PHYSICAL REVIEW E 80, 031146 共2009兲
Robustness of optimal intermittent search strategies in one, two, and three dimensions
C. Loverdo, O. Bénichou, M. Moreau, and R. Voituriez
Laboratoire de Physique Théorique de la Matière Condensée, UMR CNRS 7600, Université Pierre et Marie Curie,
4 Place Jussieu, 75252 Paris, France
共Received 2 July 2009; published 30 September 2009兲
Search problems at various scales involve a searcher, be it a molecule before reaction or a foraging animal,
which performs an intermittent motion. Here we analyze a generic model based on such type of intermittent
motion, in which the searcher alternates phases of slow motion allowing detection and phases of fast motion
without detection. We present full and systematic results for different modeling hypotheses of the detection
mechanism in space in one, two, and three dimensions. Our study completes and extends the results of our
recent letter 关Loverdo et al., Nat. Phys. 4, 134 共2008兲兴 and gives the necessary calculation details. In addition,
another modeling of the detection case is presented. We show that the mean target detection time can be
minimized as a function of the mean duration of each phase in one, two, and three dimensions. Importantly,
this optimal strategy does not depend on the details of the modeling of the slow detection phase, which shows
the robustness of our results. We believe that this systematic analysis can be used as a basis to study quantitatively various real search problems involving intermittent behaviors.
DOI: 10.1103/PhysRevE.80.031146
PACS number共s兲: 05.40.⫺a, 87.23.⫺n
I. INTRODUCTION
Search problems, involving a “searcher” and a “target,”
pop up in a wide range of domains. They have been subject
of intense work of modeling in situations as various as castaway rescue 关1兴, foraging animals 关2–9兴, or proteins reacting
on a specific DNA sequence 关10–19兴. Among the wide panel
of search strategies, the so-called intermittent strategies—
combining “slow” phases enabling detection of the targets
and “rapid” phases during which the searcher is unable to
detect the targets—have been proved to be relevant at various scales.
Indeed, at the macroscopic scale, numerous animal species have been reported to perform such kind of intermittent
motion 关20,21兴 while searching either for food, shelter, or
mate. In the case of an exploratory behavior, i.e., when the
searcher has no previous knowledge or “mental map” of the
location of targets, trajectories can be considered as random.
Actually, the observed search trajectories are often described
as a sequence of ballistic segments interrupted by much
slower phases. These slow, and sometimes even immobile
关21兴, phases are not always well characterized, but it seems
clear that they are aimed at sensing the environment and
trying to detect the targets 关22兴. On the other hand, during
the fast moving phases, perception is generally degraded so
that the detection is very unlikely. An example of such intermittent behavior is given by the C.elegans worm, which alternates between a fast and almost straight displacement
共“roaming”兲 and a much more sinuous and slower trajectory
共“dwelling”兲 关23兴. During this last phase, the worm’s head,
bearing most of its sensory organs, moves and touches the
surface nearby.
Intermittent strategies are actually also relevant at the microscopic scale, as exemplified by reaction kinetics in biological cells 关10,24兴. As the cellular environment is intrinsically out of equilibrium, the transport of a given tracer
particle which has to react with a target molecule cannot be
described as mere thermal diffusion: if the tracer particle can
1539-3755/2009/80共3兲/031146共29兲
indeed diffuse freely in the medium, it also intermittently
binds and unbinds to motor proteins, which perform an active ballistic motion powered by adenosine triphosphate
共ATP兲 hydrolysis along cytoskeletal filaments 关24–26兴. Such
intermittent trajectories of reactive particles are observed, for
example, in the case of vesicles before they react with their
target membrane proteins 关26兴. In that case, targets are not
accessible during the ballistic phases when the vesicle is
bound to motors, but only during the free diffusive phase.
As illustrated in the previous examples the search time is
often a limiting quantity whose optimization can be very
beneficial for the system—be it an animal or a single cell. In
the case of intermittent search strategies, the minimization of
the search time can be qualitatively discussed: on the one
hand, the fast but nonreactive phases can appear as a waste
of time since they do not give any chance of target detection.
On the other hand, such fast phases can provide an efficient
way to relocate and explore space. This puts forward the
following questions: is it beneficial for the search to perform
such fast but nonreactive phases? Is it possible by properly
tuning the kinetic parameters of trajectories 共such as the durations of each of the two phases兲 to minimize the search
time? These questions have been addressed quantitatively on
specific examples in 关6–9,27–30兴, where it was shown that
intermittent search strategies can be optimized. In this paper,
we perform a systematic analytical study of intermittent
search strategies in one, two, and three dimensions and fully
characterize the optimal regimes. This study completes our
previous works, and in particular our recent letter 关24兴, by
providing all calculation details and specifying the validity
domains of our approach. It also presents another case relevant to model real search problems. Overall, this systematic
approach allows us to identify robust features of intermittent
search strategies. In particular, the slow phase that enables
detection is often hard to characterize experimentally. Here
we propose and study three distinct modelings for this phase,
which allows us to assess to which extent our results are
robust and model dependent. Our analysis covers in detail
intermittent search problems in one, two, and three dimen-
031146-1
©2009 The American Physical Society
PHYSICAL REVIEW E 80, 031146 共2009兲
LOVERDO et al.
sions and is aimed at giving a quantitative basis—as complete as possible—to model real search problems involving
intermittent searchers.
We first define our model and give general notations that
we will use in this paper. Then we systematically examine
each case, studying the search problem in one, two, and three
dimensions, where for each dimension different types of motion in the slow phase are considered. Each case is ended by
a short summary, and we highlight the main results for each
dimension. Eventually we synthesize the results in Table II
where all cases, their differences, and similarities are gathered. This table finally leads us to draw general conclusions.
