Journal of Mathematical Biology

, Volume 76, Issue 7, pp 1951–1973 | Cite as

The ESS and replicator equation in matrix games under time constraints

  • József GarayEmail author
  • Ross Cressman
  • Tamás F. Móri
  • Tamás Varga


Recently, we introduced the class of matrix games under time constraints and characterized the concept of (monomorphic) evolutionarily stable strategy (ESS) in them. We are now interested in how the ESS is related to the existence and stability of equilibria for polymorphic populations. We point out that, although the ESS may no longer be a polymorphic equilibrium, there is a connection between them. Specifically, the polymorphic state at which the average strategy of the active individuals in the population is equal to the ESS is an equilibrium of the polymorphic model. Moreover, in the case when there are only two pure strategies, a polymorphic equilibrium is locally asymptotically stable under the replicator equation for the pure-strategy polymorphic model if and only if it corresponds to an ESS. Finally, we prove that a strict Nash equilibrium is a pure-strategy ESS that is a locally asymptotically stable equilibrium of the replicator equation in n-strategy time-constrained matrix games.


Evolutionary stability Monomorphic Polymorphic Replicator equation 

Mathematics Subject Classification

91A22 92D15 91A80 91A40 91A05 91A10 92D40 



This work was partially supported by the Hungarian National Research, Development and Innovation Office NKFIH [Grant Numbers K 108615 (to T.F.M.) and K 108974 and GINOP 2.3.2-15-2016-00057 (to J.G.)]. This project has received funding from the European Union Horizon 2020: The EU Framework Programme for Research and Innovation under the Marie Sklodowska-Curie Grant Agreement No. 690817 (to J.G., R.C. and T.V.). R.C. acknowledges support provided by an Individual Discovery Grant from the Natural Sciences and Engineering Research Council (NSERC).


  1. Broom M, Luther RM, Ruxton GD, Rychtar J (2008) A game-theoretic model of kleptoparasitic behavior in polymorphic populations. J Theor Biol 255:81–91MathSciNetCrossRefGoogle Scholar
  2. Broom M, Rychtar J (2013) Game-theoretical models in biology. Chapman & Hall/CRC Mathematical and Computational Biology, LondonzbMATHGoogle Scholar
  3. Charnov EL (1976) Optimal foraging: attack strategy of a mantid. Am Nat 110:141–151CrossRefGoogle Scholar
  4. Cressman R (1992) The stability concept of evolutionary game theory (a dynamic approach), vol 94. Lecture notes in biomathematics, Springer, BerlinCrossRefzbMATHGoogle Scholar
  5. Cressman R (2003) Evolutionary dynamics and extensive form games. MIT Press, CambridgezbMATHGoogle Scholar
  6. Eriksson A, Lindgren K, Lundh T (2004) War of attrition with implicit time cost. J Theor Biol 230:319–332MathSciNetCrossRefGoogle Scholar
  7. Garay J, Móri TF (2010) When is the opportunism remunerative? Commun Ecol 11:160–170CrossRefGoogle Scholar
  8. Garay J, Varga Z, Cabello T, Gámez M (2012) Optimal nutrient for aging strategy of an omnivore: Liebig’s law determining numerical response. J Theor Biol 310:31–42CrossRefzbMATHGoogle Scholar
  9. Garay J, Cressman R, Xu F, Varga Z, Cabello T (2015a) Optimal forager against ideal free distributed prey. Am Nat 186:111–122CrossRefGoogle Scholar
  10. Garay J, Varga Z, Gámez M, Cabello T (2015b) Functional response and population dynamics for fighting predator, based on activity distribution. J Theor Biol 368:74–82MathSciNetCrossRefGoogle Scholar
  11. Garay J, Csiszár V, Móri TF (2017) Evolutionary stability for matrix games under time constraints. J Theor Biol 415:1–12MathSciNetCrossRefzbMATHGoogle Scholar
  12. Hofbauer J, Sigmund K (1998) Evolutionary games and population dynamics. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  13. Holling CS (1959) The components of predation as revealed by a study of small mammal predation of the European pine sawfly. Can Entomol 9:293–320CrossRefGoogle Scholar
  14. Krivan V, Cressman R (2017) Interaction times change evolutionary outcomes: two-player matrix games. J Theor Biol 416:199–207MathSciNetCrossRefzbMATHGoogle Scholar
  15. Maynard Smith J (1982) Evolution and the theory of games. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  16. Sirot E (2000) An evolutionarily stable strategy for aggressiveness in feeding groups. Behav Ecol 11:351–356CrossRefGoogle Scholar
  17. Taylor PD, Jonker LB (1978) Evolutionary stable strategies and game dynamics. Math Biosci 40:145–156MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • József Garay
    • 1
    • 2
    Email author
  • Ross Cressman
    • 3
  • Tamás F. Móri
    • 4
  • Tamás Varga
    • 5
  1. 1.MTA-ELTE Theoretical Biology and Evolutionary Ecology Research Group and Department of Plant Systematics, Ecology and Theoretical BiologyEötvös Loránd UniversityBudapestHungary
  2. 2.MTA Centre for Ecological Research, Evolutionary Systems Research GroupTihanyHungary
  3. 3.Department of MathematicsWilfrid Laurier UniversityWaterlooCanada
  4. 4.Department of Probability Theory and StatisticsEötvös Loránd UniversityBudapestHungary
  5. 5.MTA-SZTE Analysis and Stochastics Research Group, Bolyai InstituteUniversity of SzegedSzegedHungary

Personalised recommendations