Evolutionary games and quasispecies

– We discuss a population of sequences subject to mutations and frequencydependent selection, where the fitness of a sequence depends on the composition of the entire population. This type of dynamics is crucial to understand, for example, the coupled evolution of different strands in a viral population. Mathematically, it takes the form of a reactiondiffusion problem that is nonlinear in the population state. In our model system, the fitness is determined by a simple mathematical game, the hawk-dove game. The stationary population distribution is found to be a quasispecies with properties different from those which hold in fixed fitness landscapes. Introduction. – The roles of chance and determinism are a central theme in evolutionary biology. Eigen’s quasispecies theory [1] has been pivotal as a simple quantitative model for the intertwined effects of random mutations and Darwinian selection. These forces act on individuals with a genotype defined by an L-letter sequence σ = (σ1, . . . , σL). The individuals reproduce at a rate f(σ) called the fitness of the genotype σ, and are subject to random mutations of the sequence elements at a rate μ. The population is described by a timedependent frequency distribution P (σ). The evolution of P (σ) is described by a deterministic equation in the limit of large population size, when sampling errors in the reproductionmutation process become negligible. For small μ, the evolution is dominated by reproductive success and produces a quasispecies, that is, a stationary population distribution ps(σ) peaked around the genotype σ of maximal fitness. Large mutation rates, on the other hand, wipe out fitness differences and lead to a broad distribution. These two regimes are linked by a crossover which, depending on the fitness “landscape” f(σ), may become a phase transition in the limit of infinite sequence length. The transition point is called the error threshold . Quasispecies evolution can be seen as a reaction-diffusion problem in sequence space that is conceptually related to problems in statistical physics. It takes the form of an (imaginary-time) Schrödinger equation for the population state, ∂tP = HP , with f(σ) as scalar potential and the kinetic term describing mutations [2]. This type of problem is mathematically tractable (∗) To whom correspondence should be addressed. E-mail: [email protected] (∗∗) Associato INFN, Sezione di Napoli.