II. MODEL AND NOTATIONS
A. Model
The general framework of the model relies on intermittent
trajectories, which have been put forward, for example, in
关6兴. We consider a searcher that switches between two
phases. The switching rate ␭1 共␭2兲 from phase 1 to phase 2
共from phase 2 to phase 1兲 is time independent, which assumes that the searcher has no memory and implies an exponential distribution of durations of each phase i of mean
␶ i = 1 / ␭ i.
Phase 1 denotes the phase of slow motion during which
the target can be detected if it lies within a distance from the
searcher which is smaller than a given detection radius a. a is
the maximum distance within which the searcher can get
information about target location. We propose three different
modelings of this phase in order to cover various real life
situations.
共i兲 In the first modeling of phase 1, hereafter referred to as
the “static mode,” the searcher is immobile and detects the
target with probability per unit time k if it lies at a distance
less than a.
共ii兲 In the second modeling, called the “diffusive mode,”
the searcher performs a continuous diffusive motion, with
diffusion coefficient D, and finds immediately the target if it
lies at a distance less than a.
共iii兲 In the last modeling, called the “ballistic mode,” the
searcher moves ballistically in a random direction with constant speed vl and reacts immediately with the target if it lies
at a distance less than a.
Some comments on these different modelings of the slow
phase 1 are to be made. The first two modes have already
been introduced 关8,24兴, while the analysis of the ballistic
mode has never been performed. These three modes schematically cover experimental observations of the behavior of
animals searching for food 关20,21兴, where the slow phases of
detection are often described as either static, random, or with
slow velocity. Several real situations are likely to involve a
combination of two modes. For instance the motion of a
reactive particle in a cell not bound to motors can be described by a combination of the diffusive and static modes.
For the sake of simplicity, here we treat these modes independently, and our approach can therefore be considered as a
limit of more realistic models. We note that a similar version
of the ballistic mode of detection has been discussed by
Viswanathan et al. 关2,31兴. They introduced a model searcher
performing randomly oriented ballistic movements 共and of
power law distributed duration兲, with detection capability all
along the trajectory. For this one state searcher, the search
time for a target 共which is assumed to disappear after the first
encounter兲 is minimized when the searcher performs a purely
ballistic motion and never reorientates. In fact, the ballistic
mode version of our model extends this model of one state
ballistic searcher, by allowing the searcher to switch to a
mode of faster motion, but with no perception. As it will be
discussed, adding this possibility of intermittence enables a
further minimization of the search time. Finally, combining
these three schematic modes covers a wide range of possible
motions from subdiffusive 共even static兲, diffusive, to superdiffusive 共even ballistic兲.
Phase 2 denotes the fast phase during which the target
cannot be found. In this phase, the searcher performs a ballistic motion at constant speed V and random direction, redrawn each time the searcher enters phase 2, independently
of previous phases. In real examples correlations between
successive phases could exist. If correlations are very high, it
is close to a one-dimensional problem with all phases 2 in
the same direction, a different problem already treated in 关6兴.
We consider here the limit of low correlation, which is of a
searcher with no memory skills.
We assume that the searcher evolves in d-dimensional
spherical domain of radius b, with reflective boundaries and
with one centered immobile target 共bounds can be obtained
in the case of mobile targets 关32,33兴兲. As the searcher does
not initially know the target location, we start the walk from
a random point of the d-dimensional sphere and average the
mean target detection time over the initial position. This geometry models the case of a single target in a finite domain
and also provides a good approximation of an infinite space
with regularly spaced targets. Such regular array of targets
corresponds to a mean-field approximation of random distributions of targets, which can be more realistic in some experimental situations. We note that in the one-dimensional
case, we have shown that a Poisson distribution of targets
can lead to significantly different results from the regular
distribution 关34,35兴. We expect this difference to be less in
dimensions 2 and 3, and we limit ourselves in this paper to
the mean-field treatment for the sake of simplicity.
B. Notations
ti共rជ兲 denotes the mean first-passage time 共MFPT兲 on the
target for a searcher starting in the phase i from point rជ,
where phase i = 1 is the slow motion phase with detection and
phase i = 2 is the fast motion phase without target detection.
Note that in dimension 1, the space coordinate will be denoted by x, and in the case of a ballistic mode for phase 1,
the upper index in t⫾
i stands for ballistic motion with direction ⫾x. The general method which will be used at length in
this paper consists of deriving and solving backward equations for ti共rជ兲 关36兴. These linear equations involve derivatives
with respect to the starting position rជ.
Assuming that the searcher starts in phase 1, the mean
detection time for a target is then defined as
031146-2
ROBUSTNESS OF OPTIMAL INTERMITTENT SEARCH …
PHYSICAL REVIEW E 80, 031146 共2009兲
k
tion radius a. It is the limit of a very slow searcher in the
reactive phase.
phase 1
1. Equations
Outside the target 共for x ⬎ a兲, we have the following backward equations for the mean first-passage time:
V
V
t
phase 2
x
−V
2a
1
V共⍀d兲
冕
⍀d
t1共rជ兲drជ ,
dt−2 1
+ 共t1 − t−2 兲 = − 1,
dx ␶2
冉
FIG. 1. 共Color online兲 Static mode in one dimension.
tm =
dt+2 1
+ 共t1 − t+2 兲 = − 1,
dx ␶2
冊
1 t+2 + t−2
− t1 = − 1.
2
␶1
共1兲
with ⍀d as the d-dimensional sphere of radius b and V共⍀d兲
its volume. Unless specified, we will consider the low target
density limit a Ⰶ b.
Our general aims are to minimize tm as a function of the
mean durations ␶1 , ␶2 of each phase and in particular to determine under which conditions an intermittent strategy 共with
finite ␶2兲 is faster than a usual one state search in phase 1
only, which is given by the limit ␶1 → ⬁. In the static mode,
intermittence is necessary for the searcher to move and is
therefore always favorable. In the diffusive mode, we will
compare the mean search time with intermittence tm to the
mean search time for a one state diffusive searcher tdif f and
define the gain as gain= tdif f / tm. Similarly in the ballistic
mode, we will compare tm to the mean search time for a one
state ballistic searcher tbal and define the gain as gain
= tbal / tm.
Throughout the paper, the upper index “opt” is used to
denote the value of a parameter or variable at the minimum
of tm.
III. DIMENSION 1
Besides the fact that it involves more tractable calculations, the one-dimensional case can also be interesting to
model real search problems. At the microscopic scale, tubular structures of cells such as axons or dendrites in neurons
can be considered as one dimension 关25兴. The active transport of reactive particles, which alternate diffusion phases
and ballistic phases when bound to molecular motors, can be
schematically captured by our model with diffusive mode
关24兴. At the macroscopic scale, one could cite animals such
as ants 关37兴 which tend to follow tracks or one-dimensional
boundaries.
A. Static mode
In this section we assume that the detection phase is modeled by the static mode 共Fig. 1兲. Hence the searcher does not
move during the reactive phase 1 and has a fixed reaction
rate k per unit time with the target if it lies within its detec-
共2兲
共3兲
共4兲
Inside the target 共x ⱕ a兲, the first two equations are identical,
but the third one is written as
冉 冊
1
1 t+2 + t−2
−
+ k t1 = − 1.
␶1 2
␶1
共5兲
We introduce t2 = 共t+2 + t−2 兲 / 2 and td2 = 共t+2 − t−2 兲 / 2. Then outside
the target we have the following equations:
V
V 2␶ 2
dt2 1 d
− t = 0,
dx ␶2 2
共6兲
d 2t 2 1
+ 共t1 − t2兲 = 0,
dx2 ␶2
共7兲
1
共t − t 兲 = − 1.
␶1 2 1
共8兲
Inside the target the first two equations are identical, but the
last one writes
冉 冊
1
1
t −
+ k t1 = − 1.
␶1 2
␶1
共9兲
Due to the symmetry x ↔ −x, we can restrict the study to the
part x 苸 关0 , a兴 and the part x 苸 关a , b兴. This symmetry also
implies
冏 冏
冏 冏
dtin
2
dx
dtout
2
dx
= 0,
共10兲
= 0.
共11兲
x=0
x=b
In addition, continuity at x = a for t+2 and t−2 gives
out
tin
2 共x = a兲 = t2 共x = a兲,
共12兲
d,out
td,in
共x = a兲.
2 共x = a兲 = t2
共13兲
This set of linear equations enables an explicit determination
of t1, t2, and td2 inside and outside the target.
2. Results
An exact analytical expression of the mean first-passage
time at the target is then given by
031146-3
PHYSICAL REVIEW E 80, 031146 共2009兲
LOVERDO et al.
(a)
(b)
(c)
(d)
FIG. 2. 共Color online兲 Static mode in one dimension. Exact expression of tm 关Eq. 共14兲兴 共lines兲 compared to the approximation of tm 关Eq.
opt
opt
共17兲兴 共symbols兲, both rescaled by tm
关Eq. 共20兲兴. ␶opt
1 from Eq. 共18兲; ␶2 from Eq. 共19兲. V = 1, k = 1. b / a = 10 共green, squares兲, b / a = 100 共red,
circles兲, and b / a = 1000 共blue, crosses兲.
tm =
冋
共b − a兲3 ␤共b − a兲2
␶1 + ␶2 b
+
+
b
k␶1
V␶2
3V2␶22
冉 冊
⫻coth
册
a
+ 共b − a兲␤2 ,
V ␶ 2␤
t m = 共 ␶ 1 + ␶ 2兲
共14兲
where ␤ = 冑共k␶1兲−1 + 1.
In order to determine the optimal strategy, we need to
simplify this expression, by expanding Eq. 共14兲 in the regime
b Ⰷ a,
t m = 共 ␶ 1 + ␶ 2兲
冋
冋
冉 冊 册
␶opt
1 =
冑 冉 冊
a
b
Vk 12a
共15兲
We make the further assumption a / V␶2 Ⰶ 1 and obtain using
␤⬎1
冉
冊
1
b
b2
+ 2 2 + ␤2 + ␤2 .
k␶1 3V ␶2
a
共16兲
Since ␤ ⬎ 1 and ␤ ⬎ 1 / 共k␶1兲, we obtain in the limit b Ⰷ a
共17兲
This simple expression gives a very good and convenient
approximation of the mean first-passage time at the target, as
shown in Fig. 2.
We use this approximation 关Eq. 共17兲兴 to find ␶1 and ␶2
values which minimize tm,
1
a
␤b
b2
+ 2 2+
coth
+ ␤2 .
k␶1 3V ␶2 V␶2
V ␶ 2␤
t m = 共 ␶ 1 + ␶ 2兲
冊册
冉
1
b2
b
+
+1
.
2 2
k␶1
a
3V ␶2
␶opt
2 =
a
V
冑
b
.
