Oshanin G, Vasilyev O, Krapivsky P L, Klafter J
Laboratory J.-V. Poncelet, Unité Mixte Internationale 2615, Centre National de la Recherche Scientifique, Independent University of Moscow, Bolshoy Vlasyevskiy Pereulok 11, 119002 Moscow, Russia.
Proc Natl Acad Sci U S A. 2009 Aug 18;106(33):13696-701. doi: 10.1073/pnas.0904354106. Epub 2009 Jul 29.
We study the survival of a prey that is hunted by N predators. The predators perform independent random walks on a square lattice with V sites and start a direct chase whenever the prey appears within their sighting range. The prey is caught when a predator jumps to the site occupied by the prey. We analyze the efficacy of a lazy, minimal-effort evasion strategy according to which the prey tries to avoid encounters with the predators by making a hop only when any of the predators appears within its sighting range; otherwise the prey stays still. We show that if the sighting range of such a lazy prey is equal to 1 lattice spacing, at least 3 predators are needed in order to catch the prey on a square lattice. In this situation, we establish a simple asymptotic relation ln P(ev)(t) approximately (N/V)(2)ln P(imm)(t) between the survival probabilities of an evasive and an immobile prey. Hence, when the density rho = N/V of the predators is low, rho << 1, the lazy evasion strategy leads to the spectacular increase of the survival probability. We also argue that a short-sighting prey (its sighting range is smaller than the sighting range of the predators) undergoes an effective superdiffusive motion, as a result of its encounters with the predators, whereas a far-sighting prey performs a diffusive-type motion.
我们研究了一种被(N)个捕食者追捕的猎物的生存情况。捕食者在具有(V)个格点的正方形晶格上进行独立随机游走，并且每当猎物出现在它们的视线范围内时就开始直接追捕。当一个捕食者跳到猎物所在的格点时，猎物就会被捕获。我们分析了一种懒惰的、最小努力的逃避策略的有效性，根据这种策略，猎物仅在任何一个捕食者出现在其视线范围内时才跳跃，试图避免与捕食者相遇；否则猎物保持静止。我们表明，如果这种懒惰猎物的视线范围等于(1)个晶格间距，那么在正方形晶格上至少需要(3)个捕食者才能捕获猎物。在这种情况下，我们建立了逃避猎物和静止猎物的生存概率之间的一个简单渐近关系(\ln P_{(ev)}(t) \approx (\frac{N}{V})^2 \ln P_{(imm)}(t))。因此，当捕食者的密度(\rho = \frac{N}{V})较低时，即(\rho << 1)，懒惰的逃避策略会导致生存概率显著增加。我们还认为，短视的猎物（其视线范围小于捕食者的视线范围）由于与捕食者的相遇而经历有效的超扩散运动，而远视的猎物则进行扩散型运动。
