American Scientist

Evolutionary Dynamics: Exploring the Equations of Life. Martin A. Nowak. xiv + 363 pp. Harvard University Press, 2006. $35.

Today's heroes of evolutionary biology are not just biologists—they are also applied mathematicians. In his new book, Evolutionary Dynamics: Exploring the Equations of Life, Martin A. Nowak provides a readable and at times compelling hands-on account of the growing contributions of mathematics and simulations to the understanding of evolution. He argues that the evolution both of living systems and of social processes such as language should be understood in terms of evolutionary dynamics. Milestones in this interdisciplinary area of study include Motoo Kimura's formulation of the neutral theory of molecular evolution (which holds that most of the mutations that survive in a population provide no selective advantage to the organism), William Hamilton's theory of kin selection (which offers a genetic explanation for altruism), John Maynard Smith's invention of evolutionary game theory, and Robert May's mathematical approaches to ecology and epidemiology.

The format and organization of the book highlight the value that Nowak places on understanding and pedagogy—he starts with a basic foundation and builds on it, using as simple a framework as possible. He explains that evolutionary dynamics can be studied on a "simplex"—the set of all points whose coordinates are not negative and add up to one. The structure of a population composed of n different types is modeled via the vector

and it is assumed that the total population size is constant. Here

where x_i (t) denotes the frequency of type i at time t. This setup (working within the confines of a simplex) makes it easy to explore the operation of mutation and natural selection under conditions of increasing complexity.

Nowak makes a powerful case for the critical role that mathematics and simulations play in the study of evolutionary dynamics. However, a curious student of mathematics and evolutionary biology may wonder, why do such mathematical analyses all work so well? In fact, they don't; hence Nowak should have added some cautionary notes throughout the book to point to problematic cases, or he could have at least cited articles that discuss the consequences of deviating from the standard model in particular instances.

The book covers a rather ambitious program. The initial chapters provide the mathematical groundwork and conceptual framework for the exploration of myriad situations. Topics covered include quasispecies theory, fitness landscapes, sequence spaces and evolutionary game dynamics (which arise when the fitness of an individual depends on the relative abundance of others in the population)—including prisoner's dilemma, games in finite populations, games on graphs, and manipulations on spatial grids that bring together game theory and cellular automata. Later chapters focus on applications: HIV infection within an individual host (the disease progression that leads to AIDS), the evolution of the virulence of infectious agents, the evolutionary dynamics of human cancer (how long it takes for a population of reproducing cells to activate an oncogene or inactivate a tumor suppressor gene) and language evolution (what language is, how children learn language and how language evolves in a population). Key content is revisited in a short concluding chapter that focuses on the quasispecies and replicator equations—that is, the "equations of life" advertised in the book's subtitle.

The chapter on the evolution of virulence revisits a decade-old model that incorporates the possibility of superinfection, which may occur when an already-infected host is invaded by competing parasite strains. The model explicitly assumes that, here, the most virulent strain (the one that results in the greatest rate of disease-induced mortality) will outcompete and displace the less-virulent ones. Curiously enough, this model allows the possibility of the stable coexistence of a potentially unlimited number of strains (one per infected individual) within the host population. To maintain the study of the dynamics and evolution of virulence on a simplex, Nowak assumes that the hosts that perish are immediately replaced by uninfected individuals. As the research of one of us confirms, relaxing such a constraint still results in the possibility of multistrain coexistence, but in somewhat less dramatic terms.

In the chapter on language evolution, Nowak provides a solid introduction to the ideas behind Chomskyan syntax. His grammar-learning model assumes a constant, finite population of grammars. In this model, each learner in the population selects his or her own grammar from a finite menu of possible grammars. Failure to achieve mutual understanding may lead some individuals to select a different grammar from the list.

How reasonable is this assumption? After all, grammars are not holistic objects; they are in fact composed of rules. Individuals do not learn language by hypothesizing grammars out of whole cloth. They construct them by hypothesizing rules. It is this process of creating rules from individual experience that drives language change.

Nowak's model also makes the explicit assumption that two individuals using the same grammar have perfect understanding. Although he states that this assumption can be relaxed in more complicated models, it is not the assumption of perfect understanding that is the problem so much as the assumption that individuals are capable of having identical grammars. This may be a reasonable assumption for a small number of possible grammars. But because grammars are generated from a set of possible rules that is potentially infinite, and because those rules are not inherited or directly transmitted, the idea seems somewhat problematic.

Nowak also models the evolution of grammatical rules by measuring their linguistic fitness. However, he provides no examples of real-life grammatical rules. Nor does he explain exactly how one would measure the fitness of a rule, beyond noting that the fitness advantage of a rule "can be caused by increased linguistic performance or efficiency or simply by conferring an elevated status in the eyes of the observers" and that "others want to learn the new rule." It is disappointing that Nowak does not consider cross-cultural or cross-linguistic issues relating to fitness, though; instead, he limits his discussion to situations where everyone speaks the same language.

Despite such misgivings, we found Evolutionary Dynamics to be both interesting and provocative. It is refreshing that the book explores topics outside the realm of population genetics. Students, in particular, will benefit from the broad selection of topics and from the author's pedagogical approach.