Hierarchical Meanfield Theory of Evolutionary Games on Structured Populations

author: Gergely J. Szöllősi, Department of Biological Physics, Eötvös Loránd University
published: Nov. 26, 2007,   recorded: October 2007,   views: 2961

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Evolutionary games played out in populations structured by spatial embedding or more general networks of interaction have been shown to have fundamentally different dynamics and outcomes compared to those taking place in well mixed ones. Recent experimental and theoretical work - published largely in interdisciplinary journals such as Nature and PNAS - has demonstrated that longstanding open problems in biology, sociology, and the economic sciences (ranging from the maintenance of diversity to the evolution of altruism and reciprocity) can only be understood if we look beyond well mixed populations and take into account the effects of spatial structure. The question of how one goes about choosing the relevant model to describe population structures, however, stands unanswered. Models where individuals are confined to the sites of some lattice or the nodes of some random graph have proved highly sensitive to seemingly minor changes in the implementation of the dynamics and the details of the underlying topology of interactions. In our paper we introduce a minimal model of population structure that is described by two distinct hierarchical levels of interaction. We derive the dynamics governing the evolution of such a system starting from fundamental individual level stochastic processes and find that the simple and straightforward hierarchical application of the mean-field approximation (the assumption of being well mixed) at both levels surprisingly unveils a new level of dynamical complexity. We believe that such minimal structure is more relevant in a wide range of natural systems, than more subtle setups with a delicate dependence on the details and symmetries of the model. We show that such minimal structure is sufficient for the emergence of effects previously only observed for explicit spatial embedding and demonstrate the potential of our model to identify robust effects of population structure on the dynamics and outcome of evolutionary games.

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