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The recombination equation for interval partitions. (English) Zbl 1357.92050
Summary: 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 general solution for the lattice of all partitions [M. Salamat et al., Discrete Contin. Dyn. Syst. 36, No. 1, 63–95 (2016; Zbl 1325.34064); erratum and addendum ibid. 36, No. 4, 2365–2366 (2016; Zbl 1326.34080)], 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.

MSC:
92D10 Genetics and epigenetics
34G20 Nonlinear differential equations in abstract spaces
06B23 Complete lattices, completions
60J25 Continuous-time Markov processes on general state spaces
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