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Colonization and Collapse on Homogeneous Trees

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Abstract

We investigate a metapopulation model referring to populations that are spatially structured in colonies. Each colony thrives during a random time until a catastrophe when only a random amount of individuals of that colony survives. These survivors try independently establishing new colonies at neighbour sites, randomly. If the chosen site is occupied, that individual dies, otherwise the individual founds there a new colony. Here we consider this metapopulation model subject to two schemes: (i) Poisson growth, during an exponential time, for each colony and geometric catastrophe, and (ii) Yule growth, during an exponential time, for each colony and binomial catastrophe. We study conditions on the set of parameters for these processes to survive, present relevant bounds for the probability of survival, for the number of vertices that were colonized and for the reach of the colonies compared to the starting point. As a byproduct we study convergence of sequence of branching processes.

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Acknowledgements

V.V.J. and A.R-C wish to thank Instituto de Matemática e Estatística-USP Brazil for kind hospitality. The authors are thankful for the anonymous referees for a careful reading and many suggestions and corrections that greatly helped to improve the paper.

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Authors and Affiliations

  1. Institute of Mathematics and Statistics, University of São Paulo, Rua do Matão 1010, São Paulo, SP, CEP 05508-090, Brazil

    Fábio P. Machado

  2. Instituto de Matemáticas, Universidad de Antioquia, Calle 67, no 53-108, Medellín, Colombia

    Alejandro Roldán-Correa

  3. Federal University of Goias, Campus Samambaia, Goiânia, GO, CEP 74001-970, Brazil

    Valdivino V. Junior

Authors
  1. Fábio P. Machado
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  2. Alejandro Roldán-Correa
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  3. Valdivino V. Junior
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Corresponding author

Correspondence to Fábio P. Machado.

Additional information

Research supported by CNPq (310829/2014-3 and 141046/2013-9), FAPESP (2017/10555-0), PNPD-CAPES (536114) and Universidad de Antioquia.

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Machado, F.P., Roldán-Correa, A. & Junior, V.V. Colonization and Collapse on Homogeneous Trees. J Stat Phys 173, 1386–1407 (2018). https://doi.org/10.1007/s10955-018-2161-3

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  • DOI: https://doi.org/10.1007/s10955-018-2161-3

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