Deterministic approximation for population dynamics in the presence of advantageous mutants

Abstract

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.