3a
1/4
,
共18兲
共19兲
It can be noticed than ␶opt
2 does not depend on k. Then the
expression of the minimal value of the search time tm 关Eq.
opt
共17兲兴 with ␶1 = ␶opt
1 and ␶2 = ␶2 is
031146-4
ROBUSTNESS OF OPTIMAL INTERMITTENT SEARCH …
PHYSICAL REVIEW E 80, 031146 共2009兲
共27兲
t1 = 0.
V
We introduce the variables t2 = 共t+2 + t−2 兲 / 2 and td2 = 共t+2 − t−2 兲 / 2.
This leads to the following system outside the target 共x ⬎ a兲,
phase 1
V
D
t
V 2␶ 2
phase 2
x
2a
D
FIG. 3. 共Color online兲 Diffusive mode in one dimension.
opt
tm
=
冉 冊 冋冑 冉 冊 册冋冑 冉 冊 册
b b
ak 3a
1/4
2bk 3a
3V b
1/4
2ka
3a
+
V
b
+1
dt2 1 d
= t ,
dx ␶2 2
共28兲
dt2 1
+ 共t1 − t2兲 = − 1,
dx ␶2
共29兲
d 2t 1 1
+ 共t2 − t1兲 = − 1,
dx2 ␶1
共30兲
and inside the target 共x ⱕ a兲,
1/4
.
V
dt2,in 1 d
= t2,in ,
dx
␶2
共31兲
共20兲
V 2␶ 2
3. Summary
For the static modeling of the detection phase in dimension 1, in the b Ⰷ a limit, the mean detection time is
t m = 共 ␶ 1 + ␶ 2兲
冋
冊册
冉
1
b2
b
+
+1
.
2 2
k␶1
a
3V ␶2
共21兲
Intermittence is always favorable, and the optimal strategy is
a冑 b
冑 a b 1/4 and ␶opt
realized when ␶opt
1 = Vk 共 12a 兲
2 = V 3a . Importantly,
the optimal duration of the relocation phase does not depend
on k, i.e., on the description of the detection phase.
dt2,in 1
− t2,in = − 1,
dx
␶2
共32兲
共33兲
t1 = 0.
Interestingly, this system is exactly of the same type that
what would be obtained with two diffusive phases, with
f
2
Def
2 = V ␶2 in phase 2. Boundary conditions result from continuity and symmetry,
t1共a兲 = 0,
共34兲
+
共a兲,
t+2 共a兲 = t2,in
共35兲
−
t−2 共a兲 = t2,in
共a兲,
共36兲
B. Diffusive mode
We now turn to the diffusive modeling of the detection
phase 共Fig. 3兲. The detection phase 1 is now diffusive, with
immediate detection of the target if it is within a radius a
from the searcher.
冏 冏
冏 冏
冏 冏
1. Equations
Along the same lines, the backward equations for the
mean first-passage time read outside the target 共x ⬎ a兲,
V
dt+2 1
+ 共t1 − t+2 兲 = − 1,
dx ␶2
−V
D
dt−2 1
+ 共t1 − t−2 兲 = − 1,
dx ␶2
冉
冊
d2t1 1 t+2 t−2
+
+ − t1 = − 1,
dx2 ␶1 2 2
V
dt+2 1 +
− t = − 1,
dx ␶2 2
−V
dt−2
dx
−
1 −
t = − 1,
␶2 2
共25兲
dt1
dx
x=b
= 0,
共37兲
= 0,
共38兲
共39兲
= 0.
x=0
2. Results
共23兲
and inside the target 共x ⱕ a兲,
x=b
dt2,in
dx
共22兲
共24兲
dt2
dx
Standard but lengthy calculations lead to an exact expression of mean first detection time of the target tm given in
Appendix A 1. We first studied numerically the minimum of
tm in Appendix A 2 and identified three regimes. In the first
regime 共b ⬍ DV 兲 intermittence is not favorable. For b ⬎ DV intermittence is favorable and two regimes 共bD2 / a3V2 ⬍ 1 and
bD2 / a3V2 ⬎ 1兲 should be distinguished. We now study analytically each of these regimes.
D
3. Regime where intermittence is not favorable: b ⬍ V
共26兲
If b ⬍ DV , the time spent to explore the search space is
smaller in the diffusive phase than in the ballistic phase.
031146-5
PHYSICAL REVIEW E 80, 031146 共2009兲
LOVERDO et al.
(a)
(b)
FIG. 4. 共Color online兲 Diffusive mode in one dimension. tm / tdif f , tdif f from Eq. 共42兲 and tm exact expression 关Eq. 共A1兲兴 共line兲 and
approximation in the regime of favorable intermittence and bD2 / a3V2 Ⰷ 1 关Eq. 共43兲兴 共symbols兲. a = 1 and b = 100 共green, circles兲, a = 1, b
opt
= 104 共red, crosses兲, and a = 10, b = 105 共blue, squares兲. D = 1, V = 1. ␶opt
1 is from expression 共44兲; ␶2 is from expression 共45兲.
Intermittence cannot be favorable in this regime, as confirmed by the numerical study in Appendix A 2.
Without intermittence, the searcher only performs diffusive motion and the problem can be solved straightforwardly.
The backward equations read tdif f = 0 inside the target 共x
ⱕ a兲 and outside the target 共x ⬎ a兲,
D
d2tdif f
= − 1.
dx2
共40兲
3
共b − a兲3
,
3Db
共41兲
b2
.
