Random search processes occur in many areas, from chemical

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1 évy strategies in intermittent search processes are advantageous Michael A. omholt, Tal Koren, Ralf Metzler, Joseph Klafter MEMPHYS Center for Biomembrane Physics, Department of Physics and Chemistry, University of Southern Denmark, Campusvej 55, DK-53 Odense M, Denmark; Physics Department, Technical University of Munich, James Franck Strasse, D Garching, Germany; and School of Chemistry, Tel Aviv University, Tel Aviv, IsraelB; Communicated by Joshua Jortner, Tel Aviv University, Tel Aviv, Israel, and approved April 6, 8 (received for review January 8, 8) Intermittent search processes switch between local Brownian search events and ballistic relocation phases. We demonstrate analytically and numerically in one dimension that when relocation times are évy distributed, resulting in a évy walk dynamics, the search process significantly outperforms the previously investigated case of exponentially distributed relocation times: The resulting évy walks reduce oversampling and thus further optimize the intermittent search strategy in the critical situation of rare targets. We also show that a searching agent that uses the évy strategy is much less sensitive to the target density, which would require considerably less adaptation by the searcher. random processes optimization évy walk movement ecology Random search processes occur in many areas, from chemical reactions of diffusing reactants () to the foraging behavior of bacteria and animals (, 3). Of general importance is the search efficiency. Brownian search in one and two dimensions involves frequent returns to an area, leading to oversampling. Higher efficiency, can be achieved, for instance, by facilitated diffusion in gene regulation (4) or by controlled motion in foraging (, 3). From theoretical and data analysis évy strategies, in which the searching agent performs excursions whose length is drawn from distributions with a heavy tail λ(x) x α, for <α< were shown to be advantageous (5 6); occasional long excursions assist in exploring previously unvisited areas and significantly reduce oversampling. As an alternative to évy search, intermittent strategies have been introduced to improve the efficiency of diffusive search (7 ). Intermittent search requires that the searcher occasionally shifts focus from the search and concentrates on fast relocation. The relocation phase implies that the searcher is wasting time in the short run because the target cannot be spotted during it. However, the overall search efficiency is improved by introducing the searcher to previously unexplored areas (7 ). In refs. 8 and relocation events were assumed to occur in a random direction for exponentially distributed time spans, giving rise to a Markovian process. We show here analytically and numerically in one dimension that this is only a partial solution to oversampling, as eventually the central limit theorem (CT) reduces the process to a Brownian random walk with jumps on the scale of vτ, where τ is the typical time spent in a relocation event. In practice, revisits can be reduced by adjusting the average time spent in search and relocation phases to the density of targets. évy strategies, on the other hand, fundamentally circumvent the CT, and we here demonstrate a twofold advantage of them over the exponential distribution: évy walk intermittent processes find the target faster than exponential strategies in the critical case of rare targets, and their performance is much less dependent on adapting to the target density. Intermittent Search with évy Relocations Generalizing the model from ref., we consider two phases: In the search phase the searcher scans for the target by diffusive motion with diffusivity D. With probability per time τ the searcher switches to the relocation phase, during which it moves ballistically with velocity v in a random direction (). The relocation time is drawn from the waiting time distribution ψ(t), which will be considered to be exponential or évy stable. The purpose of relocations is to move as quickly as possible away from the area that has just been searched, and thus the searcher is not scanning for the target in this phase. To compare with previous results we take a closed cell approach: the search is performed on an interval of length with periodic boundary conditions, corresponding to regularly spaced targets with density /. The model can be formulated as an equation for the probability density P(x, t) for the position x of the searcher in the search phase: P t = / dx dt W (x x, t t )P(x, t ) τ / P(x, t) + D P τ x p fa(t)δ(x). The role of the last term on the right-hand side is to remove the particle when it arrives at the target placed at x =. The density p fa (t) thus represents the first arrival time at the target, which is determined implicitly by the absorbing boundary condition P(x =, t) =. The term proportional to the diffusivity D describes the local Brownian motion in the search phase. The term τ P(x, t) removes the searcher from location x with rate τ. The searcher is then relocated according to the integral expression in which the kernel W (x, t) is the joint probability density of making a relocation of length x during a time t. It is defined by W (x, t) = ψ(t) δ( x + n vt). 