American Institute of Mathematical Sciences

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January  2016, 36(1): 63-95. doi: 10.3934/dcds.2016.36.63

The general recombination equation in continuous time and its solution

Ellen Baake 1, , Michael Baake 2, and  Majid Salamat 1,

1. 

Technische Fakultät, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany, Germany
2. 

Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld

Received  August 2014 Revised  March 2015 Published  June 2015

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.
Keywords: population genetics, lattice of partitions, Nonlinear recombination equation, measure-valued ODE, recursive solution..
Mathematics Subject Classification: 34G20, 06B23, 92D1.
Citation: Ellen Baake, Michael Baake, Majid Salamat. The general recombination equation in continuous time and its solution. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 63-95. doi: 10.3934/dcds.2016.36.63
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show all references

References:
[1]

Springer, Berlin, 1997. doi: 10.1007/978-3-642-59101-3.  Google Scholar

[2]

2nd ed., de Gryuter, Berlin, 1995. Google Scholar

[3]

in: Proceedings of the International Congress of Mathematicians, Hyderabad, India, 2010, Vol. VI, ed. J. Bhatia, Hindustan Book Agency, New Delhi (2010), 3037-3053.  Google Scholar

[4]

Bull. Math. Biol., 70 (2008), 603-624; arXiv:q-bio/0612024. doi: 10.1007/s11538-007-9270-5.  Google Scholar

[5]

Monatsh. Math., 146 (2005), 267-278 and 150 (2007), 83-84 (Addendum); arXiv:math.CA/0506099. doi: 10.1007/s00605-005-0326-z.  Google Scholar

[6]

Can. J. Math., 55 (2003), 3-41 and 60 (2008), 264-265 (Erratum); arXiv:math.CA/0210422. doi: 10.4153/CJM-2003-001-0.  Google Scholar

[7]

Markov Proc. Rel. Fields, 17 (2011), 429-446; arXiv:1105.0793.  Google Scholar

[8]

J. Math. Biol., 68 (2014), 1371-1402; arXiv:1206.0950. doi: 10.1007/s00285-013-0662-x.  Google Scholar

[9]

M. Baake and R. Speicher, in, preparation., ().   Google Scholar

[10]

Ann. Human Gen., 18 (1954), 311-317.  Google Scholar

[11]

Wiley, Chichester, 2000.  Google Scholar

[12]

Theor. Popul. Biol., 58 (2000), 1-20. doi: 10.1006/tpbi.2000.1471.  Google Scholar

[13]

Lin. Alg. Appl., 348 (2002), 115-137. doi: 10.1016/S0024-3795(01)00586-9.  Google Scholar

[14]

2nd ed., Springer, New York, 2008. doi: 10.1007/978-0-387-78168-6.  Google Scholar

[15]

M. Esser, S. Probst and E. Baake, Partitioning, duality, and linkage disequilibria in the Moran model with recombination,, submitted, ().   Google Scholar

[16]

Genetics, 87 (1977), 807-819.  Google Scholar

[17]

Vol. I, 3rd ed., Wiley, New York, 1986. doi: 10.1063/1.3062516.  Google Scholar

[18]

Ann. Math. Stat., 15 (1944), 25-57. doi: 10.1214/aoms/1177731313.  Google Scholar

[19]

Springer, Berlin, 1992. doi: 10.1007/978-3-642-76211-6.  Google Scholar

[20]

J. Math. Biol., 38 (1999), 103-133. doi: 10.1007/s002850050143.  Google Scholar

[21]

Cambridge University Press, Cambridge, 1998, reprint, 2005.  Google Scholar

[22]

J. Math. Biol., 49 (2004), 201-226; arXiv:math.DS/0402351. doi: 10.1007/s00285-004-0273-7.  Google Scholar

[23]

Lecture Notes in Computer Science, 4573 (2007), p 130, https://oeis.org/ doi: 10.1007/978-3-540-73086-6_12.  Google Scholar

[24]

IEEE Trans. Automatic Control, 46 (2001), 1028-1047; arXiv:math.DS/0002113. doi: 10.1109/9.935056.  Google Scholar

[25]

Marcel Dekker, New York, 1997.  Google Scholar

[26]

J. Math. Biol., 60 (2010), 727-760; arXiv:0906.1678. doi: 10.1007/s00285-009-0277-4.  Google Scholar

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