Abstract
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].
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References
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.
Brown J.S.: Patch use as an indicator of habitat preference, predation risk and competition. Behavioral Ecology and Sociobiology, 22:37–47, 198
Brown J.S., Rosenzweig M.L.: Habitat selection in slowly regenerating environments. Journal of Theoretical Biology, 123:151–171, 1986.
Charnov E.L.: Optimal foraging: the marginal value theorem. Theoretical Population Biology, 9:129–136, 1976.
Clark C.W., Mangel M.: The evolutionary advantages of group foraging. Theoretical Population Biology. 30:45–75, 1986.
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.
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.
Giraldeau, L.-A., Beauchamp, G.: Food exploitation: searching for the optimal joining policy. Trends in Ecology and Evolution, 14:102–106, 1999.
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.
Holling C.S.: Some characteristics of simple types of predation and parasitism. The Canadian Entomologist, 91:385–398, 1959.
Houston A.I., McNamara J.M.: Models of adaptive behavior: an approach based on state. Cambridge University Press, Cambridge, UK, 1999.
Kacelnik A., Krebs J.R., Bernstein C.: The ideal free distribution and predator-prey populations. Trends in Ecology and Evolution, 7:50–55, 1992.
Krebs J.R., Davies N.B., editors: Behavioural ecology: an evolutionary approach. Blackwell Science, Oxford, UK, 1997.
Maynard Smith J.: Evolution and the theory of games. Cambridge University Press, Cambridge, UK, 1982.
McNamara J.M., Houston A.I., Collins E.J.: Optimality models inBehavioral Biology. SIAM Review. 43: 413–466, 2001.
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.
Ruxton G.D., Fraser and Broom M.: An evolutionarily stable joining policy for group foragers. Behavioral Ecology, 16:856–864, 2005.
Sjerps M., Haccou P.: Effects of competition on optimal patch leaving: a war of attrition. Theoretical Population Biology, 3:300–318, 1994.
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.
Stephens D.W., Krebs J.R.: Foraging theory. Monographs in Behavior and Ecology, Princeton University Press, Princeton, New Jersey, USA, 1986.
Sutherland W.J.: From individual behavior to population ecology. Oxford Series in Ecology and Evolution. Oxford University Press, New York, USA, 1996.
Trezenga T.: Building on the ideal free distribution. Advances in Ecological Research, 26:253–302, 1995.
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Hamelin, F., Bernhard, P., Nain, P., Wajnberg, É. (2007). Foraging Under Competition: Evolutionarily Stable Patch-Leaving Strategies with Random Arrival Times.. In: Jørgensen, S., Quincampoix, M., Vincent, T.L. (eds) Advances in Dynamic Game Theory. Annals of the International Society of Dynamic Games, vol 9. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4553-3_16
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DOI: https://doi.org/10.1007/978-0-8176-4553-3_16
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