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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 Fuchs
  • Yi-Wen Lee
Article

Abstract

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.

Keywords

Social animals Number of groups Moments Limit laws Singularity perturbation analysis 

Mathematics Subject Classification

05A16 60F05 92B05 

Notes

Acknowledgments

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.

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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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