Occupancy of a single site by many random walkers

Survival of an evasive prey

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.

Multiparticle random walks on a deformable medium

Multiparticle random walks on a deformable medium have been investigated in (2+1) dimensions. The time evolution of the particle distribution is studied. The results show that the randomly distributed particles in the beginning will be self-organized into a cluster pattern in the intermediate stage, and then return to the random distribution pattern in the late stage. The dependence of the clustering degree on the stiffness parameter of medium alpha, stability parameter of systems beta, and average particle density rho(0) is also investigated. There exists an optimal clustering stability beta(p), at which the system has the strongest clustering ability and corresponds to a maximum clustering coefficient Gamma(*)(p). The dependence of the optimal clustering coefficient Gamma(*)(p) on the stiffness alpha and particle density rho(0) is obtained, and the landscape of the medium generated by particles is also investigated.