3a Here the δ-coupling enforces that the distance traveled in time t is vt, and the sum over n renders W (x, t) -periodic in x. ψ(t) is related to the spatial distribution of the relocations λ(x)by ψ(t) = vλ(vt). 3b The jump length distribution λ(x) is assumed to be symmetric around x = (no orientational memory). The search efficiency is quantified by the mean search time t = dt tp fa (t). 4 Author contributions: M.A.., T.K., R.M., and J.K. designed research, performed research, analyzed data, and wrote the paper. The authors declare no conflict of interest. To whom correspondence should be addressed. metz@ph.tum.de 8 by The National Academy of Sciences of the USA / cgi / doi /.73 / pnas.8375 PNAS August, 8 vol. 5 no

2 To obtain t we Fourier expand P(n, t) = / / dx e iknx P(x, t), 5 where n is an integer with corresponding wavenumber k n = πn/, and aplace transform, where P(n, u) = We find dt e ut P(n, t). 6 up(n, u) δ n, = W (n, u)p(n, u) P(n, u) τ τ Dkn P(n, u) p fa(u). 7 The initial distribution is uniform, P(x, t = ) = /, because the searcher initially has no information on the target position. Isolating P(n, u), summing over n (note that n P(n, u) = P(x =, u) = ), we find for p fa (u), { } u + ψ(u)/τ p fa (u) = u + Dkn + W (n, u)/τ. 8 In aplace space the mean search time t yields from expansion of p fa at small u because p fa (u) t u +... From the average time τ spent in a single relocation event (ψ(u) τ u +...), one obtains t = n= (τ + τ ) Dτ k n + λ(k n). 9a Here, λ(k n ) = W (n, u = ) = dx e iknx λ(x) 9b is the Fourier transform of the relocation length distribution λ(k) = dx eikx λ(x), at the discrete wavenumbers k n = πn/. We now use Eq. 9a to determine the search efficiency of (i) évy and (ii) exponentially distributed relocations: (i) For évy distributed relocations we use the symmetric évy stable law with characteristic function () λ(k) = exp{ σ α k α πvτ }, σ = Ɣ( /α). From this closed expression the asymptotic form follows. The index α is restricted to <α<so that the mean relocation time τ is finite. Fig. depicts trajectories for cases of exponential and évy relocations, distinguishing the évy case with its occasional long relocations. We introduce three approximations valid for large : (a) Assume that vτ Dτ, i.e., that the mean relocation distance is much longer than the average distance scanned in a typical search phase. We will see that this is self-consistent with the obtained optimal values of τ and τ that have the same -scaling for large. This assumption means that Dτ kn and λ(k n) are to a good approximation nonzero at different n, and we expand Dτ kn + λ(k n) Dτ kn + +. λ(k n ) (b) Assuming that the search range Dτ is much smaller than, we replace the sum over the first term on the right-hand side of Eq. by an integral, yielding n= Dτ kn + Dτ k n + dn = 4 Dτ. Fig.. x t diagram with exponential and évy relocations, with τ = 37, τ =, D =, v =., and =. (c) Because λ(k n ) σ α k n α at small values of k n (k n at n = in the limit of large ) we approximate the last two terms of Eq.. Namely, the contribution from the singularity at small n dominates the sum, λ(k n ) n= ( ) α ζ (α). 3 πσ Here ζ (α) = n= n α is the Riemann ζ function. Collecting a to c, Eq. 9a is approximated by t (τ + τ ) 4 Dτ + ( πσ ) α ζ (α). 4 For honest comparison between évy and exponential strategies, we determine the respective optimal τ and τ. Solving t / τ = and t / τ = simultaneously, we obtain from Eq. 4 that at large τ (b/a α ) /(α /), τ (b/ a) /(α /), 5 where (using + 4(α )α) a = ( + )/(α ), 6a b = Ɣ( α Dα + 3ζ (α) α α ) 6b π v such that the optimal τ i scale with as (α )/(α /). According to Eq. 4, t will then scale as (3α )/(α ), implying that for large the more efficient search will occur for α close to. However, the prefactor to the -scaling diverges as α, so the optimal choice of α will be somewhat larger than for any finite, as demonstrated in Fig.. The inset of Fig. shows the validity of the approximate t for optimal τ i. (ii) For exponentially distributed relocation with ψ(t) = τ e t/τ, 7 approximations a to c also apply, with σ = vτ. The corresponding results for t and optimal τ i obtain by replacing Ɣ( /α)byπ/ and taking α = : t τ + τ ( 6 + Dτ vτ ), 8 τ (D/8v 4 ) /3 /3 /, τ τ / cgi / doi /.73 / pnas.8375 omholt et al.

