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Species abundance distributions in neutral models with immigration or mutation and general lifetimes. (English) Zbl 1230.92043
Summary: We consider a general, neutral, dynamical model of biodiversity. Individuals have i.i.d. life time durations, which are not necessarily exponentially distributed, and each individual gives birth independently at a constant rate λ. Thus, the population size is a homogeneous, binary Crump-Mode-Jagers process (which is not necessarily a Markov process). We assume that types are clonally inherited. We consider two classes of speciation models in this setting. In the immigration model, new individuals of an entirely new species singly enter the population at a constant rate μ (e.g., from the mainland into the island). In the mutation model, each individual independently experiences point mutations in its germ line, at a constant rate θ. We are interested in the species abundance distribution, i.e., in the numbers, denoted I n (k) in the immigration model and A n (k) in the mutation model, of the species represented by k individuals, k=1,2,,n, when there are n individuals in the total population. In the immigration model, we prove that the numbers (I t (k);k1) of the species represented by k individuals at time t, are independent Poisson variables with parameters as in Fisher’s log-series. When conditioning on the total size of the population to equal n, this results in species abundance distributions given by W.J. Ewens’ [Theor. Popul. Biol. 3, 87–112 (1972; Zbl 0245.92009)] sampling formula. In particular, I n (k) converges as n to a Poisson r.v. with mean γ/k, where γ:=μ/λ. In the mutation model, as n, we obtain the almost sure convergence of n -1 A n (k) to a nonrandom explicit constant. In the case of a critical, linear birth-death process, this constant is given by Fisher’s log-series, namely n -1 A n (k) converges to α k /k, where α:=λ/(λ+θ). In both models, the abundances of the most abundant species are briefly discussed.
92D15Problems related to evolution
60J85Applications of branching processes
92D25Population dynamics (general)
60J80Branching processes
60G51Processes with independent increments; Lévy processes
[1]Athreya KB, Ney PE (1972) Branching processes. Springer-Verlag, New York
[2]Bertoin J (1996) Lévy processes. Cambridge University Press, Cambridge
[3]Champagnat N, Lambert A (2010a) Splitting trees with neutral Poissonian mutations I: small families. (submitted)
[4]Champagnat N, Lambert A (2010b) Splitting trees with neutral Poissonian mutations II: large families. (in preparation)
[5]Donnelly P, Tavaré S (1986) The ages of alleles and a coalescent. Adv Appl Probab 18: 1–19 · Zbl 0588.92013 · doi:10.2307/1427237
[6]Durrett R (2008) Probability models for DNA sequence evolution, 2nd revised edn. Springer-Verlag, Berlin
[7]Etienne RS, Alonso D, McKane AJ (2007) The zero-sum assumption in neutral biodiversity theory. J Theor Biol 248: 522–536 · doi:10.1016/j.jtbi.2007.06.010
[8]Ewens WJ (1972) The sampling theory of selectively neutral alleles. Theor Popul Biol 3: 87–112 [Erratum, p 376] · Zbl 0245.92009 · doi:10.1016/0040-5809(72)90035-4
[9]Ewens WJ (2005) Mathematical population genetics, 2nd edn. Springer-Verlag, Berlin
[10]Fisher RA (1943) A theoretical distribution for the apparent abundance of different species. J Anim Ecol 12: 54–58
[11]Fisher RA, Corbet SA, Williams CB (1943) The relation between the number of species and the number of individuals in a random sample of an animal population. J Anim Ecol 12: 42–58 · doi:10.2307/1411
[12]Geiger J, Kersting G (1997) Depth-first search of random trees, and Poisson point processes. In: Classical and modern branching processes (Minneapolis, 1994). IMA Math Appl, vol 84. Springer-Verlag, New York
[13]Haegeman B, Etienne RS (2008) Relaxing the zero-sum assumption in neutral biodiversity theory. J Theor Biol 252: 288–294 · doi:10.1016/j.jtbi.2008.01.023
[14]Hubbell SP (2001) The unified neutral theory of biodiversity and biogeography. Princeton University Press, Princeton
[15]Jagers P (1974) Convergence of general branching processes and functionals thereof. J Appl Probab 11: 471–478 · Zbl 0296.60054 · doi:10.2307/3212691
[16]Jagers P, Nerman O (1984a) The growth and composition of branching populations. Adv Appl Probab 16: 221–259 · Zbl 0535.60075 · doi:10.2307/1427068
[17]Jagers P, Nerman O (1984b) Limit theorems for sums determined by branching processes and other exponentially growing processes. Stoch Proc Appl 17: 47–71 · Zbl 0532.60081 · doi:10.1016/0304-4149(84)90311-9
[18]Karlin S, McGregor (1967) The number of mutant forms maintained in a population. In: Proc 5th Berkeley Symposium Math Statist Prob IV:415–438
[19]Kendall DG (1948) On some modes of population growth leading to R.A. Fisher’s logarithmic series distribution. Biometrika 35: 6–15
[20]Kimura M, Crow JF (1964) The number of alleles that can be maintained in a finite population. Genetics 49: 725–738
[21]Lambert A (2009) The allelic partition for coalescent point processes. Markov Proc Relat Fields 15: 359–386
[22]Lambert A (2010) The contour of splitting trees is a Lévy process. Ann Probab 38: 348–395 · Zbl 1190.60083 · doi:10.1214/09-AOP485
[23]MacArthur RH, Wilson EO (1967) The theory of island biogeography. Princeton University Press, Princeton
[24]Rannala B (1996) The sampling theory of neutral alleles in an island population of fluctuating size. Theor Popul Biol 50: 91–104 · Zbl 0856.92013 · doi:10.1006/tpbi.1996.0024
[25]Rannala B (1997) Gene genealogy in a population of variable size. Heredity 78: 417–423 · doi:10.1038/hdy.1997.65
[26]Richard M (2010) Limit theorems for splitting trees with structured immigration and applications to biogeography. Preprint available at http://hal.archives-ouvertes.fr/hal-00507443/fr/
[27]Taïb Z (1992) Branching processes and neutral evolution. In: Lecture notes in biomathematics, vol 93. Springer-Verlag, Berlin
[28]Watterson GA (1974) Models for the logarithmic species abundance distributions. Theor Popul Biol 6: 217–250 · Zbl 0292.92003 · doi:10.1016/0040-5809(74)90025-2