Department of Mathematics, London School of Economics, London WC2A 2AE, UK.
J R Soc Interface. 2011 Nov 7;8(64):1665-72. doi: 10.1098/rsif.2011.0154. Epub 2011 May 13.
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.
我们将搜索博弈的数学理论应用于捕食者搜索猎物的问题建模。有两种搜索模式可供选择:伏击和巡游搜索。有些物种可以采用任一种模式,它们在特定时间的选择传统上可以根据不同的栖息地和生理条件来解释。我们提出了另一种解释,即捕食者在这两种搜索模式之间交替的原因,这基于它们所玩的搜索博弈的动态性质:伏击的可能性降低了猎物频繁改变位置的倾向,从而使其更容易受到策略中系统巡游搜索部分的影响。这种启发式解释得到了支持,通过表明在一个新的理想化搜索游戏中,捕食者可以随时进行伏击或搜索,而猎物可以在间歇时间改变位置,最优捕食者策略需要随着时间的推移交替(或混合)伏击和巡游搜索。因此,我们的博弈是研究得很好的“公主与怪物”搜索博弈的扩展。搜索博弈是零和博弈,收益是捕获时间,搜索者和隐藏者都不知道对方的位置。当捕食者使用随时间变化的混合策略时,我们能够确定搜索模式的最佳混合比例,也能够确定当允许动态策略时模式混合如何随时间变化(伏击概率随时间增加)。特别地,我们建立了“搜索捕食的平方根法则”:主动搜索的最佳比例等于尚未探索区域的分数的平方根。