3D
共42兲
As explained in detail in Appendix A 3, we use the approximation of low target density 共b Ⰷ a兲 and we use asopt
sumptions on the dependence of ␶opt
1 and ␶2 on b and a.
These assumptions lead to the following approximation of
the mean first-passage time:
冉
gainopt =
冊
1
b
+
.
2 2
冑
3V ␶2
D␶1
冑
1 3 2b2D
=
,
2
9V4
35 b4
.
24 DV2
共46兲
tdif f
opt
tm
冑冉 冊
3
⯝
24 bV
38 D
冉 冊
2/3
⯝ 0.13
bV
D
2/3
.
共47兲
These results are in agreement with numerical minimization
of the exact tm 共Table III in Appendix A 2兲.
We start from the exact expression of tm 关Eq. 共A1兲兴. As
detailed in Appendix A 4, we make assumptions on the deopt
pendence of ␶opt
1 and ␶2 with b and a and use the assumptions that b Ⰷ a and 1 Ⰷ bD2 / a3V2. It leads to
冉
冊
a
b
ab
+ 2 2 .
t m ⯝ 共 ␶ 1 + ␶ 2兲
冑
a
a + D␶1 3V ␶2
共48兲
This expression gives a good approximation of tm, at least
around the optimum 共Fig. 5兲, which is characterized by
共43兲
We checked numerically that this expression gives a good
approximation of tm in this regime, in particular around the
optimum 共Fig. 4兲.
The simplified tm expression 关Eq. 共43兲兴 is minimized for
␶opt
1
共45兲
5. Optimization in the second regime where intermittence is
D
favorable: b ⬍ V and 1 š bD2 Õ a3V2
4. Optimization in the first regime where intermittence is
D
favorable: b ⬍ V and bD2 Õ a3V2 š 1
tm = 共␶1 + ␶2兲b
2b2D
,
9V4
This compares to the case without intermittence 关Eq. 共41兲兴
according to
which in the limit b Ⰷ a leads to
tdif f ⯝
冑
3
opt
⯝
tm
Since tdif f 共x = a兲 = 0 and dtdif f / dx 兩x=b = 0, we get tdif f 共x兲
1
= 2D
关共b − a兲2 − 共b − x兲2兴. The mean first-passage time at the
target then reads
tdif f =
冑
␶opt
2 =
共44兲
031146-6
␶opt
1 =
Db
,
48V2a
␶opt
2 =
a
V
opt
⯝
tm
冑
b
,
3a
冑 冉 冊
2a
共49兲
b
a
V 3
共50兲
3/2
,
共51兲
ROBUSTNESS OF OPTIMAL INTERMITTENT SEARCH …
PHYSICAL REVIEW E 80, 031146 共2009兲
(a)
(b)
FIG. 5. 共Color online兲 Diffusive mode in one dimension. tm / tdif f , tdif f from Eq. 共42兲 and tm exact expression 关Eq. 共A1兲兴 共line兲 and
approximation in the regime of favorable intermittence and bD2 / a3V2 Ⰶ 1 关Eq. 共48兲兴 共symbols兲. a = 10 and b = 100 共green, circles兲, a = 10,
opt
b = 1000 共red, crosses兲, and a = 100, b = 104 共blue, squares兲. D = 1, V = 1. ␶opt
1 is from expression 共49兲; ␶2 is from expression 共50兲.
gain ⯝
1 aV
2 冑3 D
冑
b
.
a
1. Equations
共52兲
The backward equations read outside the target 共x ⬎ a兲,
vl
These results are in very good agreement with numerical
data 共Table III in Appendix A 2兲. Note that the gain can be
very large at low target density.
− vl
6. Summary
We calculated explicitly the mean first-passage time tm in
the case where the detection phase is modeled by the diffusive mode. We minimized tm as a function of ␶1 and ␶2, the
mean phase durations, with the assumption a Ⰶ b. There are
three regimes:
共i兲 when b ⬍ DV , intermittence is not favorable, thus
opt
opt
2
␶1 → ⬁, ␶opt
2 → 0, and tm = tdif f ⯝ b / 3D;
D
2 3 2
共ii兲 when b ⬎ V and bD / a V Ⰷ 1, intermittence is
opt 冑
3
2
4
␶opt
and
favorable,
with
2 = 2␶1 = 2b D / 9V
opt 冑
3 5 4
4
2
tm ⯝ 共3 / 2 兲共b / DV 兲; and
共iii兲 when b ⬎ DV and bD2 / a3V2 Ⰶ 1, intermittence is favor2a b 3/2
opt
opt a 冑 b
2
able, with ␶opt
1 = Db / 48V a, ␶2 = V 3a , and tm ⯝ v冑3 共 a 兲 .
This last regime is of particular interest since the value
obtained for ␶opt
2 is the same as in the static mode 共cf. Sec.
III A 3兲.
V
冉
冊
冉
冊
冉
冊
冉
冊
dt+1 1 t+2 t−2 +
+
+ − t = − 1,
dx ␶1 2 2 1
dt−1 1 t+2 t−2 −
+
+ − t = − 1,
dx ␶1 2 2 1
dt+2 1 t+1 t−1 +
+
+ − t = − 1,
dx ␶2 2 2 2
−V
dt−2 1 t+1 t−1 −
+
+ − t = − 1.
dx ␶2 2 2 2
共53兲
共54兲
共55兲
共56兲
Defining tdi = 共t+i − t−i 兲 / 2 and ti = 共t+i + t−i 兲 / 2, we get the following equations 共and similar expressions with vl → V, t1 → t2,
t2 → t1兲:
vl
dtd1 1
+ 共t2 − t1兲 = − 1,
dx ␶1
共57兲
phase 2
phase 1
C. Ballistic mode
We now treat the case where the detection phase 1 is
modeled by the ballistic mode 共Fig. 6兲. This model schematically accounts for the general observation that speed often
degrades perception abilities. Our model corresponds to the
extreme case where only two modes are available: either the
motion is slow and the target can be found or the motion is
fast and the target cannot be found. Note that this model can
be compared with 关2兴.
