Evolutionary Dynamics in Finite Populations Mix Rapidly

10 years 18 days ago

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In this paper we prove that the mixing time of a broad class of evolutionary dynamics in ﬁnite, unstructured populations is roughly logarithmic in the size of the state space. An important special case of such a stochastic process is the Wright-Fisher model from evolutionary biology (with selection and mutation) on a population of size N over m genotypes. Our main result implies that the mixing time of this process is O(log N) for all mutation rates and ﬁtness landscapes, and solves the main open problem from [4]. In particular, it signiﬁcantly extends the main result in [18] who proved this for m = 2. Biologically, such models have been used to study the evolution of viral populations with applications to drug design strategies countering them. Here the time it takes for the population to reach a steady state is important both for the estimation of the steady-state structure of the population as well in the modeling of the treatment strength and duration. Our result, that such ...

Ioannis Panageas, Piyush Srivastava, Nisheeth K. V

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