The recombination equation for interval partitions
- 77 Downloads
The general deterministic recombination equation in continuous time is analysed for various lattices, with special emphasis on the lattice of interval (or ordered) partitions. Based on the recently constructed (Baake et al. in Discr Cont Dynam Syst 36:63–95, 2016) general solution for the lattice of all partitions, the corresponding solution for interval partitions is derived and analysed in detail. We focus our attention on the recursive structure of the solution and its decay rates, and also discuss the solution in the degenerate cases, where it comprises products of monomials with exponentially decaying factors. This can be understood via the Markov generator of the underlying partitioning process that was recently identified. We use interval partitions to gain insight into the structure of the solution, while our general framework works for arbitrary lattices.
KeywordsRecombination equation Population genetics Markov generator Interval partitions Measure-valued equations Nonlinear ODEs Closed solution
Mathematics Subject Classification34G20 06B23 92D10 60J25
It is our pleasure to thank E. Baake and M. Salamat for valuable discussions. ES is grateful to the FSPM\(^2\) of Bielefeld University for financial support during her stay in Bielefeld. This work was also supported by the German Research Foundation (DFG), within the SPP 1590.
- 1.Amann, H.: Gewöhnliche Differentialgleichungen, 2nd edn. de Gryuter, Berlin (1995)Google Scholar
- 3.Baake, E.: Deterministic and stochastic aspects of single-crossover recombination. In: Bhatia, J. (ed.) Proceedings of the International Congress of Mathematicians, Hyderabad, India, 2010, vol. VI. Hindustan Book Agency, New Delhi (2010), pp. 3037–3053. arXiv:1101.2081
- 4.Baake, E., Baake, M.: Haldane linearisation done right: Solving the nonlinear recombination equation the easy way. Discr. Cont. Dynam. Syst. A 36, 6645–6656 (2016). arXiv:1606.05175
- 5.Baake, E., Baake, M., Salamat, M.: The general recombination equation in continuous time and its solution. Discr. Cont. Dynam. Syst. A 36, 63–95 (2016). arXiv:1409.1378
- 6.Baake, M.: Recombination semigroups on measure spaces. Monatsh. Math. 146, 267–278 (2005), 150, 83–84 (2007) (Addendum). arXiv:math.CA/0506099
- 7.Baake, M., Baake, E.: An exactly solved model for mutation, recombination and selection. Can. J. Math. 55, 3–41 (2003), 60, 264–265 (2008) (Erratum). arXiv:math.CA/0210422
- 14.Norris, J.R.: Markov Chains. Cambridge University Press, Cambridge (1997), reprint (2005)Google Scholar
- 15.Popa, E.: Some remarks on a nonlinear semigroup acting on positive measures. In: Carja, O., Vrabie, I.I. (eds.) Applied Analysis and Differential Equations, pp. 308–319. World Scientific, Singapore (2007)Google Scholar