Research article Special Issues

Discrete time darwinian dynamics and semelparity versus iteroparity

  • Received: 16 November 2018 Accepted: 07 February 2019 Published: 06 March 2019
  • We derive and analyze a Darwinian dynamic model based on a general di erence equation population model under the assumption of a trade-o between fertility and survival. Both inherent and density dependent terms are functions of a phenotypic trait (subject to Darwinian evolution) and its population mean. We prove general theorems about the existence and stability of extinction equilibria and the bifurcation of positive equilibria when extinction equilibria destabilize. We apply these results, together with the Evolutionarily Stable Strategy (ESS) Maximum Principle, to the model when both semelparous and iteroparous traits are available to individuals in the population. We find that if the density terms in the population model are trait independent, then only semelparous equilibria are ESS. When density terms do depend on the trait, then in a neighborhood of a bifurcation point it is again the case that only semelparous equilibria are ESS. However, we also show by simulations that ESS iteroparous (and also non-ESS semelparous) equilibria can arise outside a neighborhood of bifurcation points when density e ects depend in a hierarchical manner on the trait.

    Citation: J. M. Cushing. Discrete time darwinian dynamics and semelparity versus iteroparity[J]. Mathematical Biosciences and Engineering, 2019, 16(4): 1815-1835. doi: 10.3934/mbe.2019088

    Related Papers:

  • We derive and analyze a Darwinian dynamic model based on a general di erence equation population model under the assumption of a trade-o between fertility and survival. Both inherent and density dependent terms are functions of a phenotypic trait (subject to Darwinian evolution) and its population mean. We prove general theorems about the existence and stability of extinction equilibria and the bifurcation of positive equilibria when extinction equilibria destabilize. We apply these results, together with the Evolutionarily Stable Strategy (ESS) Maximum Principle, to the model when both semelparous and iteroparous traits are available to individuals in the population. We find that if the density terms in the population model are trait independent, then only semelparous equilibria are ESS. When density terms do depend on the trait, then in a neighborhood of a bifurcation point it is again the case that only semelparous equilibria are ESS. However, we also show by simulations that ESS iteroparous (and also non-ESS semelparous) equilibria can arise outside a neighborhood of bifurcation points when density e ects depend in a hierarchical manner on the trait.


    加载中


    [1] D. Roff, Evolution of Life Histories: Theory and Analysis, Chapman & Hall, 1992.
    [2] S. C. Stearns, The Evolution of Life Histories, Oxford University Press, 2004.
    [3] R. A. Fisher, The Genetical Theory of Natural Selection: A Complete Variorum Edition, Oxford University Press, 1930
    [4] L. C. Cole, The population consequences of life history phenomena, Quar. Rev. Biol. 29 (1954), 103–137.
    [5] P. W. Hughes, Between semelparity and iteroparity: empirical evidence for a continuum of modes of parity, Ecol. Evol. 7 (2017), 8232–8261.
    [6] E. L. Charnov and W. M. Schaffer, Life history consequences of natural selection: Cole's result revisited, Amer. Nat. 107 (1973), 791–793.
    [7] T. Vincent and J. Brown, Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics, Cambridge University Press, 2005.
    [8] B. McGill and J. Brown, Evolutionary game theory and adaptive dynamics of continuous traits, Ann. Rev. Ecol., Evol. Syst. 38 (2007), 403–435.
    [9] R. Lande, A quantitative genetic theory of life history evolution, Ecol. 33 (1982), 607–615.
    [10] J. Lush, Animal Breeding Plans , Iowa State College Press, 1937.
    [11] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Func. Analy. 7 (1970), 487–513.
    [12] J. M. Cushing, Backward bifurcations and strong Allee effects in matrix models for the dynamics of structured populations, J. Biol. Dyn. 8 (2014), 57–73.
    [13] J. M. Cushing, On the dynamics of a class of Darwinian matrix models, Nonl. Dyn. Syst. Theory 10 (2010), 103–116.
    [14] J. M. Cushing and Simon MacCracken Stump, Darwinian dynamics of a juvenile-adult model, Math. Biosci. Engr. 10 (2013), 1017–1044.
    [15] J. M. Cushing, F. Martins, A. A. Pinto, et. al., A bifurcation theorem for evolutionary matrix models with multiple traits, J. Math. Biol. 75 (2017), 491–520.
    [16] E. P. Meissen, K. R. Salau and J. M. Cushing, A global bifurcation theorem for Darwinian matrix models, J. Diff. Eqs. Appl. 22 (2016), 1114–1136.
  • Reader Comments
  • © 2019 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)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2603) PDF downloads(643) Cited by(4)

Article outline

Figures and Tables

Figures(4)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog