Towards a general theory of group selection.

The longstanding debate about the importance of group (multilevel) selection suffers from a lack of formal models that describe explicit selection events at multiple levels. Here, we describe a general class of models for two-level evolutionary processes which include birth and death events at both levels. The models incorporate the state-dependent rates at which these events occur. The models come in two closely related forms: (1) a continuous-time Markov chain, and (2) a partial differential equation (PDE) derived from (1) by taking a limit. We argue that the mathematical structure of this PDE is the same for all models of two-level population processes, regardless of the kinds of events featured in the model. The mathematical structure of the PDE allows for a simple and unambiguous way to distinguish between individual- and group-level events in any two-level population model. This distinction, in turn, suggests a new and intuitively appealing way to define group selection in terms of the effects of group-level events. We illustrate our theory of group selection by applying it to models of the evolution of cooperation and the evolution of simple multicellular organisms, and then demonstrate that this kind of group selection is not mathematically equivalent to individual-level (kin) selection.