Ambush frequency should increase over time during optimal predator search for prey |
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Authors: | Steve Alpern Robbert Fokkink Marco Timmer Jér?me Casas |
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Affiliation: | 1.Department of Mathematics, London School of Economics, London WC2A 2AE, UK;2.Faculty of Electrical Engineering, Mathematics and Information Technology, Delft University, PO Box 5031, 2600 GA Delft, The Netherlands;3.Institut de Recherches sur la Biologie de l''Insecte, University of Tours UMR CNRS 6035, 37200 Tours, France |
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Abstract: | We advance and apply the mathematical theory of search games to model the problem faced by a predator searching for prey. Two search modes are available: ambush and cruising search. Some species can adopt either mode, with their choice at a given time traditionally explained in terms of varying habitat and physiological conditions. We present an additional explanation of the observed predator alternation between these search modes, which is based on the dynamical nature of the search game they are playing: the possibility of ambush decreases the propensity of the prey to frequently change locations and thereby renders it more susceptible to the systematic cruising search portion of the strategy. This heuristic explanation is supported by showing that in a new idealized search game where the predator is allowed to ambush or search at any time, and the prey can change locations at intermittent times, optimal predator play requires an alternation (or mixture) over time of ambush and cruise search. Thus, our game is an extension of the well-studied ‘Princess and Monster’ search game. Search games are zero sum games, where the pay-off is the capture time and neither the Searcher nor the Hider knows the location of the other. We are able to determine the optimal mixture of the search modes when the predator uses a mixture which is constant over time, and also to determine how the mode mixture changes over time when dynamic strategies are allowed (the ambush probability increases over time). In particular, we establish the ‘square root law of search predation’: the optimal proportion of active search equals the square root of the fraction of the region that has not yet been explored. |
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Keywords: | predator– prey, search game, ambush, foraging behaviour |
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