Scaling law in target-hunting processes

a r X i v :c o n d -m a t /0405080v 1 [c o n d -m a t .s t a t -m e c h ] 5 M a y 2004scaling law in target-hunting processes

Shi-Jie Yang 1

1Department of Physics,Beijing Normal University,Beijing 100875,China

We study the hunting process for a target,in which the hunter tracks the goal by smelling odors it emits.The odor intensity is supposed to decrease with the distance it di?uses.The Monte Carlo experiment is carried out on a 2-dimensional square lattice.Having no idea of the location of the target,the hunter determines its moves only by random attempts in each direction.By sorting the searching time in each simulation and introducing a variable x to re?ect the sequence of searching time,we obtain a curve with a wide plateau,indicating a most probable time of successfully ?nding out the target.The simulations reveal a scaling law for the searching time versus the distance to the position of the target.The scaling exponent depends on the sensitivity of the hunter.Our model may be a prototype in studying such the searching processes as various foods-foraging behavior of the wild animals.PACS numbers:02.50.Ng,07.05.Tp,05.10.Gg,89.20.-a In the past years,the di?usion-controlled reactions have been extensively studied through random-walk models.Such applications range from chemical processes,electronic scavenging and recombination,to electronic and vibra-tional energy transfer in condensed media[1,2,3,4,5,6,7,8,9].Many works have been devoted to the target annihilation problem,in which randomly placed targets are annihilated by random walkers,and its dual of the trap-ping problem[10,11].Other models treat hindered di?usion problems which involve random point obstacles [12].In these models,the tracer moves from site to site on a lattice and falls into wells of various depth at the sites.The tracer does not know the depth of a well before it enters.Another possibility is a mountain model,in which all sites are at zero energy and the barriers are on the bonds joining the sites[13,14].Generally,random-walk models are ideally suited for computer simulations,a practical way to obtain results,since for the vast majority of cases no purely analytical method exists.In this work,we focus on another class of random-walk problems.We study the so-called target-hunting processes,which frequently occurs in biological systems,such as a shark searching for foods by smelling the blood in the ocean,or honeybees ?ying in the countryside to locate the foraging-nectars[15,16,17],or in metabolic processes such as cell motions and chemotaxis[18,19,20,21,22].It can be viewed as the target-oriented problems,in which the hunters try to reach the targets by following some kind of behavior rules.In our model,an active hunter is trying to ?nd out a target which emits a special kind of odor.The Monte Carlo simulations are carried out on a 2-dimensional square lattice.Since neither the distance nor the direction of the target is presumedly to be known,the searcher should determine its moves by random attempts in each direction,just like a snake turns its head from side to side to test the variation of the odor intensity.There are some chance for the hunter to move in the wrong direction because of randomness.Hence it is not a traditional biased random-walk.After sorting each searching process in time sequence,we obtain a curve with a wide plateau,indicating a most probable time of successfully ?nding out the target.By ?tting the numerical results,we ?nd a scaling law for the searching time on the distance to the position of the target.The scaling exponent is found to be dependent on the sensitivity of the hunter.We consider this scaling law scarcely happens in ordinary biased random walks.The game rules are as follows:The hunter at the origin O is trying to ?nd out a target which emits a special kind of odor.Since the hunter have no way to know the location of the target,it randomly moves around its original position to test the variation of the odor intensity.z 0is the present distance of the hunter to the target while z 1is the corresponding distance of the next attempted step.The Monte Carlo steps are implemented as:if (z 0/z 1)α>ζ,where the parameter αre?ects the sensitivity of the hunter and ζis a random number,then the attempt is accepted.Otherwise it is refused.This rule implies that the intensity of the signal emitted by the target is inversely proportional to the distance of the hunter to the target.Other choices of the relation do not alter the result qualitatively.By this way,the hunter approaches the goal in a stochastic style.

Fig.1displays a typical route of the hunter searching for the target on a regular lattice.When the hunter is far away from the goal,the ratio z 0/z 1is close to 1.Most of the moving attempts are accepted,even the hunter walks in the wrong direction.The hunter appears to linger around for quite a while.Hence the motion of the hunter is nearly a Brownian random walk.As the goal is nearer,the ratio of z 0/z 1gradually approaches 0.5and the probability of being refused for the hunter moving in the wrong direction increases.Hence the searching route seems more straightforward.Fig.2shows the searching time for each simulation for a distance of z =31.4and α=6.It is understandable that the searching time are di?erent for di?erent stochastic processes.The distribution is not like a white noise.There are large ?uctuations away from the most probable searching time.In Fig.3,we plot the distribution of searching time for z =31.4,65.6,137.1,188.4,respectively.It is seen that the distribution is not of the Poissonian form.The curve

has a very long time tail.Instead,a power relation is found for the maximal value of the distribution with the most probable time,

V m～t?d m p,(1) with d m=1.05forα=6.

