Abstract
We study a class of coalescents derived from a sampling procedure out of \(N\) i.i.d. Pareto\(\left( \alpha \right) \) random variables, normalized by their sum, including \(\beta \) –size-biasing on total length effects (\(\beta <\alpha \)). Depending on the range of \(\alpha ,\) we derive the large \(N\) limit coalescents structure, leading either to a discrete-time Poisson-Dirichlet \( \left(\alpha ,-\beta \right) \Xi -\)coalescent (\(\alpha \in \left[ 0,1\right) \)), or to a family of continuous-time Beta\(\left( 2-\alpha ,\alpha -\beta \right) \Lambda -\)coalescents (\(\alpha \in \left[ 1,2\right) \)), or to the Kingman coalescent (\(\alpha \ge 2\)). We indicate that this class of coalescent processes (and their scaling limits) may be viewed as the genealogical processes of some forward in time evolving branching population models including selection effects. In such constant-size population models, the reproduction step, which is based on a fitness-dependent Poisson Point Process with scaling power-law\(\left( \alpha \right) \) intensity, is coupled to a selection step consisting of sorting out the \(N\) fittest individuals issued from the reproduction step.
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Notes
This particular way of introducing selection in a randomly evolving branching population with constant poulation size seems to appear first in Brunet et al. (2006). It has nothing to do with the way selection is classically introduced in population genetics; see Maruyama (1977), Ewens (2004) and Bürger (2000).
We abusively use the same notation \(P_{i,1}^{\left( N\right) }\) in the size-biased setup as in (1) (corresponding to \(\beta =0\)), to avoid overburden notations.
A ‘true’ coalescent process takes values in the set of equivalence relations or partitions on \(\left\{ 1,\ldots ,N\right\} \) and we rather deal here and throughout with its block-counting counterpart.
Here, because two processes are involved, the symbol \(\overset{d}{=}\) means convergence of all the finite-dimensional distributions of \(x_{\left[ t/c_{N}\right] }^{\left( N\right) }, t\ge 0\) to the ones of \(x_{t}, t\ge 0\).
\(\overset{d}{=}\) means equality in distribution between random variables.
The technique we use is inspired from the one used in Brunet and Derrida (2013) in a particular case. The author is indebted to B. Derrida for pointing this out to him.
The \(^{\prime }\) symbol indicates derivative with respect to \(x\).
We used the scaling property of Pareto\(\left( \alpha \right) \) rvs \(X\) on \( \left( 1,\infty \right) \) stating that \(X\mid X>a\overset{d}{=}aX.\)
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Acknowledgments
The author acknowledges partial support from the ANR Modélisation Aléatoire en Écologie, Génétique et Évolution (ANR-Manège- 09-BLAN-0215 project) and from the labex MME-DII (Modèles Mathématiques et Économiques de la Dynamique, de l’ Incertitude et des Interactions). The author is also indebted to his referees for pointing out some errors in an earlier version of the draft and for encouraging him to write down a more concise and complete version.
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Huillet, T.E. Pareto genealogies arising from a Poisson branching evolution model with selection. J. Math. Biol. 68, 727–761 (2014). https://doi.org/10.1007/s00285-013-0649-7
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DOI: https://doi.org/10.1007/s00285-013-0649-7
Keywords
- Pareto coalescents
- Scaling limits
- Poisson-Dirichlet
- Kingman and Beta coalescents
- Poisson point process
- Evolution model including selection