Haldane linearisation done right: Solving the nonlinear recombination equation the easy way

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December 2016, 36(12): 6645-6656. doi: 10.3934/dcds.2016088

Haldane linearisation done right: Solving the nonlinear recombination equation the easy way

Ellen Baake ^1, and Michael Baake ^2,

1.	Technische Fakultät, Universität Bielefeld, Postfach 100131, 33501 Bielefeld
2.	Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld

Received February 2016 Revised July 2016 Published October 2016

Abstract
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Under a Creative Commons license

The nonlinear recombination equation from population genetics has a long history and is notoriously difficult to solve, both in continuous and in discrete time. This is particularly so if one aims at full generality, thus also including degenerate parameter cases. Due to recent progress for the continuous time case via the identification of an underlying stochastic fragmentation process, it became clear that a direct general solution at the level of the corresponding ODE itself should also be possible. This paper shows how to do it, and how to extend the approach to the discrete-time case as well.

Keywords: closed solution., Nonlinear recombination equation, measure-valued equations, population genetics, continuous and discrete time.

Mathematics Subject Classification: 34G20, 06B23, 39A12, 92D1.

Citation: Ellen Baake, Michael Baake. Haldane linearisation done right: Solving the nonlinear recombination equation the easy way. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6645-6656. doi: 10.3934/dcds.2016088

References:

[1]	M. Aigner, Combinatorial Theory, reprint, Springer, Berlin, 1997. doi: 10.1007/978-3-642-59101-3.
[2]	H. Amann, Gewöhnliche Differentialgleichungen, 2nd ed., de Gryuter, Berlin, 1995.
[3]	E. Baake, Deterministic and stochastic aspects of single-crossover recombination, in: Proceedings of the International Congress of Mathematicians, Hyderabad, India, 2010, Vol. VI, ed. J. Bhatia, Hindustan Book Agency, New Delhi (2010), 3037-3053; arXiv:1101.2081.
[4]	E. Baake, M. Baake and M. Salamat, The general recombination equation in continuous time and its solution, Discr. Cont. Dynam. Syst. A, 36 (2016), 63-95, arXiv:1409.1378. doi: 10.3934/dcds.2016.36.63.
[5]	M. Baake, Recombination semigroups on measure spaces}, Monatsh. Math., 146 (2005), 267-278 and 150 (2007), 83-84 (Addendum); arXiv:math.CA/0506099. doi: 10.1007/s00605-005-0326-z.
[6]	M. Baake and E. Baake, An exactly solved model for mutation, recombination and selection, Can. J. Math., 55 (2003), 3-41 and 60 (2008), 264-265 (Erratum); arXiv:math.CA/0210422. doi: 10.4153/CJM-2003-001-0.
[7]	M. Baake and E. Shamsara, The recombination equation for interval partitions, preprint, arXiv:1508.04985.
[8]	J. H. Bennett, On the theory of random mating, Ann. Human Gen., 18 (1954), 311-317.
[9]	R. Bürger, The Mathematical Theory of Selection, Recombination and Mutation, Wiley, Chichester, 2000.
[10]	F. B. Christiansen, Population Genetics of Multiple Loci, Wiley, Chichester (1999).
[11]	K. J. Dawson, The decay of linkage disequilibrium under random union of gametes: How to calculate Bennett's principal components, Theor. Popul. Biol., 58 (2000), 1-20. doi: 10.1006/tpbi.2000.1471.
[12]	K. J. Dawson, The evolution of a population under recombination: How to linearise the dynamics, Lin. Alg. Appl., 348 (2002), 115-137. doi: 10.1016/S0024-3795(01)00586-9.
[13]	H. Geiringer, On the probability theory of linkage in Mendelian heredity, Ann. Math. Stat., 15 (1944), 25-57. doi: 10.1214/aoms/1177731313.
[14]	H. S. Jennings, The numerical results of diverse systems of breeding, with respect to two pairs of characters, linked or independent, with special relation to the effects of linkage. Genetics, 2 (1917), 97-154.
[15]	Y. I. Lyubich, Mathematical Structures in Population Genetics, Springer, Berlin, 1992. doi: 10.1007/978-3-642-76211-6.
[16]	S. Martínez, A probabilistic analysis of a discrete-time evolution in recombination, preprint, arXiv:1603.07201, and dto., part II: On partitions, preprint, arXiv:1604.05124.
[17]	D. McHale and G. A. Ringwood, Haldane linearisation of baric algebras, J. London Math. Soc., 28 (1983), 17-26. doi: 10.1112/jlms/s2-28.1.17.
[18]	T. Nagylaki, J. Hofbauer and P. Brunovski, Convergence of multilocus systems under weak epistasis or weak selection, J. Math. Biol., 38 (1999), 103-133. doi: 10.1007/s002850050143.
[19]	J. R. Norris, Markov Chains, Cambridge University Press, Cambridge, 1998.
[20]	R. B. Robbins, Some applications of mathematics to breeding problems III. Genetics, 3 (1918), 375-389.
[21]	U. von Wangenheim, E. Baake and M. Baake, Single-crossover recombination in discrete time, J. Math. Biol., 60 (2010), 727-760; arXiv:0906.1678. doi: 10.1007/s00285-009-0277-4.

