Erratum and addendum to: The general recombination equation in continuous time and its solution

Abstract

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 finitedimensional problem recursively for generic sets of parameters and briefly discuss the singular cases, and how to extend the solution to this situation.

A probabilistic analysis of a discrete-time evolution in recombination

“…Recently, in [4], the continuous-time evolution was studied in a framework of general partitions other than dyadic partitions. The extension of our results to the analogous framework but for discrete-time, deserves a different study.…”

“…Richer discussions on the interpretation of the above equation in a broader perspective of recombination in population genetics, are given in the introductory sections of references [2], [3], [4] and [13].…”

Entropy production in nonlinear recombination models

“…In general there are many other conservation laws in the system (2.4); see part 3 of Proposition 2.3 below. Existence and uniqueness of the solution of (2.4) for any p ∈ P(X ) can be established in a standard way; see, e.g., [1]. From reversibility (2.2), setting f (σ) = p(σ)/µ(σ) one finds…”

“…where A, B, C form a partition of [n], and ν is a probability over such partitions. Generalized recombination models of this kind have been recently considered in [1]. Further generalizations (which are not necessarily even mass-preserving) can be found in the field of "mass action kinetics" introduced in [10,14], which remains a very active area today.…”

“…It is not difficult to see that the map (1.1) preserves the single site marginals, so that p t,i = p 0,i for all t 0 and for all i ∈ [n]. The study of this model goes back to the pioneering work of Geiringer [12]; see also [21,20,1,17] for more recent accounts. As emphasized in [21,20] this model is a special case of the much larger class of so-called "quadratic dynamical systems", which provides a rich family of discrete analogues of Boltzmann's equation from statistical physics.…”