Optimizing persistent random searches
OOptimizing persistent random searches
Vincent Tejedor,
1, 2
Raphael Voituriez, and Olivier B´enichou Physics Department, Technical University of Munich,James Franck Strasse, 85747 Garching, Germany Laboratoire de Physique Th´eorique de la Mati`ere Condens´ee,CNRS/Universit´e Pierre et Marie Curie, 4 Place Jussieu, 75005 Paris
We consider a minimal model of persistent random searcher with short range memory. We cal-culate exactly for such searcher the mean first-passage time to a target in a bounded domain andfind that it admits a non trivial minimum as function of the persistence length. This reveals anoptimal search strategy which differs markedly from the simple ballistic motion obtained in thecase of Poisson distributed targets. Our results show that the distribution of targets plays a crucialrole in the random search problem. In particular, in the biologically relevant cases of either a singletarget or regular patterns of targets, we find that, in strong contrast with repeated statements in theliterature, persistent random walks with exponential distribution of excursion lengths can minimizethe search time, and in that sense perform better than any Levy walk.
PACS numbers:
The random search problem addresses the question ofdetermining the time it takes a searcher performing arandom walk to find a target [1]. At the microscopicscale, search processes naturally occur in the context ofchemical reactions, for which the encounter of reactivemolecules is a required first step. An obvious historicalexample is the theory of diffusion—controlled reactions,which has regained interest in the last few years in thecontext of genomic transcription in cells [1]. Interest-ingly, the random search problem has also proved in thelast decades to be relevant at the macroscopic scale, as inthe case of animals searching for a mate, food, or shelter[2–9].In all these examples, the time needed to discover atarget is a limiting quantity, and consequently the min-imization of this search time often appears as essential.In this context Levy walks, which are defined as ran-domly reoriented ballistic excursions whose length l isdrawn from a power law distribution P ( l ) ∝ l →∞ /l µ with 0 < µ ≤
2, have been suggested as potential candi-dates of optimal strategies [5]. In fact, Levy walks havebeen shown mostly numerically to optimize the searchefficiency, but only in the particular case where the tar-gets are distributed in space according to a Poisson law,and are in addition assumed to regenerate at the samelocation after a finite time. Conversely, in the case of adestructive search where each target can be found onlyonce the optimal strategy proposed in [5] is not anymoreof Levy type, but reduces to a trivial ballistic motion.Given these restrictive conditions of optimization, thepotential selection by evolution of Levy trajectories asoptimal search strategies is disputable, and in fact thefield observation of Levy trajectories for foraging animalsis still elusive and controversial [10–12].From the theoretical point of view, the search timecan be quantified as the first-passage time of the ran-dom searcher to the target [13]. In the case of a singletarget in a bounded domain, or equivalently of infinitelymany regularly spaced targets, asymptotic results for the mean first-passage time (MFPT) and the full distributionof the first-passage time have been obtained for Marko-vian scale invariant random walks [14, 15]. These re-sults apply in particular directly to Brownian particlesthat are subject to thermal fluctuations, and thereforeto diffusion–limited reactions in general. At larger scaleshowever, most examples of searchers – even if random –have at least short range memory skills and show per-sistent motions, as is the case for bacteria [16] or largerorganisms [2], which cannot be described as Markovianscale invariant processes. The study of persistent randomwalks therefore appears as crucial to assess the efficiencyof many search processes, and has actually also provedto be important in various fields such as neutron or lightscattering [17–19]. In this context exact results have beenderived that characterize the diffusion properties of per-sistent walks in infinite space [19–21], or mean returntimes in bounded domains [17, 18, 22, 23]. The ques-tion of determining first-passage properties of persistentwalks has however remained unanswered so far.
XTargetStart ~l p FIG. 1: (Color online) Example of search trajectory for apersistent random searcher in a bounded domain.
