The genealogy of an exactly solvable ornstein–uhlenbeck type branching process with selection

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Abstract

We study the genealogy of an exactly solvable population model with N particles on the real line, which evolves according to a discrete-time branching process with selection. At each time step, every particle gives birth to children around a times its current position, where a > 0 is a parameter of the model. Then, the N rightmost newborn children are selected to form the next generation. We show that the genealogy of the process converges toward a Beta coalescent as N → ∞. The process we consider can be seen as a toy model version of a continuous-time branching process with selection, in which particles move according to independent Ornstein–Uhlenbeck processes. The parameter a is akin to the pulling strength of the Ornstein–Uhlenbeck process.

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Cortines, A., & Mallein, B. (2018). The genealogy of an exactly solvable ornstein–uhlenbeck type branching process with selection. Electronic Communications in Probability, 23, 1–13. https://doi.org/10.1214/18-ECP197

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