exponential growth (if birth rates exceed mortality rates) or extinction (in the opposite case) is a fundamental principle: its applicability in biology, physics, chemistry, as well as simple finance, is central.

An important refinement of the Malthus model was proposed in 1838 to explain why most populations do not experience exponential growth indefinitely. The refinement was the idea of the density-dependent growth law, now known as the logistic growth model.^{6} Though simple, the Verhulst model is still used widely to represent population growth in many biological examples. Both Malthus and Verhulst models relate observed trends to simple underlying mechanisms; neither model is fully accurate for real populations, but deviations from model predictions are, in themselves, informative, because they lead to questions about what features of the real systems are worthy of investigation.

More recent examples of this sort abound. Nonlinear dynamics has elucidated the tendency of excitable systems (cardiac tissue, nerve cells, and networks of neurons) to exhibit oscillatory, burst, and wave-like phenomena. The understanding of the spread of disease in populations and its sensitive dependence on population density arose from simple mathematical models. The same is true of the discovery of chaos in the discrete logistic equation (in the 1970s). This simple model and its mathematical properties led to exploration of new types of dynamic behavior ubiquitous in natural phenomena. Such biologically motivated models often cross-fertilize other disciplines: in this case, the phenomenon of chaos was then found in numerous real physical, chemical, and mechanical systems.

Simple conceptual models can be used to uncover new mechanisms that experimental science has not yet encountered. The discovery of chaos mentioned above is one of the clearest examples of this kind. A second example of this sort is Turing’s discovery that two chemicals that interact chemically in a particular way (activate and inhibit one another) and diffuse at unequal rates could give rise to “peaks and valleys” of concentration. His analysis of reaction-diffusion (RD) systems showed precisely what ranges of reaction rates and rates of diffusion would result in these effects, and how properties of the pattern (e.g., distance between peaks and valleys) would depend on those microscopic rates. Later research in the mathematical community also uncovered how other interesting phenomena (traveling waves, oscillations) were generated in such systems and how further details of patterns (spots, stripes, etc.) could be affected by geometry, boundary conditions, types of chemical reactions, and so on.

Turing’s theory was later given physical manifestation in artificial chemical systems, manipulated to satisfy the theoretical criteria of pattern formation regimes. And, although biological systems did not produce simple examples of RD pattern formation, the theoretical framework originating in this work motivated later more realistic and biologically based modeling research.

Simple conceptual models can be used to gain insight, develop intuition, and understand “how something works.” For example, the Lotka-Volterra model of species competition and predator-prey^{7} is largely conceptual and is recognized as not being very realistic. Nevertheless, this and similar models have played a strong role in organizing several themes within the discipline: for example, competitive exclusion, the tendency for a species with a slight advantage to outcompete, dominate, and take over from less advantageous species; the cycling behavior in predator-prey interactions; and the effect of