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We present a new theoretical framework to represent the dynamics of structured populations through time and across geographic space. We show (i) that the mechanisms by which populations evolve lead to combinatorial structures, and (ii) that measures of gene flow and geographical structure lead to linear systems. These characteristics determine two polytope complexes that encodes all feasible migration scenarios. Analysis of these polytope complexes demonstrates how systems of structured populations can be classified consistently, and how population histories can be represented as paths on a concrete mathematical space, which in turn promises to simplify the search space required for reconstructing past migration processes from population genetic data.

A flexible theoretical representation for the temporal dynamics of structured populations as paths on polytope complexes