Logarithmic Gromov-Witten invariants

The goal of this paper is to give a general theory of logarithmic Gromov-Witten invariants. This gives a vast generalization of the theory of relative Gromov-Witten invariants introduced by Li-Ruan, Ionel-Parker, and Jun Li, and completes a program first proposed by the second named author in 2002. One considers target spaces X carrying a log structure. Domains of stable log curves are log smooth curves. Algebraicity of the stack of such stable log maps is proven, subject only to the hypothesis that the log structure on X is fine, saturated, and Zariski. A notion of basic stable log map is introduced; all stable log maps are pull-backs of basic stable log maps via base-change. With certain additional hypotheses, the stack of basic stable log maps is proven to be proper. Under these hypotheses and the additional hypothesis that X is log smooth, one obtains a theory of log Gromov-Witten invariants.Comment: 58 pages, 5 figure

Consequences of fluctuating group size for the evolution of cooperation

Studies of cooperation have traditionally focused on discrete games such as the well-known prisoner's dilemma, in which players choose between two pure strategies: cooperation and defection. Increasingly, however, cooperation is being studied in continuous games that feature a continuum of strategies determining the level of cooperative investment. For the continuous snowdrift game, it has been shown that a gradually evolving monomorphic population may undergo evolutionary branching, resulting in the emergence of a defector strategy that coexists with a cooperator strategy. This phenomenon has been dubbed the 'tragedy of the commune'. Here we study the effects of fluctuating group size on the tragedy of the commune and derive analytical conditions for evolutionary branching. Our results show that the effects of fluctuating group size on evolutionary dynamics critically depend on the structure of payoff functions. For games with additively separable benefits and costs, fluctuations in group size make evolutionary branching less likely, and sufficiently large fluctuations in group size can always turn an evolutionary branching point into a locally evolutionarily stable strategy. For games with multiplicatively separable benefits and costs, fluctuations in group size can either prevent or induce the tragedy of the commune. For games with general interactions between benefits and costs, we derive a general classification scheme based on second derivatives of the payoff function, to elucidate when fluctuations in group size help or hinder cooperation.Comment: 22 pages, 5 figure