Survival of an evasive prey

TL;DR

This paper analyzes how a minimal-effort evasion strategy affects prey survival against multiple predators performing random walks, revealing conditions under which the prey's survival probability significantly increases.

Contribution

It introduces a simple model of prey evasion with a minimal effort strategy and derives analytical relations for survival probabilities on a lattice.

Findings

At least 3 predators are needed to catch a prey with sighting range 1.

The survival probability increases dramatically at low predator densities.

Short-sighting prey exhibit superdiffusive motion, while far-sighting prey behave diffusively.

Abstract

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(Pev)(t) \sim (N/V)2ln(Pimm(t)) between the survival probabilities of an evasive and an immobile prey. Hence, when the density of the predators is low N/V<<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.