Capture of a diffusive prey by multiple predators in confined space

Indrani Nayak, Amitabha Nandi, and Dibyendu Das

Phys. Rev. E 102, 062109 – Published 4 December 2020

Abstract

The first passage search of a diffusing target (prey) by multiple searchers (predators) in confinement is an important problem in the stochastic process literature. While the analogous problem in open space has been studied in some detail, a systematic study in confined space is still lacking. In this paper, we study the first passage times for this problem in one, two, and three dimensions. Due to confinement, the survival probability of the target takes a form $\sim e^{- t / τ}$ at large times $t$ . The characteristic capture timescale $τ$ associated with the rare capture events are rather challenging to measure. We use a computational algorithm that allows us to estimate $τ$ with high accuracy. We study in detail the behavior of $τ$ as a function of the system parameters, namely, the number of searchers $N$ , the relative diffusivity $r$ of the target with respect to the searcher, and the system size. We find that $τ$ deviates from the $\sim 1 / N$ scaling seen in the case of a static target, and this deviation varies continuously with $r$ and the spatial dimensions.

Received 31 August 2020
Accepted 12 November 2020

DOI:https://doi.org/10.1103/PhysRevE.102.062109

Physics Subject Headings (PhySH)

Diffusion Random walks Stochastic processes

Extreme event statistics Fokker–Planck equation

Statistical Physics & Thermodynamics

Authors & Affiliations

Indrani Nayak ^*, Amitabha Nandi ^†, and Dibyendu Das^‡

Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India

^*inayak21@phy.iitb.ac.in
^†amitabha@phy.iitb.ac.in
^‡dibyendu@phy.iitb.ac.in

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Vol. 102, Iss. 6 — December 2020

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Images

Figure 1
Schematic of $N$ noninteracting searchers or lions [green (light gray)] are foraging for a diffusive target or the lamb [red (dark gray)] in $d$ -dimensional confined geometry. The lamb has diffusivity $D_{l}$ , and all the lions have equal diffusivity $D_{L}$ . The confined region is (a) a line segment of length $R$ in $d = 1$ , (b) a circle in $d = 2$ , and (c) a sphere in $d = 3$ of radius $R$ . The lamb is considered to be a circle and a sphere of radius $a$ in $d = 2, 3$ respectively. In $d = 1$ , the system is bounded by two reflecting (rigid) boundaries at $x = 0$ and $R$ . In $d = 2$ and 3, the system is bounded by a radially symmetric reflecting (rigid) boundary at radius $R$ .
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Figure 2
Illustration of the cloning algorithm to compute the survival probability $S (t)$ of the lamb [red (dark gray)] in the presence of $N$ lions [green (light gray)] in a $d = 1$ setup. At $t = 0$ , one starts with $M$ realizations of the system. Here $M = 4$ and $N = 1$ . According to this algorithm, when capture happens at time $t_{1}$ in half of the realizations $[S (t_{1}) = 1 / 2]$ , we replace them with the replications of the survived ones (Enrichment-1). A similar step (Enrichment-2) is performed at time $t_{2}$ when $S (t_{2}) = 1 / 4$ . We depict the captured lion by changing its color from green to red.
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Figure 3
$S (t)$ of a lamb inside a $d = 1$ box with $R = 4.0$ (arb. units) and in presence of 50 lions is plotted in a semilog scale. Three curves correspond to three distinct initial positions $x_{0} = 2.5$ (purple lower line), 3 (blue middle line), and 3.5 (cyan upper line) (arb. units) of the pack of lions. The initial position of the lamb is $x = 1$ (arb. units) for all the cases. Although the curves are distinct from each other, asymptotically they are all exponentials with the same $τ$ ; numerically we obtained $τ^{- 1} = 11.98$ , 11.94, and 11.79 for the three different cases, respectively, which are very close. A plot of $- d (ln S) / d t$ vs $t$ (corresponding to the blue middle line) is shown in the inset, and its saturation value $τ^{- 1}$ (in the steady state) is indicated by a black dashed line.
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Figure 4
$τ D_{L}$ vs $N$ is plotted in the log-log scale for (a) $d = 1$ , (b) $d = 2$ , and (c) $d = 3$ setups (see Fig. 1). Each color of the above figures represents $τ D_{L}$ for different ratios $r = 0, 0.2, 1, 5, 20, 80$ (see the key) of diffusivities of the lamb and lions ( $D_{l} / D_{L}$ ). Static lamb or $r = 0$ (navy blue solid square) curves give the exact power law $τ D_{L} \sim N^{- 1}$ (navy blue solid line). For moving lamb or $r \neq 0$ , the curves show departures from the power law $N^{- 1}$ , and $τ D_{L}$ nontrivially decreases with $N$ . At smaller $N$ , the power of $N$ changes faster in a lower dimension than in the higher dimension upon increasing $r$ (top to bottom). For $r = 80$ (cyan hollow triangle), curves in (a) $d = 1$ shows $τ D_{L} \sim N^{- 0.3}$ , (b) $d = 2$ shows $τ D_{L} \sim N^{- 0.52}$ , and (c) $d = 3$ shows $τ D_{L} \sim N^{- 0.8}$ at smaller $N$ . However, at larger $N$ for different $r$ , the curves seem to approach a parallel trend by coming closer to each other. The above numerics are done with $a = 0.4$ (arb. units) and $R = 8, 4, 4$ (arb. units) in $d = 1, 2, 3$ respectively.
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Figure 5
Comparison of $R^{2} / τ D_{L}$ obtained numerically (dark red squares) with the approximate analytical prediction in Eq. (6) (light red circles) for different values of $r$ .
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Figure 6
(a) $τ D_{L}$ vs $N$ is plotted for $d = 2$ setup [see Fig. 1] for $r = 5, 20$ in log-log scale. $τ D_{L}$ is a function of $r$ and does not depend upon the individual diffusivities $D_{l}$ or $D_{L}$ . For a fixed $r$ , $τ D_{L}$ collapses for any pair of diffusivities $D_{l}, D_{L}$ . (b) $τ D_{L} / g_{d}$ vs $R$ is plotted for $r = 1$ with fixed $a = 0.4, N = 5$ in $d = 1, 2, 3$ . Here, $g_{d} (R, a)$ corresponds to the scaling of $τ D_{L}$ for the static lamb case, in $d$ dimensions. (c) $τ D_{L} / g_{2}$ vs $N$ is plotted for three distinct radii $a = 0.2, 0.3, 0.4$ of the lamb, in log-log scale with fixed $R = 0.4$ . We show this for two different relative diffusivities $r = 1 (D_{l} = 3, D_{L} = 3)$ and $5 (D_{l} = 5, D_{L} = 1)$ [see Fig. 1 setup].
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