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The general recombination equation in continuous time and its solution. (English) Zbl 1325.34064
Discrete Contin. Dyn. Syst. 36, No. 1, 63-95 (2016); erratum and addendum ibid. 36, No. 4, 2365-2366 (2016).
Summary: The process of recombination in population genetics, in its deterministic limit, leads to a nonlinear ODE in the Banach space of finite measures on a locally compact product space. It has an embedding into a larger family of nonlinear ODEs that permits a systematic analysis with lattice-theoretic methods for general partitions of finite sets. We discuss this type of system, reduce it to an equivalent finite-dimensional nonlinear problem, and establish a connection with an ancestral partitioning process, backward in time. We solve the finite-dimensional problem recursively for generic sets of parameters and briefly discuss the singular cases, and how to extend the solution to this situation.

MSC:
34C60 Qualitative investigation and simulation of ordinary differential equation models
06B23 Complete lattices, completions
34A33 Ordinary lattice differential equations
34G20 Nonlinear differential equations in abstract spaces
92D10 Genetics and epigenetics
Software:
OEIS
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