An Evolutionary Beverton-Holt Model

  • J. M.  CushingEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 102)


The classic Beverton-Holt (discrete logistic) difference equation, which arises in population dynamics, has a globally asymptotically stable equilibrium (for positive initial conditions) if its coefficients are constants. If the coefficients change in time, then the equation becomes nonautonomous and the asymptotic dynamics might not be as simple. One reason the coefficients can change in time is their evolution by natural selection. If the model coefficients are functions of a heritable phenotypic trait subject to natural selection then, by standard methods for modeling evolution, the model becomes a planar system of coupled difference equations, consisting of a Beverton-Holt type equation for the population dynamics and a difference equation for the dynamics of the mean phenotypic trait. We consider a case when the trait equation uncouples from the population dynamic equation and obtain criteria under which the evolutionary system has globally asymptotically stable equilibria or periodic solutions.


Traitement Par Population Dynamics Equations Global Asymptotic Stability (GAS) Equilibrium Selection Critical Traits 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



J.M. Cushing was supported by NSF grant DMS 0917435.


  1. 1.
    Abrams, P.A.: Modelling the adaptive dynamics of traits involved in inter- and intraspecific interactions: an assessment of three methods. Ecol. Lett. 4, 166–175 (2001)CrossRefGoogle Scholar
  2. 2.
    Cushing, J.M.: Periodically forced nonlinear systems of difference equations. J. Differ. Equ. Appl. 3, 547–561 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Cushing, J.M.: A bifurcation theorem for Darwinian matrix models. Nonlinear Stud. 17(1), 1–13 (2010)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Dercole, F., Rinaldi, S.: Analysis of Evolutionary Processes: The Adaptive Dynamics Approach and Its Applications. Princeton University Press, Princeton (2008)Google Scholar
  5. 5.
    Lande, R.: Natural selection and random genetic drift in phenotypic evolution. Evolution 30, 314–334 (1976)CrossRefGoogle Scholar
  6. 6.
    Leslie, P.H.: Further notes on the use of matrices in population mathematics. Biometrika 35, 213–245 (1948)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Pielou, E.C.: Mathematical Ecology. Wiley Interscience, New York (1967)Google Scholar
  8. 8.
    Poole, R.W.: An Introduction to Quantitative Ecology. McGraw-Hill Series in Population Biology, McGraw-Hill Book Company, New York (1974)Google Scholar
  9. 9.
    Skellam, J.F.: Studies in statistical ecology. 1. Spatial pattern. Biometrika 39, 346–362 (1952)zbMATHGoogle Scholar
  10. 10.
    Utida, S.: Damped oscillation of population density at equilibrium. Res. Popul. Ecol. 9, 1–9 (1967)CrossRefGoogle Scholar
  11. 11.
    Vincent, T.L., Brown, J.S.: Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics. Cambridge University Press, Cambridge (2005)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Mathematics and The Interdisciplinary Program in Applied MathematicsUniversity of ArizonaTucsonUSA

Personalised recommendations