031146-7
V
t
vl
x
2a
FIG. 6. 共Color online兲 Ballistic mode in one dimension.
PHYSICAL REVIEW E 80, 031146 共2009兲
LOVERDO et al.
vl
dt1 1 d
− t = 0,
dx ␶1 1
共58兲
⌫
=
which eventually lead to the following system:
v2l ␶1
V 2␶ 2
d 2t 1 1
+ 共t2 − t1兲 = − 1,
dx2 ␶1
共59兲
d 2t 2 1
+ 共t1 − t2兲 = − 1,
dx2 ␶2
共60兲
共冑␣ − L1兲共L1 + L2兲 + 冑␣共L2 − L1 + 冑␣兲X + X2L2共L2 − L1兲
关共L1 + 冑␣兲X2 + 共L1 − 冑␣兲兴共冑␣ + L2 − L1兲
共72兲
X = e2a/L2 ,
共73兲
␣ = L21 + L22 ,
共74兲
L2 = V␶2 ,
共75兲
L 1 = v l␶ 1 .
共76兲
together with
td1 = vl␶1
dt1
,
dx
共61兲
td2 = V␶2
dt2
.
dx
共62兲
−,in
Inside the target 共x ⱕ a兲, one has t+,in
1 共x兲 = t1 共x兲 = 0 and
V
dt+,in
1
2
− t+,in
= − 1,
dx
␶2 2
−V
共64兲
Finally, the boundary conditions read
冏 冏
冏 冏
dt2
dx
dt1
dx
3. Regime without intermittence: ␶1 \ ⴥ
In this regime, there is no intermittence. The searcher
starts either inside the target 共x in 关−a , a兴兲 and immediately
finds the target or it starts at a position x outside the target.
We can therefore take x 苸 关a , b兴. If the searcher goes in the
−x direction, it finds its target after T = 共x − a兲 / vl. If the
searcher goes in the +x direction, it finds its target after T
= 关共b − x兲 + 共b − a兲兴 / vl. This leads to
共65兲
= 0,
tbal =
x=b
共66兲
= 0,
共67兲
−
t−2 共a兲 = t2,in
共a兲,
共68兲
冏 冏
共69兲
= 0,
共70兲
tm =
where
共␶1 + ␶2兲b
␣3/2
冋冉 冊冑
b
+ L1
3
册
␣ + ⌫L2共冑␣ + L2兲 ,
共71兲
b
a
lim tm =
␶1→0
b−a
共b − a兲2
dx =
.
bvl
vl
共77兲
冉
冊
e2a/L2 + 1
b b
+ 2a/L
.
V 3L2 e 2 − 1
共78兲
Taking the derivative with respect to L2 yields
d
共 lim tm兲 ⬀ 12ae2a/L2 + 2be2a/L2 − b − be4a/L2 , 共79兲
dL2 ␶1→0
which has only one positive root,
2. Results
The exact expression of tm 共cf. Appendix B兲 is obtained
through lengthy but standard calculations. To simplify this
expression, we consider the small density limit a / b → 0 and
finally obtain the following very good approximation of tm
共Fig. 7兲:
冕
We take the limit ␶1 → 0 in the expression of tm 关Eq. 共71兲兴
and obtain
x=0
t−1 共a兲 = 0.
1
b
4. Intermittent regime
x=b
+
t+2 共a兲 = t2,in
共a兲,
dt2,in
dx
A numerical analysis indicates 共Fig. 7 and Table I兲 that,
depending on the parameters, there are two possible optimal
strategies:
共i兲 ␶1 → ⬁, intermittence is not favorable and
共ii兲 ␶1 → 0 , ␶2 = ␶opt
2 , intermittence is favorable.
We now study analytically these regimes.
共63兲
dt−,in
1
2
− t−,in
= − 1.
dx
␶2 2
,
Lopt
2 =
2a
ln共1 + 6a/b + 2冑3a/b + 9a2/b2兲
In the limit b Ⰷ a it leads to
␶opt
2 =
a
3V
冑
b
,
a
.
共80兲
共81兲
which is in agreement with numerical minimization of the
exact mean detection time shown in Table I.
The mean first-passage time at the target is minimized in
the intermittent regime for ␶1 → 0 and ␶2 = ␶opt
2 . We replace ␶2
031146-8
ROBUSTNESS OF OPTIMAL INTERMITTENT SEARCH …
PHYSICAL REVIEW E 80, 031146 共2009兲
(a)
(b)
(c)
(d)
FIG. 7. 共Color online兲 Ballistic mode in one dimension. Comparison between low density approximation 关Eq. 共71兲兴 共symbols兲 and the
opt
exact expression of tm 关Eq. 共B1兲兴 共line兲. 共a兲 and 共b兲: tm as a function of ␶1, with ␶2 = 0.1␶opt
2 共red, crosses兲, ␶2 = ␶2 共blue, squares兲, and ␶2
opt
opt
= 10␶2 共green, circles兲. 共c兲 and 共d兲: tm as a function of ␶2 / ␶2 , with ␶1 = 0 共red, crosses兲, ␶1 = 10 共violet, circles兲, ␶1 = 100 共blue, squares兲, and
␶1 = 10 000 共green, diamonds兲. 共a兲 and 共c兲: vl = 0.12⬎ vcl : intermittence is not favorable. 共b兲 and 共d兲: vl = 0.06⬍ vcl : intermittence is favorable.
