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 lnP ev(t) ∼ (N/V)2lnP imm(t) between the survival probabilities of an evasive and an immobile prey. Hence, when the density ρ = N/V of the predators is low, ρ ≪ 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.

[1]  J. Klafter,et al.  On the joint residence time of N independent two-dimensional Brownian motions , 2003 .

[2]  O Bénichou,et al.  Lattice theory of trapping reactions with mobile species. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  D. Weihs,et al.  Optimal avoidance and evasion tactics in predator-prey interactions , 1984 .

[4]  Pai-Chi Li,et al.  Photoacoustics for molecular imaging and therapy. , 2009, Physics today.

[5]  O Bénichou,et al.  Pascal principle for diffusion-controlled trapping reactions. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  Occupancy of a single site by many random walkers , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[7]  Shlomo Havlin,et al.  On the survival probability of a random walk in a finite lattice with a single trap , 1985 .

[8]  Paul Erdös,et al.  Some Problems on Random Walk in Space , 1951 .

[9]  S. Redner,et al.  Capture of the lamb: Diffusing predators seeking a diffusing prey , 1999, cond-mat/9905299.

[10]  Richard A Blythe,et al.  Formal solution of a class of reaction-diffusion models: reduction to a single-particle problem. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  A. Berezhkovskii,et al.  Kinetics of escape through a small hole , 2002 .

[12]  K. Shuler,et al.  Order statistics for first passage times in diffusion processes , 1983 .

[13]  S. Redner,et al.  Kinetics of a Diiusive Capture Process: Lamb Besieged by a Pride of Lions , 2022 .

[14]  J. Hoogenboom,et al.  Beyond quantum jumps: Blinking nanoscale light emitters , 2009 .

[15]  S. Redner,et al.  Kinetics of the 'scavenger' reaction , 1984 .

[16]  Exact asymptotics for one-dimensional diffusion with mobile traps. , 2002, Physical review letters.

[17]  H. Hilhorst,et al.  Covering of a finite lattice by a random walk , 1991 .

[18]  Z. Schuss,et al.  The narrow escape problem for diffusion in cellular microdomains , 2007, Proceedings of the National Academy of Sciences.

[19]  J. A. Bather.  Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization , 1966 .

[20]  M. Tachiya.  Theory of diffusion-controlled reactions: Formulation of the bulk reaction rate in terms of the pair probability , 1983 .

[21]  J. Klafter,et al.  Target annihilation by random walkers , 1984 .

[22]  R A Blythe,et al.  Survival probability of a diffusing particle in the presence of Poisson-distributed mobile traps. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  O. Bénichou,et al.  Trapping reactions with randomly moving traps: exact asymptotic results for compact exploration. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  How rare are diffusive rare events , 2008, 0804.1165.

[25]  Moreau,et al.  Kinetics of stochastically gated diffusion-limited reactions and geometry of random walk trajectories , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[26]  O. Bénichou,et al.  Narrow-escape time problem: time needed for a particle to exit a confining domain through a small window. , 2008, Physical review letters.

[27]  S. B. Yuste,et al.  Survival probability of a particle in a sea of mobile traps: a tale of tails. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.