M ATHEMATICS IN B IOLOGY S PECIAL against HIV is one notable example. At the edge the usefulness of statistical recipes to help the problem are misty or a vital parameter is same time, however, there is as yet no agreed design and analyze experiments. More familiar uncertain to within, at best, an order of magni- explanation for why there is so long, and so in some areas than others are the benefits of tude. It makes no sense to convey a beguiling variable, an interval between infection with mathematical studies that underpin pattern- sense of “reality” with irrelevant detail, when S HIV and onset of AIDS. Indeed, I guess that seeking and other software that is indispensable other equally important factors can only be ECTION many researchers in this field do not even think in elucidating genomes, and ultimately in un- guessed at. Above all, remember Einstein’s dic- about this question. But I suspect the answer derstanding how living things assemble them- tum: “models should be as simple as possible, may necessarily involve understanding how selves. Very generally useful are still-unfolding but not more so.” whole populations of different strains of HIV advances that illuminate the frequently counter- interact with whole populations of different intuitive behavior of nonlinear dynamical sys- References and Notes kinds of immune system cells, within infected tems of many kinds. 1. R. A. Fisher, The Genetical Theory of Natural Selection individuals. And understanding the nonlinear Mathematics, however, does not have the (Dover [reprint], New York, 1958). dynamics of such a system will require mathe- long-standing relation to the life sciences that it 2. G. H. Hardy, Science 28, 49 (1908). matical models with similarities to and differ- does to the physical sciences and engineering. It 3. For amore full discussion see R. M. Anderson and R. M. May [Infectious Diseases of Humans (Oxford ences from those that have helped us under- is therefore not surprising to find occasional Univ. Press, 1991), chap. 11]. stand population-level problems in ecology and abuses. Some have been sketched above. Par- 4. J. Bongaarts, Stat. Med. 8, 103 (1989). infectious diseases (8). It may even be that the ticularly tricky are instances in which con- 5. R. MacArthur, Geographical Ecology (Harper & Row, New York, 1972). design of effective vaccines against protean ventional statistical packages (often based on 6. For further discussion, see G. Farmelo (Ed.) [It Must agents like HIV or malaria will require such assumptions of an underlying Gaussian distri- be Beautiful: Great Equations of Modern Science population-level understanding. As yet, this bution—the central limit theorem) are applied (Granta, London, 2002), pp. 212–229] and (7). 7. J. Gleick, Chaos: Making a New Science (Viking, New mathematically theoretical aspect of immunol- to situations involving highly nonlinear dynam- York, 1987). ogy is even less to be found in textbooks than ical processes (which can often lead to situa- 8. M. A. Nowak, R. M. May, Virus Dynamics: Mathemat- were mathematical models in ecology texts a tions in which “rare events” are significantly ical Principles of Immunology and Virology (Oxford generation ago. I venture to predict that the more common than Gaussian distributions sug- Univ. Press, Oxford, 2000). 9. N. Goldenfield, L. P. Kadanoff, Science 284,87 corresponding immunology texts will indeed gest) (9). Perhaps most common among abuses, (1999). look different in 20, or even 10, years’ time. and not always easy to recognize, are situations 10. M. J. Keeling et al., Nature 421, 136 (2003). In short, mathematical models have proved where mathematical models are constructed 11. I thank many colleagues, particularly S. Levin, A. Grafen, D. Rubenstein, and M. Stumpf for helpful to have many uses and to take many forms in with an excruciating abundance of detail in conversations, and the Leverhulme Trust and the the life sciences. We all, by this time, acknowl- some aspects, whilst other important facets of Nuffield Foundation for their support. REVIEW Evolutionary Dynamics of Biological Games Martin A. Nowak1* and Karl Sigmund2,3 Darwinian dynamics based on mutation and selection form the core of mathe- can be part of the fitness landscape of the matical models for adaptation and coevolution of biological populations. The other. A host’s successful immune response evolutionary outcome is often not a fitness-maximizing equilibrium but can to a pathogen, for instance, will exert se- include oscillations and chaos. For studying frequency-dependent selection, lection pressure leading to adapted strains game-theoretic arguments are more appropriate than optimization algorithms. of pathogens, and vice versa (3–5). But Replicator and adaptive dynamics describe