show all references

References:

[1]	M. Aigner, Combinatorial Theory, reprint, Springer, Berlin, 1997. doi: 10.1007/978-3-642-59101-3.
[2]	H. Amann, Gewöhnliche Differentialgleichungen, 2nd ed., de Gryuter, Berlin, 1995.
[3]	E. Baake, Deterministic and stochastic aspects of single-crossover recombination, in: Proceedings of the International Congress of Mathematicians, Hyderabad, India, 2010, Vol. VI, ed. J. Bhatia, Hindustan Book Agency, New Delhi (2010), 3037-3053; arXiv:1101.2081.
[4]	E. Baake, M. Baake and M. Salamat, The general recombination equation in continuous time and its solution, Discr. Cont. Dynam. Syst. A, 36 (2016), 63-95, arXiv:1409.1378. doi: 10.3934/dcds.2016.36.63.
[5]	M. Baake, Recombination semigroups on measure spaces}, Monatsh. Math., 146 (2005), 267-278 and 150 (2007), 83-84 (Addendum); arXiv:math.CA/0506099. doi: 10.1007/s00605-005-0326-z.
[6]	M. Baake and E. Baake, An exactly solved model for mutation, recombination and selection, Can. J. Math., 55 (2003), 3-41 and 60 (2008), 264-265 (Erratum); arXiv:math.CA/0210422. doi: 10.4153/CJM-2003-001-0.
[7]	M. Baake and E. Shamsara, The recombination equation for interval partitions, preprint, arXiv:1508.04985.
[8]	J. H. Bennett, On the theory of random mating, Ann. Human Gen., 18 (1954), 311-317.
[9]	R. Bürger, The Mathematical Theory of Selection, Recombination and Mutation, Wiley, Chichester, 2000.
[10]	F. B. Christiansen, Population Genetics of Multiple Loci, Wiley, Chichester (1999).
[11]	K. J. Dawson, The decay of linkage disequilibrium under random union of gametes: How to calculate Bennett's principal components, Theor. Popul. Biol., 58 (2000), 1-20. doi: 10.1006/tpbi.2000.1471.
[12]	K. J. Dawson, The evolution of a population under recombination: How to linearise the dynamics, Lin. Alg. Appl., 348 (2002), 115-137. doi: 10.1016/S0024-3795(01)00586-9.
[13]	H. Geiringer, On the probability theory of linkage in Mendelian heredity, Ann. Math. Stat., 15 (1944), 25-57. doi: 10.1214/aoms/1177731313.
[14]	H. S. Jennings, The numerical results of diverse systems of breeding, with respect to two pairs of characters, linked or independent, with special relation to the effects of linkage. Genetics, 2 (1917), 97-154.
[15]	Y. I. Lyubich, Mathematical Structures in Population Genetics, Springer, Berlin, 1992. doi: 10.1007/978-3-642-76211-6.
[16]	S. Martínez, A probabilistic analysis of a discrete-time evolution in recombination, preprint, arXiv:1603.07201, and dto., part II: On partitions, preprint, arXiv:1604.05124.
[17]	D. McHale and G. A. Ringwood, Haldane linearisation of baric algebras, J. London Math. Soc., 28 (1983), 17-26. doi: 10.1112/jlms/s2-28.1.17.
[18]	T. Nagylaki, J. Hofbauer and P. Brunovski, Convergence of multilocus systems under weak epistasis or weak selection, J. Math. Biol., 38 (1999), 103-133. doi: 10.1007/s002850050143.
[19]	J. R. Norris, Markov Chains, Cambridge University Press, Cambridge, 1998.
[20]	R. B. Robbins, Some applications of mathematics to breeding problems III. Genetics, 3 (1918), 375-389.
[21]	U. von Wangenheim, E. Baake and M. Baake, Single-crossover recombination in discrete time, J. Math. Biol., 60 (2010), 727-760; arXiv:0906.1678. doi: 10.1007/s00285-009-0277-4.

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