3 Fig.. Solid lines, optimal α and ratio η of search times for optimal α vs. exponential strategy, as function of. Dashed lines, ratio η = t,τi ()/ t,τi ( ) of optimal vs. fixed τ i search times as a function of for exponential and α =.5 évy strategies ( = 5 4 ). The values are calculated using asymptotic Eqs. 4 and 8, and corresponding optimal τ and τ (allowing the optimal α to range between and ). (Inset) Convergence of the asymptotic t a (Eqs. 4 and 8) toward the exact t e (Eq. 9a) with for asymptotically optimal τ i. D =, v = for all curves. These expressions agree with those of ref.. Performance of évy Intermittent Search The search time t for exponential strategies scales as 4/3 for optimal τ and τ, proving that évy strategies with <α< are increasingly more efficient than the exponential strategies for decreasing target density. In Fig. 3 we show t as function of relocation time τ. An additional advantage of évy strategies is due the scaling τ i (α )/(α /) of the optimal τ i : for α close to unity the optimal strategy becomes insensitive to the target density. This means that it is less important for the searcher to have advance knowledge of the density of targets if it follows a small α évy strategy, because it can choose τ i that are almost optimal over a broad range of densities. This point is illustrated in Fig.. To understand better the α-dependence of the évy strategy we study the first arrival density p fa (t) for large, where again vτ Dτ. We consider times much longer than one relocation search cycle such that ψ(u) τ u+..., and rewrite Eq. 8 as p fa (u) τ u τ + τ W (u), where we have introduced the term W (u) = u + Dk n + W (n, u)/τ. The last expression can be simplified by following similar approximations as for t before. The separation of length scales leading to approximation a allows us to write W (u) τ Dτ kn + + τ u + W (n, u). For the last two terms in Eq. the contributions at small n again dominate the sum (approximation c); expanding W (n, u) at small k n and u produces W (n, u) σ α k n α τ u. Collecting the results, we find W (u) τ Dτ kn (τ + τ )u + σ α k n α We focus on times short enough such that the -periodicity of the problem does not yet play a role, so that aplace space u (σ α k n α )/(τ +τ )atn =. In this approximation we replace the sum by the integral dk n/(π), obtaining W (u) Dτ + τ /α sin(π/α)σ. 4 u(τ + τ ) /α For shorter times (corresponding to larger u) we discard the subdominant second term in Eq. 4. aplace inversion of Eq. then produces p fa (t) Dτ /(τ + τ ). 5 At later times (smaller u) the second term in Eq. 4 dominates, and the plateau 5 turns into p fa (t) α sin ( π α ) vτ. 6 (τ + τ ) /α t /α The crossover between these two regimes occurs when the values of expressions 5 and 6 become equal, i.e., at t (τ + τ ){αsin(π/α) vτ /4 Dτ } α/(α ). 7 Discussion In Eq. 5, Dτ is the average length scanned in a search event. Division by yields the probability to find the target during this phase, and /(τ + τ ) is the rate at which the search phase itself occurs. A crucial part in this interpretation is that the probability of searching in a previously scanned area is negligible. This assumption will break down at some point because of the searcher s lack of orientational memory. The searcher will then begin to revisit explored regions with a reduced probability of finding the target Note that there is a typographical error in the expression for t in equation 5 of ref. : coth (/α) should be coth(α/). Fig. 3. Mean search time for évy (α =.35) and exponential strategies as function of τ at asymptotically optimal τ (τ = 37. for évy and τ = 4 for exponential). We chose = 5, D =, v =. Simulations versus exact (Eq. 9a) and asymptotic (Eqs. 4 and 8) theory are shown. omholt et al. PNAS August, 8 vol. 5 no. 3 57

4 Fig. 4. First arrival density versus time. The crosses are simulation data, and the straight lines are the intermediate regimes of Eq. 5 and Eq. 6. Parameters are τ = 35, τ = 5, = 4, α =.75, v =, and D =. as a result. This causes the crossover to the power-law behavior 6. Fig. 4 shows the turnover from plateau to inverse power-law of the first arrival. At even longer times, finite size effects cause a turnover to an exponential decay. From Eq. 6 the advantage of having α close to unity at large becomes evident: the presence of rare but long relocation events reduces the risk of rescanning already visited areas, which will be important for large. However, the downside to choosing an α-value too close to is that an increased amount of very long relocations implies