A dynamical model of two-level selection

Question: How do continuous-time evolutionary trajectories of two-level selection behave? Approach: Construct and solve a dynamical model of two-level selection capable of predicting evolutionary trajectories and equilibrium configurations. Mathematical methods: Evolutionary birth–death processes, simulation, large population asymptotics, numerical solutions of hyperbolic PDEs. Key assumptions: Environment composed of distinct groups of individuals. Individuals’ birth and death rates are differentiable functions of the state of the environment. Groups’ fissioning and extinction rates are integrable functions of the state of the environment. Main results: A continuous-time, discrete-state, stochastic model of two-level selection that can be simulated exactly. A continuous-time, continuous-state, deterministic (PDE) model of two-level selection that can be solved numerically. A mathematical connection between the stochastic and deterministic models. Equilibrium configurations of the environment in models of the evolution of cooperation by two-level selection often consist of complicated mixtures of groups of varying sizes, ages, and levels of cooperation.