# Stochastic analysis of the extra clustering model for animal grouping

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## 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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