Game theory and evolutionary ecology Evolutionary Games & Population Dynamics

T he liaison between game theory and evolutionary ecology has become a serious and intimate affair during the past few years. The book by Hofbauer and Sigmund will convince even the sceptical of this liaison. Evolutionary Games & Population Dynamics, a thorough revision and extension of the successful 1988 textbook entitled Theory of Evolution and Dynamical Systems by the same authors, makes the conceptual and the mathematical links between these two disciplines obvious. Perhaps the most striking evidence is that the dominant model types of evolutionary game dynamics (replicator equations) and of population dynamics (Lotka–Volterra equations) are, in many instances, mathematically isomorphic. Behind their mathematical equivalence lies a deep, substantial homology between repli-cator dynamics and population dynamics, namely the fundamental, 'like begets like' assumption that they share. From the modeller's perspective, it makes little difference whether the differential properties of the multiplying entities are called strategies or species: the common essence of the replicator-dynamic and the population-dynamic approach is that a set of populations (each assigned a specific heritable trait) compete, or cooperate, for a set of limiting resources. Resources are sometimes explicit, but more often they are implicit, components of the models. Before leading the unsuspecting reader to the wrong conclusion that there are two identical methodologies, one of which is superfluous, let me stress the difference between the two sets of models that is perhaps the most important. The usual state variable of game-theoretical models in general , and replicator dynamics in particular, is a vector of relative replicator frequencies that add up to one for the whole system. This implies a complete disregard of information about absolute population sizes on the one hand, and the relative frequency dependence of the interactions among strategies on the other. From the biologist's viewpoint, the latter is a fundamental difference compared with the usual models of ecological interactions. The state variables of Lotka–Volterra models are density vectors that are not constrained on the unit simplex: each density can take any non-negative real value. Consequently, ecological interactions are assumed to be density dependent (i.e. it is not only the relative but also the absolute size of a population that determines the intensity of its interactions). The cost of this additional detail is one of a methodological nature: density dependence adds an extra dimension to the corresponding system of differential equations, which can make certain analytical techniques difficult to apply. Frequency and density …