short- and long-term evolution in even within a single population, the fitness phenotype space and have found applications ranging from animal behavior and of a trait often depends on the prevalence of ecology to speciation, macroevolution, and human language. Evolutionary game that trait: The selective advantage of a giv- theory is an essential component of a mathematical and computational approach en tree height, for example, depends on the to biology. heights of neighboring trees. Similarly, the success of a given sex ratio depends on the Evolution through natural selection is often population would therefore be expected to overall sex ratio in the population. understood to imply improvement and increase over time. This is often pictured as Therefore, the fitness landscape is progress. A heritable trait that confers to its a steady ascent on a so-called fitness land- shaped by the phenotypic distributions of bearer a higher fitness will spread within scape. The landscape metaphor suggests the involved populations. As the population the population. The average fitness of the some solid ground over which the popula- moves through the fitness landscape, new tion moves. This paradigm (1), which is peaks and valleys form, channeling its fur- also widespread in the theory of genetic ther motion. This viewpoint affects not 1Program for Evolutionary Dynamics, Department of Mathematics, Department of Organismic and algorithms (2), neglects one-half of the only the intuition of evolutionary biologists Evolutionary Biology, Harvard University, 1 Brattle evolutionary mechanism: Although the en- but also their theoretical tools. The proper Square, Cambridge, MA 02138, USA. 2Faculty for vironment selects the adaptations, these ad- technique for describing uphill motion on Mathematics, University of Vienna, Nordberggasse aptations can shape the environment. By solid ground is optimization theory, a set of 15, A-1090 Vienna, Austria. 3International Institute for Applied Systems Analysis (IIASA), A-2361 Lax- moving across a fitness landscape, popula- mathematical techniques developed in the enburg, Austria. tions change that landscape (Fig. 1). past 300 years, mostly to solve physical or *To whom correspondence should be addressed. This is particularly clear if several pop- technical problems. If the adaptive steps, E-mail: [email protected] ulations interact, because each population however, imply changes in the environ- www.sciencemag.org SCIENCE VOL 303 6 FEBRUARY 2004 793 M ATHEMATICS IN B IOLOGY ment, eventually necessitating new adapta- root structure or tree height are problems of infect the same host or when rapid evolution tions, then game theory is the appropriate resource allocation (15). Conflicts concern- generates many different parasite mutants in ECTION framework. This technique originated more ing mate choice (16), sibling rivalry (17), and any one infected individual (33). Lack of coop- S than 50 years ago to tackle economic and parent-offspring antagonism (18) are a rich eration among parasites can lead to short- social problems involving interdependen- mine of game-theoretic models; so are social sighted, maladapted levels of excessive viru- cies among several agents. Evolutionary foraging, dispersal, and habitat selection (19). lence harming both host and parasite. biologists soon understood its potential and The arms races between predators and prey, The growth in the range of applications PECIAL started applying it to evolutionary problems or between parasites and their hosts, offer demanded an extension of classical game the- S (6–8). The success of a strategy in a game many examples of games between distinct ory, away from the prevalent static doctrine depends on the co-player’s strategy, much populations (20). Communication in its dominated by the equilibrium notion of Nash as the fitness of a phenotype depends on the widest sense, including alarm calls, threat and by the quest for a “unique solution” to composition of the population. Roughly displays, or sexual advertisement, lead to rational play. The concepts of “unbeatable speaking, game theory is the mathematical game-theoretic problems concerning bluff strategy” (6) and “evolutionary stability” (7) toolbox for methodological individualism, and honest signaling (21). Acquisition and implicitly assumed some underlying popula- the systematic attempt to found social the- performance of human language in a heter- tion dynamics describing the potential suc- ory on the actions and needs of individual ogeneous population can be studied as an cess of invading mutants and, more generally, agents (9, 10). For outcomes shaped by evolutionary game (22). Increasingly, evo- the interplay of mutation with frequency- “selfish genes” or by the selfish “homo