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Statistics of branched populations split into different types. (English) Zbl 1456.92118

Summary: Some population is made of \(n\) individuals that can be of \(P\) possible species (or types) at equilibrium. How are individuals scattered among types? We study two random scenarios of such species abundance distributions. In the first one, each species grows from independent founders according to a Galton-Watson branching process. When the number of founders \(P\) is either fixed or random (either Poisson or geometrically-distributed), a question raised is: given a population of \(n\) individuals as a whole, how does it split into the species types? This model is one pertaining to forests of Galton-Watson trees. A second scenario that we will address in a similar way deals with forests of increasing trees. Underlying this setup, the creation/annihilation of clusters (trees) is shown to result from a recursive nucleation/aggregation process as one additional individual is added to the total population.

MSC:

92D25 Population dynamics (general)
60J85 Applications of branching processes
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[25] Appendix · Zbl 0589.33003
[26] We briefly recall a general transfer result of singularity analysis (see Flajolet and Odlyzko (1990))
[27] for generating functions with power-logarithmic singularities of given ordersaandb.
[28] LetΦ (z)be any analytic function in the indented domain defined by D={z:|z| ≤z1,|Arg(z−zc)|> π/2−η},
[29] wherezc,z1> zc, andηare positive real numbers. Assume that, withσ(x) =xalogbx,aand
[30] bany real number (respectively the power and logarithmic singularity exponents or orders ofΦat
[31] 800T. Huillet
[32] zc), we have 1 · Zbl 0939.81522
[33] for some real constantsκ1andκ2.Then,
[34] - ifa /∈ {0,−1,−2, ...}the coefficients in the expansion ofΦ (z)satisfy [zn] Φ (z)∼κ1+κ2zc−n·σ(n)1asn→ ∞,(62)
[35] whereΓ (a)is the Euler function.Φ (z)presents a power-logarithmic singularity atz=zc.If
[36] b= 0,Φ (z)presents a pure power singularity atz=zcof ordera(with power exponenta).
[37] - ifa∈ {0,−1,−2, ...}, the singularityz=zcis purely logarithmic and 0
[38] involving the derivative of the reciprocal Euler function ata.
[39] Thus, for power-logarithmic singularities with ordersaandb, the asymptotics of the coefficients
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