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

Scramble Competition
  • Frédéric Hamelin
  • Pierre Bernhard
  • Philippe Nain
  • 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 [14] that is supposed to drive the behavior of foragers competing for a common patchily distributed resource [16]. Compared to [18], the innovation lies in the fact that random arrival times are allowed.

In this first part, we investigate scramble competition: the game still yields simple Charnov-like strategies [4]. Thus we attempt to compute the optimal longterm mean rate γ* [11] at which resources should be gathered to achieve the maximum expected fitness: the assumed symmetry among foragers allows us to express γ* as a solution of an implicit equation, independent of the probability distribution of arrival times.

A digression on a simple model of group foraging shows that γ*N can be simply computed via the classical graph associated to the marginal value theorem—N is the size of the group. An analytical solution allows us to characterize the decline in efficiency due to group foraging, as opposed to foraging alone: this loss can be relatively low, even in a “bad world,” provided that the handling time is relatively long.

Back to the original problem, we then assume that the arrivals on the patch follow a Poisson process. Thus we find an explicit expression of γ* that makes it possible to perform a numerical computation: Charnov’s predictions still hold under scramble competition.

Finally, we show that the distribution of foragers among patches is not homogeneous but biased in favor of bad patches. This result is in agreement with common observation and theoretical knowledge [1] about the concept of ideal free distribution [12, 22].


Intake Rate Handling Time Interarrival Time Patch Quality Scramble Competition 
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  1. [1]
    Bernstein Kacelnik A., Krebs J.R.: Individual decisions and the distribution of predators in a patchy environment II: the influence of travel costs and structure of the environment. Journal of Animal Ecology, 60:205–225, 1991.CrossRefGoogle Scholar
  2. [2]
    Brown J.S.: Patch use as an indicator of habitat preference, predation risk and competition. Behavioral Ecology and Sociobiology, 22:37–47, 198CrossRefGoogle Scholar
  3. [3]
    Brown J.S., Rosenzweig M.L.: Habitat selection in slowly regenerating environments. Journal of Theoretical Biology, 123:151–171, 1986.CrossRefGoogle Scholar
  4. [4]
    Charnov E.L.: Optimal foraging: the marginal value theorem. Theoretical Population Biology, 9:129–136, 1976.CrossRefGoogle Scholar
  5. [5]
    Clark C.W., Mangel M.: The evolutionary advantages of group foraging. Theoretical Population Biology. 30:45–75, 1986.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Clark C.W., Mangel M.: Dynamic state variable models in Ecology, methods and applications. Oxford Series in Ecology and Evolution. Oxford University Press, New York, USA, 2000.Google Scholar
  7. [7]
    Corless R.M., Gonnet G.H., Hare D.E.G., Jeffrey D J., Knuth D.E.: On the Lambert W function. Advances in Computational Mathematics, 5:329–359, 1996.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    Giraldeau, L.-A., Beauchamp, G.: Food exploitation: searching for the optimal joining policy. Trends in Ecology and Evolution, 14:102–106, 1999.CrossRefGoogle Scholar
  9. [9]
    Hamelin F., Bernhard P., Shaiju A.J., Wajnberg E.: Foraging under competition: evolutionarily stable patch-leaving strategies with random arrival times. 2. Interference competition. Annals of Dynamic Games, this volume, Birkhauser, pp. 349–366, 2007.Google Scholar
  10. [10]
    Holling C.S.: Some characteristics of simple types of predation and parasitism. The Canadian Entomologist, 91:385–398, 1959.CrossRefGoogle Scholar
  11. [11]
    Houston A.I., McNamara J.M.: Models of adaptive behavior: an approach based on state. Cambridge University Press, Cambridge, UK, 1999.Google Scholar
  12. [12]
    Kacelnik A., Krebs J.R., Bernstein C.: The ideal free distribution and predator-prey populations. Trends in Ecology and Evolution, 7:50–55, 1992.CrossRefGoogle Scholar
  13. [13]
    Krebs J.R., Davies N.B., editors: Behavioural ecology: an evolutionary approach. Blackwell Science, Oxford, UK, 1997.Google Scholar
  14. [14]
    Maynard Smith J.: Evolution and the theory of games. Cambridge University Press, Cambridge, UK, 1982.MATHGoogle Scholar
  15. [15]
    McNamara J.M., Houston A.I., Collins E.J.: Optimality models inBehavioral Biology. SIAM Review. 43: 413–466, 2001.MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    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
  17. [17]
    Ruxton G.D., Fraser and Broom M.: An evolutionarily stable joining policy for group foragers. Behavioral Ecology, 16:856–864, 2005.CrossRefGoogle Scholar
  18. [18]
    Sjerps M., Haccou P.: Effects of competition on optimal patch leaving: a war of attrition. Theoretical Population Biology, 3:300–318, 1994.CrossRefGoogle Scholar
  19. [19]
    Spiegel M.R.: Shaum’s outline of theory and problems of Laplace transforms, Shaum’s Outline Series, McGraw-Hill Book Company, New York, USA, 1965.Google Scholar
  20. [20]
    Stephens D.W., Krebs J.R.: Foraging theory. Monographs in Behavior and Ecology, Princeton University Press, Princeton, New Jersey, USA, 1986.Google Scholar
  21. [21]
    Sutherland W.J.: From individual behavior to population ecology. Oxford Series in Ecology and Evolution. Oxford University Press, New York, USA, 1996.Google Scholar
  22. [22]
    Trezenga T.: Building on the ideal free distribution. Advances in Ecological Research, 26:253–302, 1995.CrossRefGoogle Scholar

Copyright information

© Birkhäuser Boston 2007

Authors and Affiliations

  • Frédéric Hamelin
    • 1
  • Pierre Bernhard
    • 1
  • Philippe Nain
    • 2
  • Éric Wajnberg
    • 3
  1. 1.CNRS and University of Nice Sophia Antipolis-I3SÉcole Polytechnique de l’Université de Nice Sophia AntipolisSophia AntipolisFrance
  2. 2.INRIASophia AntipolisFrance
  3. 3.INRASophia Antipolis CedexFrance

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