␶opt
2 is from the analytical prediction 关Eq. 共81兲兴. a = 1, V = 1, and b = 100.
by Eq. 共81兲 in expression 共78兲 and take b Ⰷ a to finally obtain
opt
=
tm
2 b
冑3 V
冑
b
,
a
共82兲
gain =
冑3 V
2 vl
冑
a
.
b
This shows that the gain is larger than 1 for vl ⬍ vcl = V 2 冑 ba ,
which defines the regime where intermittence is favorable.
冑3
TABLE I. Ballistic mode in one dimension. Numerical minimization of the exact tm 关Eq. 共B1兲兴. Values of
␶1 and ␶2 at the minimum. Comparison with theoretical ␶2. a = 0.5, V = 1.
vl = 1
b=5
vl = 0.1
␶1 → ⬁
vl = 0.01
vl = 0.001
␶1 → 0, ␶opt
2 = 0.86
␶opt,th
关Eq. 共81兲兴
2
␶opt,th
= 0.91
2
b = 50
␶1 → ⬁
␶1 → 0, ␶opt
2 = 2.9
␶opt,th
= 2.9
2
b = 500
␶1 → ⬁
␶1 → 0, ␶opt
2 = 9.1
␶opt,th
= 9.1
2
b = 5000
共83兲
␶1 → ⬁
␶1 → 0, ␶opt
2 = 29
031146-9
␶opt,th
= 29
2
PHYSICAL REVIEW E 80, 031146 共2009兲
LOVERDO et al.
5. Summary
In the case where phase 1 is modeled by the ballistic
mode in dimension 1, we calculated the exact mean firstpassage time tm at the target. tm can be minimized as a function of ␶1 and ␶2, yielding two possible optimal strategies:
冑3
共i兲 for vl ⬎ vcl = V 2 冑 ba , intermittence is not favorable, thus
opt
␶opt
1 → ⬁ and ␶2 → 0;冑
3
共ii兲 for vl ⬍ vcl = V 2 冑 ba , intermittence is favorable, with
a 冑b
opt
opt
␶1 → 0 and ␶2 = 3V a
Note that the model studied in 关2兴 shows that when targets
are not revisitable, the optimal strategy for a one state
searcher is to perform a straight ballistic motion. This strategy corresponds to ␶1 → ⬁ in our model. Our results show
that if a faster phase without detection is allowed, this
straight line strategy can be outperformed.
D. Conclusion in dimension 1
Intermittent search strategies in dimension 1 share similar
features for the static, diffusive, and ballistic detection
modes. In particular, all modes show regimes where intermittence is favorable and lead to a minimization of the search
time. Strikingly, the optimal duration of the nonreactive relocation phase 2 is quite independent of the modeling of the
a 冑b
reactive phase: ␶opt
2 = 3V a for the static mode, for the ballistic mode 共in the regime vl ⬍ vcl ⯝ V2 冑 3ba 兲, and for the diffusive
b
mode 共in the regime b ⬎ DV and a Ⰷ DV 冑 a 兲. This shows the
opt
robustness of the optimal value ␶2 .
y
phase 2
x
A. Static mode
We study here the case where the detection phase is modeled by the static mode 共Fig. 8兲: the searcher does not move
during the detection phase and has a finite reaction rate with
the target if it is within its detection radius a.
a
k
FIG. 8. 共Color online兲 Static mode in two dimension.
ជ · ⵜrt2共rជ兲 −
V
1
关t 共rជ兲 − t1共rជ兲兴 = − 1.
␶2 2
共85兲
The function Ia is defined by Ia共rជ兲 = 1 inside the target 共if
兩rជ兩 ⱕ a兲 and Ia共rជ兲 = 0 outside the target 共if 兩rជ兩 ⬎ a兲. In the
present form, these integrodifferential equations 共completed
with boundary conditions兲 do not seem to allow for an exact
resolution with standard methods. t2 is the mean first-passage
ជ,
time on the target, starting from rជ in phase 2, with speed V
of angle ␪v, and with projections on the axes Vx, Vy. i and j
can take either x or y as a value. We use the following decoupling assumption
具ViV jt2典␪V ⯝ 具ViV j典␪V具t2典␪V
共86兲
and finally obtain the following approximation of the mean
search time, which can be checked by numerical simulations:
再
IV. DIMENSION 2
The two-dimensional problem is particularly well suited
to model animal behaviors. It is also relevant to the microscopic scale since it mimics, for example, the case of cellular
traffic on membranes 关25兴. The results for the static and diffusive modes, already treated in 关8,9兴, are summarized here
for completeness. While in dimension 1 the mean search
time can be calculated analytically, we introduce in dimension 2 共and later dimension 3兲 approximation schemes,
which we check by numerical simulations. In these numerical simulations, diffusion was simulated using variable step
lengths, as in 关38兴, and we used square domains instead of
disks for numerical convenience 共it was checked numerically
that results are not affected as soon as b Ⰷ a兲.
phase 1
V
tm =
␶1 + ␶2 1
I0共x兲 1 2
共1 + k␶1兲共y 2 − x2兲2
+ 兵8y + 共1 + k␶1兲
2
2k␶1y x
I1共x兲 4
冎
⫻关4y 4 ln共y/x兲 + 共y 2 − x2兲共x2 − 3y 2 + 8兲兴其 ,
共87兲
where
x=
冑
2k␶1 a
1 + k␶1 V␶2
and
y=
冑
2k␶1 b
.
