196 CATALYZING INQUIRY
Box 5.23
Genetic Complexities in Evolutionary Processes
The dynamics of alleles at single loci are well understood, but the dynamics of alleles at two loci are still not
completely understood, even in the deterministic case. As a rule, two-locus models require the use of a variety
of computational approaches, from straightforward simulation to more complex analyses based on optimiza-
tion or the use of computer algebra systems. Three-locus models can be understood only through numerical
approaches, except for some very special cases.
Compare these analytical capabilities to the fact that the number of loci exhibiting genetic variation in popu-
lations of higher organisms is well into the thousands. Thus, the number of possible genotypes can be much
larger than the population. In such a situation, the detailed population simulation (i.e., a detailed consider-
ation of events at each locus) leads to problems of substantial computational difficulty.
An alternative is to represent the population as phenotypes—that is, in terms of traits that can be directly
observed and described. For example, certain traits of individuals are quantitative in the sense that they
represent the sum of multiple small effects. Efforts have been undertaken to integrate statistical models of the
dynamics of quantitative traits with more mechanistic genetic approaches, though even under simplifying
assumptions concerning the relation between genotype and phenotype, further approximations are required
to obtain a closed system of equations.
Frequency dependence in evolution refers to the phenomenon in which the fitness of an individual
depends both on its own traits and on the traits of other individuals in the population—that is, selection is
dependent on the frequency with which certain traits appear in the population, not just on pressures from
the environment.
This point arises most strongly in understanding how cooperation (altruism) can evolve through individual
selection. The simplest model is the game of prisoner’s dilemma, in which the game-theoretic solution for a
single encounter between parties is unconditional noncooperation. However, in the iterated prisoner’s dilem-
ma, the game theoretic solution is a strategy known as “tit-for-tat,” which begins with cooperation and then
uses the strategy employed by the other player in the previous interaction. (In other words, the iterated prison-
er’s dilemma stipulates repeated interactions over time between players.)
Although the iterated prisoner’s dilemma yields some insight into how cooperative behavior might emerge
under some circumstances, it is a highly and perhaps oversimplified model. Most importantly, it does not
account for possible spatial localizations of individuals—a point that is important in light of the fact that
individuals who are spatially separated have low probabilities of interacting. Because the evolution of traits
dependent on population frequency requires knowledge of which individuals are interacting, more realistic
models introduce some explicit spatial distribution of individuals—and, for these, simulations are required to
dynamical understanding. These more realistic models suggest that spatial localization affects the evolution of
both cooperative and antagonistic behaviors.
SOURCE: Adapted from S.A. Levin, B. Grenfell, A. Hastings, and A.S. Perelson, “Mathematical and Computational Challenges in
Population Biology and Ecosystems Science,” Science 275(5298):334-343, 1997. (References in the original.)
