relationship (Crowther 1924), predator-prey relationships (Lotka 1925), and national economy (Tinbergen 1937). An early example of the level of sophistication possible in computational models is Arrhenius’s climate model for assessing the greenhouse effect (Arrhenius 1896). Arrhenius’s model is a seasonal, spatially disaggregated climate model that relies on a numerical solution to a set of differential equations that represent surface energy balance. The numerical computations required months of hand calculations (Weart 2003), similar to many early numerical models. The computa-

1

The committee decided to use the term “computational model” rather than “mathematical model.” These terms are synonymous. The committee considers the the term computational model to be a better descriptor in the era when these models are solved on computers. However, as noted in the text, computational models emerged long before the invention of the digital computer.



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