In this paper, we consider a minimal model of per- a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b sistent random searcher – called persistent random walkmodel hereafter – with short range memory character-ized by an exponential distribution of the length of itssuccessive ballistic excursions P ( l ) ∝ l →∞ e − αl/l p , where l p is the persistence length of the walk and α a numeri-cal factor. We calculate exactly for such persistent ran-dom walker the MFPT to a target in a bounded domain,which corresponds implicitly to the case of a destructivesearch since the target can be discovered only once, andfind that it admits a non trivial minimum as a functionof l p , thus revealing an optimal search strategy which isvery different from the simple ballistic motion obtained inthe case of Poisson distributed targets. In addition, weshow numerically that such optimal persistent randomwalk strategy is more efficient than any Levy walk ofparameter µ ∈ ]0 , d –dimensional cubic lattice L of vol-ume V = X d , where a single target site is located. Inpractice we take d = 2 or d = 3, make use of peri-odic boundary conditions and consider the dilute regime X (cid:29)
1. This geometry encompasses both cases of asingle target centered in a confined domain, and of reg-ularly spaced targets in infinite space with concentra-tion 1 /V . The latter situation can be seen as a limitingcase of target distribution with strong correlations, asopposed to the Poissonian case, and is biologically mean-ingful for example in the case of repulsive interactionsbetween targets [2]. The case of a target of arbitraryposition in a domain with reflective boundary conditionscan also be solved exactly using similar techniques andhas been checked to yield analogous results; analyticalexpressions are however much more complicated in thiscase and are omitted here for clarity. Note that herethe lattice step size corresponds to the target size and isset to 1, which defines the unit length of the problem.At each time step, the random searcher has a probabil-ity p to continue in the same direction, p to go back-ward, and p to choose an orthogonal direction, so that p = (1 − p − p ) / (2 d − p = p + (cid:15) and p = p − δ , and set in what follows δ = 0for the sake of simplicity. The probability of a ballisticexcursion of l consecutive steps with unchanged directionis then P ( l ) = (1 − p ) p l − , and the persistence length ofthe walk can be defined as l p = (cid:80) ∞ l =1 lP ( l ) = 1 / (1 − p )where p = (1 + (2 d − (cid:15) ) / (2 d ), so that eventually l p = (2 d/ (2 d − / (1 − (cid:15) ). In what follows we calcu-late analytically the search time (cid:104) T (cid:105) , defined here as the MFPT to the target averaged over all possible startingpositions and velocities of the searcher, and analyze itsdependence on the persistence length l p (or equivalently (cid:15) ) and the volume X d .While the position process alone is non Markovian, thejoint process of the position and velocity of the searcher isMarkovian. One can therefore derive an exact backwardequation for the MFPT T ( r , e i ) to the target of position r T , for a random searcher starting from r with initialvelocity e i , where B = { e , . . . , e d } defines a basis of thelattice : T ( r , e i ) = p T ( r + e i , e i ) + p T ( r − e i , − e i )+ p (cid:88) e j ∈B ,j (cid:54) = i (cid:0) T ( r + e j , e j ) + T ( r − e j , − e j ) (cid:1) + 1 . (1)Note that this equation holds for all sites r (cid:54) = r T . Indeed,by definition for r = r T the lhs of of Eq. (1) yields T ( r T , e i ) = 0, while the rhs gives the mean return timeto site r T , which is exactly equal to V in virtue of atheorem due to Kac [24]. We next introduce the Fouriertransform (cid:101) f ( q ) = (cid:80) r ∈L f ( r ) e − ı q . r of a function f ( r ),where q i = 2 πn i /X with n i ∈ [0 , X − r = r T then yields : (cid:101) T ( q , e i ) + V e − ı q . r T = (cid:15) (cid:101) T ( q , e i ) e ı q . e i + V δ ( q ) + p g ( q )(2)where g ( q ) = (cid:88) e j ∈B (cid:16) (cid:101) T ( q , e j ) e ı q . e j + (cid:101) T ( q , − e j ) e − ı q . e j (cid:17) , (3)and δ ( q ) is the d -dimensional Kronecker function. Wethus obtain: (cid:101) T ( q , e i ) = V ( δ ( q ) − e − ı q . r T ) + p g ( q )1 − (cid:15)e ı q . e i . (4)Summing Eq.