Deterministic approximation for population dynamics in the presence of advantageous mutants

Spatial stochastic simulations of evolutionary processes are computationally expensive. Here, based on spatially explicit decoupling approximations (SEDA) introduced in [1], we derive a deterministic approximation to a spatial stochastic birth-death process in the presence of two types: the less advantageous resident type and a more advantageous mutant. At the core of this technique are two essential steps: (1) a system of ODEs that approximate spatial interactions among neighboring individuals must be solved; (2) the time-variable has to be rescaled with a factor (called "") that depends on the kinetic parameters of the wild type and mutant individuals. An explicit formula for is derived, which is a power law of division and death rates of the two types. The method is relatively fast and provides excellent time-series agreement with the stochastic simulation results for the spatial agent-based model. The methodology can be used to describe hard selective sweep events, including the expansion of driver mutations in carcinogenesis, bacterial evolution, and aspects of resistance dynamics.