There is another power relation between the distance of the target and the most probable searching-time,

t p～z d p,(2) where d p=1.77forα=6.

In Fig.4we redistribute the data in Fig.2by sorting with increasing time.Fig.4(a)is for various distances z from the origin,with z=31.4,65.6,137.1,188.4from bottom to top andα=6.The horizonal axis is the sequence of searching time represented in percentage.A wide plateau is formed in the intermediate range.Fig.4(b)shows that after proper displacement,all of the curves collapse into one,implying these curves be parallel to each other.Hence each curve can be described by a single function f(x,α)plus a z-dependent functionφ(z,α),

ln t(x,z,α)=f(x,α)+φ(z,α)(3) with x the sequence of searching time represented in percentage.The function f(x)is of Arabic Ogive-like.It can be checked that

ln t(√2(ln t(z1,x)+ln t(z2,x).(4) Formula(3)can be written as

ln t(x,z,α)=f(x,α)+η(α)ln z.(5) In?gure5we studied the dependence of the searching-time with respect to the sensitivity parameterα.Fig.5(a) shows the curves forα=4,8,12,16,20.After properly rescaling the curves in(a)by times ln t with a coe?cientαβ, whereβis determined below,all of the curves become parallel.From Fig.5(b),we deduceαβln t=?f(x)+?φ(z,α). By comparing with eq.(5),one gets

ln t(x,z,α)=α?βf(x)+η(α)ln z.(6) The indexβcan be derived by considering the dependence of the slope k2of the plateau in?gure5(a)on parameter α.There is a good linear relation between ln k2and lnα,as show in the inset of?gure6,

ln k2～?βlnα.(7) We measuredβ=0.623.It is noteworthy thatβis a constant independent of sensitivity parameterα.It results from the stochastic process.

Finally,we try to found out the relation betweenηandα.Fig.7(a)depicts the relation of ln t versus ln z at x=0.6 for various parameterα.From?gure7(b),

lnη(α)～?δα(8) withδ=0.01.

Combining all the above factors,we consequently obtain a complete relation of the searching-time with respect to the distance as well as sensitivity parameterα,

ln t(x,z,α)=α?βf(x)+c0e?δαln z,(9) or

t(x,z,α)=e f(x)/αβ·z d.(10) We?nd that there is a generalized scaling-law between t and z with exponent

d=c0e?δα.(11)

From formula(2),α=6,d=d p=1.77.We get c0=1.88.It shows that the power-law exponent isα-dependent.As αincreases from zero to in?nity,the exponent decreases from1.88to zero.In Eq.(10),the contributions of variable x,which sorts the searching time in each simulation,are completely merged into a prefactor and the scaling exponent is x-independent.It should be noted that it is a functional relation between the searching time and the distance in such stochastic processes.

In summery,we introduced a variable x to denote the sequence of searching time.We plot a curve with a wide plateau,indicating a most probable time of successfully?nding out the goal.In stead of calculating the mean square root,we introduce a sort parameter x to?gure out an analytical expression.The simulations reveal a scaling law for the searching time versus the distance to the position of the target.The scaling exponent is dependent on the sensitivity of the hunter.We believe that our treatment of the statistical data may be useful in other cases.The existence of the scaling law may have implications with the possibility for the hunter to walk in a wrong direction or stay at the same place for quite a while.It scarcely happens in an ordinary biased random walk.We point out that the results are valid not only on the square lattice,but also for continuous moving(with?xed step length)in the two-dimensional plane.However,the explicit form of function f(x)is still lack.It is also desirable to deduce an analytical express of Eq.(10)from the?rst principle of statistics.

We suggest that the scaling law in the hunting process may be an additional behavior rule in the foods-foraging processes of wild animals,which has not caught much attention.In turn,veri?cations of the law from direct obser-vations by zoologists or entomologists are also expected.Our target-oriented model may be a prototype in studying the foods-foraging processes in wildlife as well as in other searching games.

[1]G.H.Weiss and R.J.Rubin,Adv.Chem.Phys.52,363(1982).

