Journal of Mathematical Biology

, Volume 73, Issue 1, pp 123–159 | Cite as

Stochastic analysis of the extra clustering model for animal grouping

  • Michael Drmota
  • Michael FuchsEmail author
  • Yi-Wen Lee


We consider the extra clustering model which was introduced by Durand et al. (J Theor Biol 249(2):262–270, 2007) in order to describe the grouping of social animals and to test whether genetic relatedness is the main driving force behind the group formation process. Durand and François (J Math Biol 60(3):451–468, 2010) provided a first stochastic analysis of this model by deriving (amongst other things) asymptotic expansions for the mean value of the number of groups. In this paper, we will give a much finer analysis of the number of groups. More precisely, we will derive asymptotic expansions for all higher moments and give a complete characterization of the possible limit laws. In the most interesting case (neutral model), we will prove a central limit theorem with a surprising normalization. In the remaining cases, the limit law will be either a mixture of a discrete and continuous law or a discrete law. Our results show that, except of in degenerate cases, strong concentration around the mean value takes place only for the neutral model, whereas in the remaining cases there is also mass concentration away from the mean.


Social animals Number of groups Moments Limit laws Singularity perturbation analysis 

Mathematics Subject Classification

05A16 60F05 92B05 



An extended abstract (Drmota et al. 2014) and talk with preliminary materials of this paper were presented at the 25th International Meeting on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms which took place in Paris from June 16 to June 20, 2014. The authors thank the reviewers of this abstract and the participants of the meeting for valuable input. Moreover, the authors also thank the referees of the present paper for helpful comments leading to an improvement of the presentation. M. Drmota acknowledges partial support by the Austrian Science Foundation, SFB F50 “Algorithmic and Enumerative Combinatorics”. M. Fuchs was partially supported by the Ministry of Science and Technology, Taiwan under the Grant MOST-103-2115-M-009-007-MY2.


  1. Beals R, Wong R (2010) Special functions: a graduate text. Cambridge studies in advanced mathematics, vol 126. Cambridge University Press, CambridgeGoogle Scholar
  2. Billingsley P (1995) Probability and measure, 3rd edn. Wiley series in probability and mathematical statistics. A Wiley-Interscience Publication/Wiley, New YorkGoogle Scholar
  3. Blum MGB, François O (2005) Minimal clade size and external branch length under the neutral coalescent. Adv Appl Probab 37(3):647–662MathSciNetCrossRefzbMATHGoogle Scholar
  4. Blum MGB, François O, Janson S (2006) The mean, variance and limiting distributions of two statistics sensitive to phylogenetic tree balance. Ann Appl Probab 16(4):2195–2214MathSciNetCrossRefzbMATHGoogle Scholar
  5. Chang H, Fuchs M (2010) Limit theorems for patterns in phylogenetic trees. J Math Biol 60(4):481–512MathSciNetCrossRefzbMATHGoogle Scholar
  6. Chern H-H, Fuchs M, Hwang H-K (2007) Phase changes in random point quadtrees. ACM Trans Algorithms 3(2):12MathSciNetCrossRefzbMATHGoogle Scholar
  7. Durand E, Blum MGB, Franco̧is O (2007) Prediction of group patterns in social mammals based on a coalescent model. J Theor Biol 249(2):262–270MathSciNetCrossRefGoogle Scholar
  8. Durand E, Franco̧is O (2010) Probabilistic analysis of a genealogical model of animal group patterns. J Math Biol 60(3):451–468MathSciNetCrossRefzbMATHGoogle Scholar
  9. Drmota M, Fuchs M, Lee Y-W (2014) Limit laws for the number of groups formed by social animals under the extra clustering model. In: Proceedings of the 25th international meeting on probabilistic, combinatorial and asymptotic methods for the analysis of algorithms. Discrete mathematics and theoretical computer science proceedings, pp 73–85Google Scholar
  10. Drmota M, Iksanov A, Möhle M, Rösler U (2009) A limiting distribution for the number of cuts needed to isolate the root of a random recursive tree. Random Struct Algorithms 34(3):319–336MathSciNetCrossRefzbMATHGoogle Scholar
  11. Fill JA, Flajolet P, Kapur N (2004) Singularity analysis, Hadamard products, and tree recurrences. J Comput Appl Math 174(2):271–313MathSciNetCrossRefzbMATHGoogle Scholar
  12. Fill JA, Kapur N (2004) Limiting distributions for additive functionals on Catalan trees. Theor Comput Sci 326(1–3):69–102MathSciNetCrossRefzbMATHGoogle Scholar
  13. Flajolet P, Gourdon X, Martinez C (1997) Patterns in random binary search trees. Random Struct Algorithms 11(3):223–244MathSciNetCrossRefzbMATHGoogle Scholar
  14. Flajolet P, Lafforgue T (1994) Search costs in quadtrees and singularity perturbation asymptotics. Discrete Comput Geom 12(2):151–175MathSciNetCrossRefGoogle Scholar
  15. Flajolet P, Sedgewick R (2009) Analytic combinatorics. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  16. Hwang H-K, Neininger R (2002) Phase change of limit laws in the quicksort recurrences under varying toll functions. SIAM J Comput 31(6):1687–1722MathSciNetCrossRefzbMATHGoogle Scholar
  17. Janson S (2014) Maximal clades in random binary search trees. Electron J Combin 22(1):31MathSciNetGoogle Scholar
  18. Janson S, Kersting G (2011) On the total external length of the Kingman coalescent. Electron J Probab 16:2203–2218MathSciNetCrossRefzbMATHGoogle Scholar
  19. Kingman JFC (1982) The coalescent. Stoch Process Appl 13(3):235–248MathSciNetCrossRefzbMATHGoogle Scholar
  20. Mahmoud HM (1992) Evolution of random search trees. Wiley-Interscience, New YorkzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Institute for Discrete Mathematics and GeometryTechnical University of ViennaViennaAustria
  2. 2.Department of Applied MathematicsNational Chiao Tung UniversityHsinchuTaiwan

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