COMPUTATIONAL MODELING AND SIMULATION AS ENABLERS FOR BIOLOGICAL DISCOVERY 121
exponential growth (if birth rates exceed mortality rates) or extinction (in the opposite case) is a funda-
mental 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 popula-
tions 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 accu-
rate for real populations, but deviations from model predictions are, in themselves, informative, be-
cause 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 mathemati-
cal 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.
5.2.3 Models Uncover New Phenomena or Concepts to Explore,
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.
5.2.4 Models Identify Key Factors or Components of a System,
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
(^6) P.F. Verhulst, “Notice sur la loi que la population suit dans son accroissement,” Correspondence Mathématique et Physique, 1838.
(^7) A.J. Lotka, Elements of Physical Biology, Williams & Wilkins Co., Baltimore, MD, 1925; V. Volterra, “Variazioni e fluttuazioni
del numero d’individui in specie animali conviventi,” Mem. R. Accad. Naz. dei Lincei., Ser. VI, Vol. 2, 1926. The Lotka-Volterra
model is a set of coupled differential equations that relate the densities of prey and predator given parameters involving the
predator-free rate of prey population increase, the normalized rate at which predators can successfully remove prey from the
population, the normalized rate at which predators reproduce, and the rate at which predators die.