Abstract
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.
Highlights
Many demographic factors, including movements of both individuals and entire populations, can alter the nature of linkages between populations over time
We show that the key characteristics of these migration dynamics—the intensity of migration, pathways of migration and the spatial distribution of populations—give rise to various mathematical structures that allow us to view populations and migration between them as geometric objects, and their dynamics as pathways on a polytope complex K
In the subsection on linear systems and polytopes, we will show that two linear systems of equations/inequalities can be associated with each migration graph, and indicate how our approach leads to two Euclidean polytope complexes that encode all of the information provided by the migration pattern
Summary
Many demographic factors, including movements of both individuals and entire populations, can alter the nature of linkages between populations over time. This basic summary raises the question of whether there is a general description of the space of all possible migration scenarios, within which populations can move, split and merge through time. Population movements and individual migration are interrelated, and the Voronoi cell tessellation must represent both characteristics jointly. With this general framework in mind, we define the concept of a migration pattern . We are not aware of prior applications to migration theory or analysis, especially using combinatorics
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