Capture of the Lamb: Diffusing Predators Seeking a Diffusing Prey

What is the survival probability of a diffusing lamb which is hunted by N hungry lions? Although this capture process is appealingly simple to define ~see Fig. 1!, its long-time behavior poses a theoretical challenge because of the delicate interplay between the positions of the lamb and the closest lion. This model also illustrates a general feature of nonequilibrium statistical mechanics: life is richer in low dimensions. For spatial dimension d.2, it is known that the capture process is ‘‘unsuccessful’’ ~in the terminology of Ref. 1!, as there is a nonzero probability for the lamb to survive to infinite time for any initial spatial distribution of the lions. This result is a consequence of the transience of diffusion for d.2, which means that two nearby diffusing particles in an unbounded d.2 domain may never meet. For d52, capture is ‘‘successful,’’ as the lamb dies with certainty. However, diffusing lions in d52 are such poor predators that the average lifetime of the lamb is infinite! Also, the lions are essentially independent, so that the survival probability of a lamb in the presence of N lions in two dimensions is SN(t)}S1(t) , where S1(t), the survival probability of a lamb in the presence of a single lion, decays as (ln t). Lions are more efficient predators in d51 because of the recurrence of diffusion, which means that two diffusing particles are certain to meet eventually. The d51 case is also special because there are two distinct generic cases. When the lamb is surrounded by lions, the survival probability at a fixed time decreases rapidly with N because the safe zone which remains unvisited by lions at fixed time shrinks rapidly in N . This article focuses on the more interesting situation of N lions all to one side of the lamb ~Fig. 1!, for which the lamb survival probability decays as a power law in time with an exponent that grows only logarithmically in N . We begin by considering a lamb and a single stationary lion in Sec. II. The survival probability of the lamb S1(t) is closely related to the first-passage probability of onedimensional diffusion and leads to S1(t);t . It is also instructive to consider general lion and lamb diffusivities. We treat this two-particle system by mapping it onto an effective single-particle diffusion problem in two dimensions with an absorbing boundary to account for the death of the lamb when it meets the lion, and then solving the twodimensional problem by the image method. We apply this approach in Sec. III by mapping a diffusing lamb and two diffusing lions onto a single diffusing particle within an absorbing wedge whose opening angle depends on the particle diffusivities, and then solving the diffusion problem in this absorbing wedge by classical methods. In Sec. IV, we study N@1 diffusing lions. An essential feature of this system is that the motion of the closest ~‘‘last’’! lion to the lamb is biased towards the lamb, even though each lion diffuses isotropically. The many-particle system can be recast as a two-particle system consisting of the lamb and an absorbing boundary which, from extreme statistics, moves to the right as A4DLt ln N, where DL is the lion diffusivity. Because this time dependence matches that of the lamb’s diffusion, the survival probability depends intimately on these two motions, with the result that SN(t);t 2bN and bN}ln N. The logarithmic dependence of bN on N reflects the fact that each additional lion poses a progressively smaller marginal peril to the lamb—it matters little whether the lamb is hunted by 99 or 100 lions. Amusingly, the value of bN implies an infinite lamb lifetime for N<3 and a finite lifetime otherwise. In the terminology of Ref. 1, the capture process changes from successful to ‘‘complete’’ when N>4. We close with some suggestions for additional research on this topic.