an increased amount of very short ones too, because the average distance is fixed by vτ (4). This means that the crossover to the less favorable situation described by Eq. 6 happens earlier, so that larger α becomes more efficient for shorter search times relevant at smaller. Intermittent strategies are beneficial when purely diffusive search would slow down over time due to the increasing returns to previously scanned areas (oversampling). Choosing an exponential strategy for relocations, however, only partially solves this problem: At times t τ, the CT governs, leading to oversampling on a typical scale vτ. Conversely, évy-intermittent strategies are not bound to the CT, rendering them a more amenable solution to reduce oversampling and therefore advantageous in the search for rare targets. Although less pronounced, the problem of oversampling still occurs in two-dimensional search studied in ref. 9. évy strategies are expected to improve the search efficiency in this case, as well; however, as to what extent remains to be established quantitatively. On the basis of our results we advocate that intermittent strategies should not be thought of as alternatives to évy strategies. In contrast, the synergistic combination of intermittent search and évy relocation strategies turns out to be beneficial. Moreover, a given évy walk intermittent search strategy (with fixed τ i )is almost optimal over a wide range of sparse target densities, which might be a strategic advantage for creatures that have limited abilities to adjust their search parameters. We note, however, that the small scaling exponent of t with for the évy strategy is not a result of the évy part of the strategy alone. To explain what we mean by this we will define the pure évy strategy as a strategy where the searcher only quickly tests his immediate neighborhood for the target at the end of each relocation. Thus we assume that τ has a small finite value (alternatively the target could have a small finite size and τ = ) and only consider optimization of the strategy with respect to τ. Doing this, we find from our analytic asymptotic result that the optimal τ scales with as /α and that this results in a scaling t /α,a scaling that increases faster with for any α> compared with the result where τ is also optimized. And it is only an improvement over the optimized exponential strategy when α<3/. Without any optimization the évy strategy would result in t α,a scaling that that is still better than for the optimized exponential strategy when α<4/3. A remark on the recent discussion about the empirical observation of évy distributions of relocation lengths in animal foraging is in order. Thus, while the original publications provided evidence of long-tailed relocation lengths in accordance with theoretical considerations (6 8), a reanalysis of the data reveals that the original data contained few extreme events for the flight times, after removal of which the data no longer unequivocally allow an interpretation as évy pattern (3). In that paper also a few other previous claims of évy foraging patterns were invalidated. This has caused some uncertainty about the general relevance of évy search patterns in animal foraging (4). Among the recent criticisms of ref. 3 we refer to the consideration of finite size effects of real trajectories in ref. 5 that were shown to reestablish the validity of a évy-based search mechanism for the albatross flight. It is our belief that évy search models show a distinct advantage over strategies governed by the central limit theorem. However, it will require considerably larger data sets to be able to tell for sure whether typically animals use a specific search strategy. The value of this and similar theoretical studies is to provide a framework for the analysis of data that are being collected now or in the future. The robustness of the search efficiency of évy strategies to changing target densities, as demonstrated here, appears to be a key concept in the discussion of search mechanisms, and potentially an important evolutionary advantage. Our analysis relies on the assumption that each relocation is pointed toward a random direction. This will be a good model for nonintelligent search, similar to bacterial movement in the absence of chemical or temperature gradients, during which tumbling motion changes with directed motion (). Intelligent creatures will improve the target search by partial or complete memory, avoiding previously visited locations. 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