[2]E.M.Montroll and M.F.Shlesinger,in The Mathematics and Physics of of Disordered Media,edited by B.D.Hughes and

R.M.Hochstrasser(North-Holland,Amsterdam,1983),p.139.

[3]J.Klafter and A.Blumen,in Random-Walks and their Applications in the Physical and Biological Sciences,edited by M.F.

Shlesinger and B.J.West(AIP,New York,1984),p.173.

[4]A.Blumen,G.Zumofen,and J.Klafter,Phys.Rev.B30,5379(1984).

[5]H.E.Stanley,K,Kang,S.Redner,and D.L.Blumberg,Phys.Rev.Lett.51,1223(1983).

[6]A.Szabo,R.Zwanzig,and N.Agmon,Phys.Rev.Lett.61,2496(1988).

[7]Z.Koza and H.Taitelbaum,Phys.Rev.E57,237(1998).

[8]A.M.Berezhkovskii,D.J.Bicout,and G.H.Weiss,J.Chem.Phys.110,1112(2000).

[9]N.Agmon and I.V.Gopich,J.Chem.Phys.112,2863(2000).

[10]G.H.Weiss,Aspects and Applications of the Random Walk(North-Holland,Amsterdam,1994).

[11]F.Jasch and A.Blumen,Phys.Rev.E63,041108(2001).

[12]M.J.Saxton,Single-particle tracking:e?ect ofcorrals.Biophys.J.60,389(1995);ibid,70,1250(1996).

[13]A.Bunde,Solid State Ionics,Di?usion&Reactions(1988).

[14]M.J.Saxton,J.Biophys.72,1744(1997).

[15]T.S.Collett,https://www.wendangku.net/doc/f017314866.html,p.Physio.163,355(1988).

[16]P.Kareiva and G.Odell,American Naturalist,130,233(1987).

[17]L.M.Marsh and R.E.Jones,J.Theor.Bio.133,113(1988).

[18]W.Alt,J.Math.Biophys.15,311(1980).

[19]E.F.Keller and L.A.Segel,J.Theor.Bio.30,225(1971).

[20]G.Maheshwari and https://www.wendangku.net/doc/f017314866.html,u?enburger,Microscopy Research and Technique,43,358(1998).

[21]A.D.Shenderov and M.P.Sheetz,Biophys.J.72,2382(1997).

[22]J.A.Sherratt,E.H.Sage,and J.D.Murray,J.Theor.Bio.162,23(1993).

Figure Captions

Figure1A typical hunting route on a square lattice.The start point is at the origin and the target is at(25,19). Figure2Searching time for10000simulations.The original distance to the target is z=31.4and the sensitivity parameterα=6.

Figure3Searching time distribution for distances z=31.4,65.6,137.1,188.4,respectively.α=6.

Figure4Time-sorted curves for10000simulations.The horizontal axis x is the sequence of searching time repre-sented in percentage and the vertical axis is logarithmic time.α=6.(a)is for various distances z from the origin, with z=31.4,65.6,137.1,188.4from bottom to top.Evidently,the curves consist of three parts,and a wide linear region is formed.These curves is parallel to each other.(b)shows that after proper displacement,all curves collapse into one.

Figure5Time-sorted curves for10000simulations for various parameter valuesα.The horizontal axis x is the sequence of searching time represented in percentage and the vertical axis is logarithmic time.The original distance is?xed at z=31.4.(a)is forα=4,8,12,16,20from top to bottom.(b)shows the rescaled curves of(a)for

α=4,8,12,16,20from bottom to top.These curves are parallel to each other.

Figure6The dependence of the slope of the plateau in?gure4on parameterα.The distance z=31.4.Inset:a linear relation of the logarithmic slope k2with logarithmicα.

Figure7(a)A plot of ln t versus ln z at x=0.6for various parameterα.(b)A linear relation of lnη(α)with respect toα.

Y

-50510152025

18000

15000

12000

9000

t

6000

3000

0200040006000800010000

Simulations

05000100001500020000

d i s t r i b u t i o n t

0204060801004

5

6

7

8

9

10

11

1213

020*********

4

5

6

7891011

(a)

l n t (b)t r a n s l a t e d l n t

0204060801004

5

6

7

8

9

1011

020*********

10

15

20

25

30

35

40

45

(a)

l n t (b)

a

l n t

246810121416182022

6.0x10

8.0x10

1.0x101.2x10

1.4x10

1.6x101.8x10

2.0x10

k 2

3.3 3.6 3.9

4.2 4.5 4.8

5.1 5.4

246810121416182022

l n t ln z

l n h (a )