American Institute of Mathematical Sciences

Advanced
2007, 4(4): 573-594. doi: 10.3934/mbe.2007.4.573

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

Jesus R. Artalejo 1, , A. Economou 2, and  M.J. Lopez-Herrero 3,

1. 

Department of Statistics and O.R., Faculty of Mathematics, Complutense University of Madrid, Madrid 28040, Spain
2. 

Department of Mathematics, University of Athens, Panepistemiopolis, Athens 15784, Greece
3. 

School of Statistics, Complutense University of Madrid, Madrid 28040, Spain

Received  March 2007 Revised  June 2007 Published  August 2007

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.
Keywords: extinction time, metapopulation dynamics, survival time., maximum population size, immigration process, persistence time, environmental catastrophes, Disasters, Markov chain.
Mathematics Subject Classification: Primary: 92D25; Secondary: 60J9.
Citation: Jesus R. Artalejo, A. Economou, M.J. Lopez-Herrero. Evaluating growth measures in an immigration process subject to binomial and geometric catastrophes. Mathematical Biosciences & Engineering, 2007, 4 (4) : 573-594. doi: 10.3934/mbe.2007.4.573
[1]

Keng Deng, Yixiang Wu. Extinction and uniform strong persistence of a size-structured population model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 831-840. doi: 10.3934/dcdsb.2017041
[2]

Hal L. Smith, Horst R. Thieme. Persistence and global stability for a class of discrete time structured population models. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4627-4646. doi: 10.3934/dcds.2013.33.4627
[3]

Xi Zhu, Meixia Li, Chunfa Li. Consensus in discrete-time multi-agent systems with uncertain topologies and random delays governed by a Markov chain. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020111
[4]

Mustapha Mokhtar-Kharroubi, Quentin Richard. Spectral theory and time asymptotics of size-structured two-phase population models. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 2969-3004. doi: 10.3934/dcdsb.2020048
[5]

Cecilia Cavaterra, M. Grasselli. Robust exponential attractors for population dynamics models with infinite time delay. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1051-1076. doi: 10.3934/dcdsb.2006.6.1051
[6]

Tristan Roget. On the long-time behaviour of age and trait structured population dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2551-2576. doi: 10.3934/dcdsb.2018265
[7]

Lakhdar Aggoun, Lakdere Benkherouf. A Markov modulated continuous-time capture-recapture population estimation model. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 1057-1075. doi: 10.3934/dcdsb.2005.5.1057
[8]

Dan Zhang, Xiaochun Cai, Lin Wang. Complex dynamics in a discrete-time size-structured chemostat model with inhibitory kinetics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3439-3451. doi: 10.3934/dcdsb.2018327
[9]

Angelica Pachon, Federico Polito, Costantino Ricciuti. On discrete-time semi-Markov processes. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020170
[10]

Alfonso C. Casal, Jesús Ildefonso Díaz, José M. Vegas. Finite extinction time property for a delayed linear problem on a manifold without boundary. Conference Publications, 2011, 2011 (Special) : 265-271. doi: 10.3934/proc.2011.2011.265
[11]

Mattia Turra. Existence and extinction in finite time for Stratonovich gradient noise porous media equations. Evolution Equations & Control Theory, 2019, 8 (4) : 867-882. doi: 10.3934/eect.2019042
[12]

Yoshikazu Giga, Robert V. Kohn. Scale-invariant extinction time estimates for some singular diffusion equations. Discrete & Continuous Dynamical Systems - A, 2011, 30 (2) : 509-535. doi: 10.3934/dcds.2011.30.509
[13]

Piotr Oprocha. Chain recurrence in multidimensional time discrete dynamical systems. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 1039-1056. doi: 10.3934/dcds.2008.20.1039
[14]

Bradley G. Wagner, Brian J. Coburn, Sally Blower. Increasing survival time decreases the cost-effectiveness of using "test & treat'' to eliminate HIV epidemics. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1673-1686. doi: 10.3934/mbe.2013.10.1673
[15]

Francisco de la Hoz, Anna Doubova, Fernando Vadillo. Persistence-time estimation for some stochastic SIS epidemic models. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2933-2947. doi: 10.3934/dcdsb.2015.20.2933
[16]

Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020199
[17]

H. L. Smith, X. Q. Zhao. Competitive exclusion in a discrete-time, size-structured chemostat model. Discrete & Continuous Dynamical Systems - B, 2001, 1 (2) : 183-191. doi: 10.3934/dcdsb.2001.1.183
[18]

Zaidong Zhan, Shuping Chen, Wei Wei. A unified theory of maximum principle for continuous and discrete time optimal control problems. Mathematical Control & Related Fields, 2012, 2 (2) : 195-215. doi: 10.3934/mcrf.2012.2.195
[19]

Arno Berger, Doan Thai Son, Stefan Siegmund. Nonautonomous finite-time dynamics. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 463-492. doi: 10.3934/dcdsb.2008.9.463
[20]

Cristina Anton, Alan Yong. Stochastic dynamics and survival analysis of a cell population model with random perturbations. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1077-1098. doi: 10.3934/mbe.2018048

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (22)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences