Artalejo, Jesús R. and Economou, A. and Lopez-Herrero, M. J.
(2007)
*Evaluating growth measures in an immigration process subject to binomial and geometric catastrophes.*
Mathematical Biosciences and Engineering , 4
(4).
pp. 573-594.
ISSN 1547-1063

Official URL: https://www.aimsciences.org/journals/displayArticles.jsp?paperID=2767

## Abstract

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.

Item Type: | Article |
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Uncontrolled Keywords: | Disasters; Environmental catastrophes; Extinction time; Immigration process; Markov chain; Maximum population size; Metapopulation dynamics; Persistence time; Survival time |

Subjects: | Sciences > Mathematics > Stochastic processes |

ID Code: | 15316 |

References: | J. Abate and W. Whitt, Numerical inversion of Laplace transforms of probability distributions, ORSA J. Comp., 7 (1995), 36–43. F.B. Adler and B. Nüernberger, Persistence in patchy irregular landscapes, Theor. Popul. Biol., 45 (1994), 45–71. L.J.S. Allen, "An Introduction to Stochastic Processes with Applications to Biology," Prentice Hall, Upper Saddle River, NJ, 2003. J.R. Artalejo, G -networks: A versatile approach for work removal in queueing networks, Europ. J. Operat. Res., 126 (2000), 223–249. J.R. Artalejo and S.R. Chakravarthy, Algorithmic analysis of the maximum level length in general-block two-dimensional Markov processes, Math. Probl. Eng., Paper ID 53570 (2006). M.S. Barlett, "Stochastic Population Models in Ecology & Epidemiology," Methuen's Monographs on Applied Probability and Statistics, London, 1960. R. Bartoszynski, W.J. Bühler, W. Chan and D.K. Pearl, Population processes under the influence of disasters occurring independently of population size, J. Math. Biol., 27 (1989), 167–178. L. Belhadji and N. Lanchier, Individual versus cluster recoveries within a spatially structured population, Ann. Appl. Probab., 16 (2006), 403–422. P.J. Brockwell, J. Gani and S.I. Resnick, Birth, immigration and catastrophe processes, Adv. Appl. Prob., 14 (1982), 709–731. B. Cairns and P.K. Pollet, Evaluating persistence times in populations that are subject to local catastrophes, in "MODSIM 2003 International Congress on Modelling and Simulation" (ed. D.A. Post), Modelling and Simulation Society of Australia and New Zealand, (2003), 747–752. B. Cairns and P.K. Pollet, Extinction times for a general birth, death and catastrophe process, J. App. Prob., 41 (2004), 1211–1218. R. Casagrandi and M. Gatto, Habitat destruction, environmental catastrophes and metapopulation extinction, Theor. Popul. Bio., 61 (2002), 127–140. X. Chao, M. Miyazawa and M. Pinedo, "Queueing Networks: Customers, Signals and Product Form Solutions," John Wiley & Sons, Chichester, 1999. Ch. A. Charalambides, "Enumerative Combinatorics," Chapman and Hall, Boca Raton, 2002. A. Economou, The compound Poisson immigration process subject to binomial catastrophes, J. Appl. Prob., 41 (2004), 508–523. L. Flatto, The waiting time distribution for the random order service M/M/1 queue, Ann. Appl. Probab., 7 (1997), 382–409. H.R. Gail, S.L. Hantler and B.A. Taylor, Use of characteristics roots for solving infinite state Markov chains, in "Combinatorial Probability" (ed. W.K. Grassmann), Kluwer, Boston, (2000), 205–255. A. Irle, Stochastic ordering for continuous-time processes, J. Appl. Prob., 40 (2003), 361–375 H.K. Kang, S.M. Krone and C. Neuhauser, Stepping-stone models with extinction and recolonization, Ann. Appl. Probab., 5 (1995), 1025–1060. B.E. Kendall, Estimating the magnitude of environmental stochasticity in survivorship data, Ecol. Appl., 8 (1998), 184–193. B.M. Kirstein, Monotonicity and comparability of time-homogeneous Markov processes with discrete state space, Math. Operationsforsch. Statist., 7 (1976), 151–168. S.M. Krone, The two-stage contact process, Ann. Appl. Probab., 9 (1999), 331–351. V.G. Kulkarni, "Modeling and Analysis of Stochastic Systems," Chapman and Hall, London, 1995. E.G. Kyriakidis, Stationary probabilities for a simple immigration-birth-death process under the influence of total catastrophes, Statist. Prob. Letters, 20 (1994), 239–240. E.G. Kyriakidis, Transient solution for a simple immigration birth-death catastrophe process, Prob. Eng. Inform. Sci., 18 (2004), 233–236. A. Müller and D. Stoyan, "Comparison Methods for Stochastic Models and Risks," John Wiley & Sons, Chichester, 2002. A.G. Pakes, Limit theorems for the population size of a birth and death process allowing catastrophes, J. Math. Biol., 25 (1987), 307–325. P.K. Pollet, Quasi-stationarity in populations that are subject to large-scale mortality or emigration, Environ. Int., 27 (2001), 231–236. R.B. Schinazi, On the role of social clusters in the transmission of infectious diseases, Theor. Popul. Biol., 61 (2002), 163–169. R.B. Schinazi, Mass extinctions: An alternative to the Allee effect, Ann. Appl. Probab., 15 (2005), 984–991. D. Stirzaker, Disasters, Math. Scient., 26 (2001), 59–62. R.J. Swift, Transient probabilities for a simple birth–death-immigration process under the influence of total catastrophes, Int. J. Math. Sci., 25 (2001), 689–692. D.M. Walker, The expected time until absorption when absorption is not certain, J. Appl. Prob., 35 (1998), 812–823. R.W. Wolff, Poisson arrivals see time averages, Operat. Res., 30 (1982), 223–231. |

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