(4) times e ı q . e i over all e i yields a closedequation for g ( q ), which is solved by: g ( q ) = γ ( q , (cid:15) ) V ( δ ( q ) − e − ı q . r T )1 − p γ ( q , (cid:15) ) (5)where γ ( q , (cid:15) ) = 2 (cid:88) e j ∈B cos( q . e j ) − (cid:15) (cid:15) − (cid:15) cos( q . e j ) . (6)Substituting this expression of g ( q ) in Eq.(4) then leadsto an explicit expression of (cid:101) T ( q , e i ). After Fourier in-version and averaging over all possible starting positionsand velocities we finally obtain after some algebra: (cid:104) T (cid:105) = − (cid:15) ( V − − (cid:15) + 1 + (cid:15) − (cid:15) (cid:88) q (cid:54) = − h ( q , (cid:15) ) (7)where h ( q , (cid:15) ) = ( (cid:15) − d (cid:88) e j ∈B cos( q . e j )1 + (cid:15) − (cid:15) cos( q . e j ) (8)and (cid:80) q (cid:54) = denotes the sum over all possible vectors q defined above except q = . This exact expression of thesearch time for a non Markovian searcher constitutes thecentral result of this paper. We discuss below its physicalimplications, based on two useful approximations.We first consider the case where (cid:15) (cid:28)
1, which impliesthat the persistence length is of the same order as thetarget size ( l p = O (1)). In this regime the search timereads: (cid:104) T (cid:105) = (cid:15) (cid:28) A ( (cid:15), V )( V −
1) + 1 D ( (cid:15) ) (cid:104) T (cid:105) , (9)where (cid:104) T (cid:105) is the search time of a non persistent randomwalk ( (cid:15) = 0) which is known exactly [25]. The quantity A ( (cid:15), V ) writes A ( (cid:15), V ) = ( B d ( V ) − (cid:15) + O ( (cid:15) ) where B d ( V ) depends on V as follows: B d ( V ) = 2 V (cid:88) q (cid:54) = d (cid:88) e j ∈B (1 − cos(2 π q . e j )) d (cid:88) e j ∈B − cos(2 π q . e j ) . (10)In the dilute regime ( V → ∞ ), B d has a finite limit(for example B (cid:39) .
72) and Eq.(9) provides a use-ful approximate of the search time. In this expression, D ( (cid:15) ) = (1 + (cid:15) ) / (1 − (cid:15) ) is the diffusion coefficient of thepersistent random walk normalized by the diffusion coef-ficient of the non persistent walk (case (cid:15) = 0) [20]. Hence,in Eq. (11), (cid:104) T (cid:105) /D is the search time expected for a nonpersistent random searcher of same normalized diffusioncoefficient D . Note that the persistence property yields anon trivial additive correction which scales linearly withthe volume, and therefore should not be neglected; thiscould be related to the ”residual” mean first passage timedescribed in [26]. As shown in Fig. 2, the approximationof Eq. (9) is accurate as long as l p is small (that is (cid:15) (cid:28) l p (cid:29)
1, orequivalently (cid:15) →
1. In this regime the search time readsin the case d = 2 : (cid:104) T (cid:105) = 2( X − − (cid:15) + ( X − − (cid:15) ) ( X − X + 3)( X − O (cid:0) (1 − (cid:15) ) (cid:1) . Fig. 2 shows that this expression provides a good approx-imation of the exact result of Eq. (7) for l p (cid:29)
1. Notethat the search time diverges for l p → ∞ (or (cid:15) →
1) be-cause the searcher can then be trapped in extremely longunsuccessful ballistic excursions. p /X01 < T > / < T > FIG. 2: (Color online) Search time for a 2–dimensional persis-tent random walker (cid:104) T (cid:105) normalized by the search time for anon persistent walker ( (cid:104) T (cid:105) ) as a function of the rescaled per-sistence length, for X = 10 (upper set of curves) and X = 100(lower set of curves). The red line stands for the exact resultof Eq. (7), the dotted lines for the approximation (cid:15) (cid:28) (cid:15) → l p = (2 d/ (2 d − / (1 − (cid:15) ). × × × × V00.20.40.60.81 < T > l p * / < T >
