The Physical Process of Two-Level Selection

A dynamical (time-dependent) evolutionary population model featuring two levels of organization (individuals and groups) is studied. The dynamical model is represented by a partial differential equation (PDE) that governs the state of the environment as it evolves in time. The PDE is derived from an underlying stochastic model of two-level selection based on evolutionary birth-death processes. The PDE can be solved numerically to find evolutionary trajectories and equilibrium configurations. A number of important examples of social evolution fit nicely into the modeling framework described here, including virulence, social insect colonies, hunter-gatherer tribes, and other examples where distinct groups of individuals undergo internal evolution, compete with other groups, occasionally fission, and eventually die. The present work is unique in the literature on social evolution because it provides a full solution of a continuous and time-dependent mathematical model, allowing the modeler to predict and study the evolutionary trajectories and equilibrium configurations of the process. The full dynamical solution is possible because complete model generality (e.g., the Price equation) is sacrificed for a slightly less general framework with a more fully exploitable mathematical structure. The analysis here sheds light on some of the philosophical issues surrounding kin selection and group selection explanations of social evolution, but the primary purpose of this paper is to mathematically model the physical (dynamical) process of two-level selection as generally and as accurately as possible. In this way, the underlying mathematical structure of the phenomenon is revealed: the intra-group evolutionary changes (including migration) are reflected in the partial derivative terms of the PDE, while the group level changes are reflected in the other terms.