Foraging Under Competition: Evolutionarily Stable Patch-Leaving Strategies with Random Arrival Times

2. Interference Competition
  • Frédéric Hamelin
  • Pierre Bernhard
  • A. J. Shaiju
  • Éric Wajnberg
Part of the Annals of the International Society of Dynamic Games book series (AISDG, volume 9)


our objective is to determine the evolutionarily stable strategy [13] that is supposed to drive the behavior of foragers competing for a common patchily distributed resource [15]. Compared to [17], the innovation lies in the fact that random arrival times are allowed.

In this second part, we add interference to the model: it implies that a “passive” Charnov-like strategy can no longer be optimal. A dynamic programming approach leads to a sequence of wars of attrition [13] with random end times. This game is solved in Appendix A. Under some conditions that prevail in our

model, the solution is independent of the probability law of the horizon. As a consequence, the solution of the asynchronous foraging problem investigated here, expressed as a closed loop strategy on the number of foragers, is identical to that of the synchronous problem [17].

Finally, we discuss the biological implications such as a possible connection with the genetic variability in the susceptibility to interference observed in [22].


Nash Equilibrium Mixed Strategy Pure Strategy Replicator Dynamic Relative Interior 
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.


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  1. [1]
    Bishop D.T., Cannings C.: A generalized war of attrition. Journal of Theoretical Biology, 70:85–124, 1978.CrossRefMathSciNetGoogle Scholar
  2. [2]
    Bishop D.T., Cannings C., Maynard Smith J.: The war of attrition with random rewards. Journal of Theoretical Biology, 74:377–388, 1978.CrossRefGoogle Scholar
  3. [3]
    Cressman R.: Stability of the replicator equation with continuous strategy space. Mathematical Social Sciences, 50:127–147, 2005.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Cressman R., Hofbauer J.: Measure dynamics on a one-dimensional trait space: theoretical foundations for adaptive dynamics. Theoretical Population Biology, 67:47–59, 2005.MATHCrossRefGoogle Scholar
  5. [5]
    Doebeli M., Hauert C.: Models of cooperation based on the prisonner’s dilemma and the snowdrift game. Ecology Letters. 8:748–766, 2005.CrossRefGoogle Scholar
  6. [6]
    Dubois F., Giraldeau L.-A.: The forager’s dilemma: food sharing and food defense as risk-sensitive foraging options. The American Naturalist, 162:768–779, 2003.CrossRefGoogle Scholar
  7. [7]
    Goubault M., Outreman Y., Poinsot D., Cortesero A.M.: Patch exploitation strategies of parasitic wasps under intraspecific competition. Behavioral Ecology, 16: 693–701, 2005.CrossRefGoogle Scholar
  8. [8]
    Haigh J., Cannings: Then-person warof attrition. Acta Applicandae Mathematicae, 14:59–74, 1989.CrossRefMathSciNetGoogle Scholar
  9. [9]
    Hamelin F., Bernhard P., Nain P., Wajnberg E.: Foraging under competition: evolutionarily stable patch-leaving strategies with random arrival times. 1. Scramble competition. Annals of Dynamics Games, this volume, Birkhauser, pp. 327–348, 2007.Google Scholar
  10. [10]
    Hofbauer J., Sigmund K.: Evolutionary games and population dynamics. Cambridge University Press, Cambridge, UK, 1998.MATHGoogle Scholar
  11. [11]
    Kuhn H.W.: Contribution to the theory of games. Volume 24 of Annals in Mathematics Studies, Princeton University Press, Princeton, New Jersey, USA, 1950.Google Scholar
  12. [12]
    McNamaraJ. M., Houston A.I., Collins E.J.: Optimality models in behavioral biology. SI AM Review. 43: 413–466, 2001.CrossRefGoogle Scholar
  13. [13]
    Maynard Smith J.: Evolution and the theory of games. Cambridge University Press, Cambridge, UK, 1982.MATHGoogle Scholar
  14. [14]
    Maynard Smith J., Price G.R.: The logic of animal conflict. Nature, 246: 15–18, 1973.CrossRefGoogle Scholar
  15. [15]
    Parker G.A., Stuart. R.A.: Animal behaviour as a strategy optimizer: evolution of resource assessment strategies and optimal emigration thresholds. The American Naturalist, 110:1055–1076, 1976.CrossRefGoogle Scholar
  16. [16]
    Samuelson L.: Evolutionary games and equilibrium selection. MIT Press, Cambridge, Massachusetts, USA, 1997.MATHGoogle Scholar
  17. [17]
    Sjerps M., Haccou P.: Effects of competition on optimal patch leaving: a war of attrition. Theoretical Population Biology, 3:300–318, 1994.CrossRefGoogle Scholar
  18. [18]
    Van Der Meer J., Ens B.J.: Models of interference and their consequences for the spatial distribution of ideal and free predators. Journal of Animal Ecology, 66:846–858, 1997.CrossRefGoogle Scholar
  19. [19]
    Vincent T.: Evolutionary games. Journal of Optimization Theory and Applications, 46:605–612, 1985.MATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    Vincent T., Fisher M.E.: Evolutionary stable strategies in differential and differences equations models. Evolutionary Ecology, 2:321–337, 1988.CrossRefGoogle Scholar
  21. [21]
    Von Neumann J., Morgenstern O.: The theory of games and economic behaviour. Princeton University Press, Princeton, New Jersey, USA, 1944.Google Scholar
  22. [22]
    Wajnberg E., Curty C., Colazza S.: Genetic variation in the mechanisms of direct mutual interference in a parasitic wasp: consequences in terms of patch-time allocation. Journal of Animal Ecology, 73:1179–1189, 2004.CrossRefGoogle Scholar

Copyright information

© Birkhäuser Boston 2007

Authors and Affiliations

  • Frédéric Hamelin
    • 1
  • Pierre Bernhard
    • 1
  • A. J. Shaiju
    • 1
  • Éric Wajnberg
    • 2
  1. 1.CNRS and University of Nice Sophia Antipolis-I3SÉcole Polytechnique de l’Université de Nice Sophia AntipolisSophia AntipolisFrance
  2. 2.INRASophia Antipolis CedexFrance

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