50 100 150X0102030 l p* FIG. 3: (Color online) Optimal search time scaled by the nonpersistent case as the function of the domain volume V for d = 2. The black line stands for the numerical optimizationof Eq. (7), the red line for the analytical optimization ofEq. (11), and the green dashed line for a fit A/ ln( V ), wherewe used the identity l p = (2 d/ (2 d − / (1 − (cid:15) ). Inset: Theblack line stands for the persistence length at the minimum, l ∗ p , obtained by a numerical optimization of Eq. (7), as afunction of X . The dashed red line is a linear fit of this curve( l p (cid:39) . X + 3 . Both asymptotics (cid:15) → (cid:15) → (cid:15) orequivalently l p , as seen in Fig. 2. The minimum can beobtained from the analysis of the exact expression (7),and reveals that the search time is minimized in the case d = 2 for l p = l ∗ p ∼ X →∞ λ X with λ (cid:39) . ... . Note thatthe asymptotic expression (11) yields a good analyticalapproximate of this minimum. This defines the optimalstrategy for a persistent random searcher, which is real-ized when the persistence length has the same order ofmagnitude as the typical system size. In particular, forlarge system sizes the optimal persistence length becomesmuch larger than the target size. We stress however thatthe numerical factor λ is non trivial and notably small.This optimal strategy can be understood as follows. Inthe regime l p (cid:28) X , the random walk behaves as a regulardiffusion and is therefore recurrent for d = 2. The explo-ration of space is therefore redundant and yields a searchtime that scales in this regime as V ln V [27]. On thecontrary for l p (cid:29) l p and therefore less redundant. As soon as l p ∼ X onetherefore expects the search time to scale as V [27]. Tak-ing l p too large however becomes unfavorable since thesearcher can be trapped in extremely long unsuccessfulballistic excursions, so that one indeed expects an opti-mum in the regime l p ∼ X . This argument suggests thefollowing scaling of the optimal search time scaled by thenon persistent case in the case d = 2 : (cid:104) T (cid:105) l ∗ p (cid:104) T (cid:105) ∝ / ln( V ) , (12)which can indeed be derived from the asymptotic expres-sion (11) (see also Fig. 3). This shows the efficiency ofthe optimal persistent search strategy in the large vol-ume limit, as compared to the non persistent Brownianstrategy. Note that for d = 3 a similar analysis applies.In particular the search time is minimized for a value ofthe persistence length that again grows linearly with thesystem size l ∗ p ∼ X →∞ λ X with however a slightly differentnumerical value of the coefficient λ (cid:39) . ... . Addition-ally, since for d = 3 one has (cid:104) T (cid:105) ∝ V , the scaled optimalsearch time then tends to a constant.Last, we compare the efficiency of the persistent andLevy walk strategies. More precisely we consider a Levywalker such that the distribution of the length of its suc-cessive ballistic excursions follows a symmetric Levy lawof index µ and scale parameter c restricted to the positiveaxis, defined by the Fourrier transform (cid:98) P ( k ) = e − c | k | µ ,so that P ( l ) ∝ l →∞ /l µ . For 0 < µ ≤
1, the persistencelength is infinite, yielding in turn an infinite search time.We therefore focus on the regime 1 < µ ≤ µ and l p (which is set by c ). Figure 4 shows that thesearch time can be minimized as a function of l p for all µ ∈ ]1 , µ is increased. In particular the search time for the Levystrategy is minimized when µ = 2, i–e when the lengthof the ballistic excursions has a finite second moment sothat the walk is no longer of Levy type. As seen in Fig.4 the optimal persistent random walk strategy thereforeyields a search time shorter than any Levy walk strategy.This optimal persistent search strategy is in markedcontrast with the simple ballistic motion obtained in the p < T > / V FIG. 4: (Color online) Numerical computation of the searchtime for a Levy walk on a 2D lattice ( X = 50). Plots withcircles stand for the search time for the following values of µ (from top to bottom): µ = 1 .
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