Deterministic and stochastic aspects of single-crossover recombination

Baake E. Deterministic and stochastic aspects of single-crossover recombination. In: Proceedings of the International Congress of Mathematicians, Hyderabad 2010, ed. R. Bhatia, Hindustan Book Agency, New Delhi, Vol. IV, pp. 3038--3053. 2010.This contribution is concerned with mathematical models for the dynamics ofthe genetic composition of populations evolving under recombination.Recombination is the genetic mechanism by which two parent individuals createthe mixed type of their offspring during sexual reproduction. The correspondingmodels are large, nonlinear dynamical systems (for the deterministic treatmentthat applies in the infinite-population limit), or interacting particle systems(for the stochastic treatment required for finite populations). We reviewrecent progress on these difficult problems. In particular, we present a closedsolution of the deterministic continuous-time system, for the important specialcase of single crossovers; we extract an underlying linearity; we analyse howthis carries over to the corresponding stochastic setting; and we provide asolution of the analogous deterministic discrete-time dynamics, in terms of itsgeneralised eigenvalues and a simple recursion for the correspondingcoefficients