Evaluating growth measures in an immigration process subject to binomial and geometric catastrophes

  • Received: 01 March 2007 Accepted: 29 June 2018 Published: 01 August 2007
  • MSC : Primary: 92D25; Secondary: 60J99.

  • Populations are often subject to the effect of catastrophic events that cause mass removal. In particular, metapopulation models, epidemics, and migratory flows provide practical examples of populations subject to disasters (e.g., habitat destruction, environmental catastrophes). Many stochastic models have been developed to explain the behavior of these populations. Most of the reported results concern the measures of the risk of extinction and the distribution of the population size in the case of total catastrophes where all individuals in the population are removed simultaneously. In this paper, we investigate the basic immigration process subject to binomial and geometric catastrophes; that is, the population size is reduced according to a binomial or a geometric law. We carry out an extensive analysis including first extinction time, number of individuals removed, survival time of a tagged individual, and maximum population size reached between two consecutive extinctions. Many explicit expressions are derived for these system descriptors, and some emphasis is put to show that some of them deserve extra attention.

    Citation: Jesus R. Artalejo, A. Economou, M.J. Lopez-Herrero. Evaluating growth measures in an immigration process subject to binomial and geometric catastrophes[J]. Mathematical Biosciences and Engineering, 2007, 4(4): 573-594. doi: 10.3934/mbe.2007.4.573

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  • Populations are often subject to the effect of catastrophic events that cause mass removal. In particular, metapopulation models, epidemics, and migratory flows provide practical examples of populations subject to disasters (e.g., habitat destruction, environmental catastrophes). Many stochastic models have been developed to explain the behavior of these populations. Most of the reported results concern the measures of the risk of extinction and the distribution of the population size in the case of total catastrophes where all individuals in the population are removed simultaneously. In this paper, we investigate the basic immigration process subject to binomial and geometric catastrophes; that is, the population size is reduced according to a binomial or a geometric law. We carry out an extensive analysis including first extinction time, number of individuals removed, survival time of a tagged individual, and maximum population size reached between two consecutive extinctions. Many explicit expressions are derived for these system descriptors, and some emphasis is put to show that some of them deserve extra attention.
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    © 2007 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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