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Dynamic Selection Systems and Replicator Equations

  • Zdeněk PospíšilEmail author
Conference paper
  • 807 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 102)

Abstract

The dynamic replicator equation is inferred from a generalized Kolmogorov-type dynamic system that is called selection system. This way, both systems have the same dimension. The main result shows that the replicator equation is in a certain sense equivalent to a selection system of lower dimension. Corollaries demonstrating connections with known results are also presented. The equations are interpreted as models of biological evolution on different time scales. Hence, the results show a link between ecology and evolution at least on the level of mathematical models.

Keywords

Selection System Game Dynamic Dynamic Replicator Replicator Equation Forward Jump 
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.

Notes

Acknowledgments

This research was supported by the grant No. GAP201/10/1032 of the Czech Grant Agency.

References

  1. 1.
    Bohner, M., Peterson, A.: Dynamic Equations on Time Scales. Birkhauser, Boston (2001)CrossRefzbMATHGoogle Scholar
  2. 2.
    Hofbauer, J.: On the occurrence of limit cycles in the Volterra-Lotka equation. Nonlinear Anal. 5, 1003–1007 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Hofbauer, J., Sigmund, K.: Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge (2002)Google Scholar
  4. 4.
    Karev, G.P.: On mathematical theory of selection: continuous time population dynamics. J. Math. Biol. 60, 107–129 (2010)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Murray, J.: Mathematical Biology I. Springer, Berlin (2002)Google Scholar
  6. 6.
    Murray, J.: Mathematical Biology II. Springer, Berlin (2003)Google Scholar
  7. 7.
    Nowak, M.A.: Evolutionary Dynamics. Harvard University Press, Cambridge (2006)zbMATHGoogle Scholar
  8. 8.
    Schuster, P., Sigmund, K.: Coyness, philandering and stable strategies. Anim. Behavior 29, 186–192 (1981)CrossRefGoogle Scholar
  9. 9.
    Sigmund, K.: The Calculus of Selfishness. Princeton University Press, Princeton (2010)zbMATHGoogle Scholar
  10. 10.
    Svirezhev, Y.M., Logofet, D.O.: Stability of Biological Communities. MIR, Moscow (1983)Google Scholar
  11. 11.
    Taylor, P.D., Jonker, L.: Evolutionary stable strategies and game dynamics. Math. Biosci. 40, 145–156 (1978)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMasaryk UniversityBrnoCzech Republic

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