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Mathematical Biosciences and Engineering

2007, Volume 4, Issue 2: 133-157. doi: 10.3934/mbe.2007.4.133
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Comparison between stochastic and deterministic selection-mutation models

  • 1.
    Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504-1010
  • 2.
    Center for Research in Scientific Computation, North Carolina State University, Raleigh, North Carolina 27695-8205
  • Received: 01 April 2006 Accepted: 29 June 2018 Published: 01 February 2007

  • MSC : 92D25, 34D23, 65C30.

  • We present a deterministic selection-mutation model with a discrete trait variable. We show that for an irreducible selection-mutation matrix in the birth term the deterministic model has a unique interior equilibrium which is globally stable. Thus all subpopulations coexist. In the pure selection case, the outcome is known to be that of competitive exclusion, where the subpopulation with the largest growth-to-mortality ratio will survive and the remaining subpopulations will go extinct. We show that if the selection-mutation matrix is reducible, then competitive exclusion or coexistence are possible outcomes. We then develop a stochastic population model based on the deterministic one. We show numerically that the mean behavior of the stochastic model in general agrees with the deterministic one. However, unlike the deterministic one, if the differences in the growth-to-mortality ratios are small in the pure selection case, it cannot be determined a priori which subpopulation will have the highest probability of surviving and winning the competition.

    Citation: Azmy S. Ackleh, Shuhua Hu. Comparison between stochastic and deterministic selection-mutation models[J]. Mathematical Biosciences and Engineering, 2007, 4(2): 133-157. doi: 10.3934/mbe.2007.4.133

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  • We present a deterministic selection-mutation model with a discrete trait variable. We show that for an irreducible selection-mutation matrix in the birth term the deterministic model has a unique interior equilibrium which is globally stable. Thus all subpopulations coexist. In the pure selection case, the outcome is known to be that of competitive exclusion, where the subpopulation with the largest growth-to-mortality ratio will survive and the remaining subpopulations will go extinct. We show that if the selection-mutation matrix is reducible, then competitive exclusion or coexistence are possible outcomes. We then develop a stochastic population model based on the deterministic one. We show numerically that the mean behavior of the stochastic model in general agrees with the deterministic one. However, unlike the deterministic one, if the differences in the growth-to-mortality ratios are small in the pure selection case, it cannot be determined a priori which subpopulation will have the highest probability of surviving and winning the competition.


  • This article has been cited by:

    1. T. Bayen, F. Mairet, Optimisation of strain selection in evolutionary continuous culture, 2017, 90, 0020-7179, 2748, 10.1080/00207179.2016.1264631
    2. Sylvie Méléard, Michael Rera, Tristan Roget, A birth–death model of ageing: from individual-based dynamics to evolutive differential inclusions, 2019, 79, 0303-6812, 901, 10.1007/s00285-019-01382-z
    3. Azmy S. Ackleh, Baoling Ma, Paul L. Salceanu, Persistence and global stability in a selection–mutation size-structured model, 2011, 5, 1751-3758, 436, 10.1080/17513758.2010.538729
    4. Azmy S. Ackleh, Robert J. Sacker, Paul Salceanu, On a discrete selection–mutation model, 2014, 20, 1023-6198, 1383, 10.1080/10236198.2014.933819
    5. Azmy S. Ackleh, Keng Deng, Qihua Huang, Stochastic juvenile–adult models with application to a green tree frog population, 2011, 5, 1751-3758, 64, 10.1080/17513758.2010.498924
    6. Dirk Langemann, Otto Richter, Antje Vollrath, Multi-gene-loci inheritance in resistance modeling, 2013, 242, 00255564, 17, 10.1016/j.mbs.2012.11.010
    7. Daniel S. Maynard, Carlos A. Serván, José A. Capitán, Stefano Allesina, John Drake, Phenotypic variability promotes diversity and stability in competitive communities, 2019, 22, 1461-023X, 1776, 10.1111/ele.13356
    8. Tor Flå, Florian Rupp, Clemens Woywod, 2013, Chapter 11, 978-3-0348-0450-9, 221, 10.1007/978-3-0348-0451-6_11
    9. Arnaud Z. Dragicevic, Stochastic network Price identity, 2020, 50, 13675788, 294, 10.1016/j.arcontrol.2020.05.002
    10. John Cleveland, Azmy S. Ackleh, Evolutionary game theory on measure spaces: Well-posedness, 2013, 14, 14681218, 785, 10.1016/j.nonrwa.2012.08.002
    11. Azmy S. Ackleh, Sankar Sikder, Aijun Zhang, 2023, Chapter 1, 978-3-031-25224-2, 1, 10.1007/978-3-031-25225-9_1
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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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