1 + k␶1 V␶2
共88兲
In that case, intermittence is trivially necessary to find the
target: indeed, if the searcher does not move, the MFPT is
infinite. In the regime b Ⰷ a, the optimization of the search
time 关Eq. 共87兲兴 leads to
␶opt
1 =
冉 冊冉
a
Vk
1/2
2 ln共b/a兲 − 1
8
冊
1/4
,
共89兲
1. Equations
a
1/2
␶opt
2 = 关ln共b/a兲 − 1/2兴 ,
V
The mean first-passage time 共MFPT兲 at a target satisfies
the following backward equations 关39兴:
1
2␲␶1
冕
2␲
0
关t2共rជ兲 − t1共rជ兲兴d␪Vជ − kIa共rជ兲t1共rជ兲 = − 1,
共84兲
共90兲
and the minimum search time is given in the large volume
limit by
031146-10
ROBUSTNESS OF OPTIMAL INTERMITTENT SEARCH …
opt
tm
=
PHYSICAL REVIEW E 80, 031146 共2009兲
21/4 共a2 − 4b2兲ln共b/a兲 + 2b2 − a2
b2
2 − 冑
ak
关2 ln共b/a兲 − 1兴3/4
Vka3
−
冑2
共96b2a2 − 192b4兲ln2共b/a兲 + 共192b4 − 144a2b2兲ln共b/a兲 + 46a2b2 − 47b4 + a4
.
关2 ln共b/a兲 − 1兴3/2
48ab2V
共91兲
冑
2. Summary
In the case of a static detection mode in dimension 2, we
obtained a simple approximate expression of the mean firstpassage time tm at the target. With the static detection mode,
intermittence is always favorable and leads to a single optimal intermittent strategy. The minimal search time is realized
a 1/2
a
1/4
for ␶opt
and ␶opt
1 = 共 Vk 兲 (关2 ln共b / a兲 − 1兴 / 8)
2 = V 关ln共b / a兲
1/2
− 1 / 2兴 . Importantly, the optimal duration of the relocation
phase does not depend on k, i.e., on the description of the
detection phase, as in dimension 1.
L⫾ = I0共a/ D̃␶2兲关I1共b␣兲K1共a␣兲
冑
冑
− I1共a␣兲K1共b␣兲兴 ⫾ ␣ D̃␶2I1共a/ D̃␶2兲关I1共b␣兲K0共a␣兲
+ I0共a␣兲K1共b␣兲兴
and
冑
− 4关a2冑D̃␶2/␣共b2 − a2兲2兴I1共a/冑D̃␶2兲关I1共b␣兲K1共a␣兲
M = I0共a/ D̃␶2兲关I1共b␣兲K0共a␣兲 + I0共a␣兲K1共b␣兲兴
− I1共a␣兲K1共b␣兲兴
B. Diffusive mode
where
We now assume that the searcher diffuses during the detection phase 共Fig. 9兲. For this process, the mean firstpassage time at the target satisfies the following backward
equation 关39兴:
Dⵜ2r t1共rជ兲 +
1
2␲␶1
冕
ជ · ⵜrt2共rជ兲 −
V
2␲
关t2共rជ兲 − t1共rជ兲兴d␪V = − 1,
共92兲
0
1
关t 共rជ兲 − t1共rជ兲兴 = − 1,
␶2 2
−
再
and
D̃ = v2␶2 .
We then minimize this time as a function of ␶1 and ␶2.
1. a ⬍ b ™ D Õ V: Intermittence is not favorable
共93兲
with t1共rជ兲 = 0 inside the target 共r ⱕ a兲. We use the same decoupling assumption than for the static case 关Eq. 共86兲兴. It
eventually leads to the following approximation of the mean
search time:
t m = 共 ␶ 1 + ␶ 2兲
␣ = 关1/共D␶1兲 + 1/共D̃␶2兲兴1/2
1 − a2/b2
M
L−
a␣共b2/a2 − 1兲
−
共 ␣ 2D ␶ 1兲 2
2L+ L+
冎
␣2D␶1 关3 − 4 ln共b/a兲兴b4 − 4a2b2 + a4
,
b2 − a2
8D̃␶2
In that regime, intermittence is not favorable. Indeed, the
typical time required to explore the whole domain of radius b
is of order b2 / D for a diffusive motion, which is shorter than
the corresponding time b / V for a ballistic motion. As a consequence, it is never useful to interrupt the diffusive phases
by mere relocating ballistic phases. We use standard methods
to calculate the mean first-passage time at the target in this
optimal regime of diffusion only,
冉 冊
D d dt
r
= − 1.
r dr dr
共94兲
The boundary conditions t共a兲 = 0 and
with
tdif f =
phase 1
tdif f =
V
x
a
dt
dr 共r = b兲 = 0
lead to
冊
1
b
4a2b2 − a4 − 3b4 + 4b4 ln ,
8b Def f
a
2
and we find in the limit b Ⰷ a
phase 2
y
冉
共95兲
冉
冊
b2
b
− 3 + 4 ln .
8Def f
a
共96兲
共97兲
D
FIG. 9. 共Color online兲 Diffusive mode in two dimension.
2. a ™ D Õ V ™ b: First regime of intermittence
In this second regime, one can use the following approximate formula for the search time:
031146-11
PHYSICAL REVIEW E 80, 031146 共2009兲
LOVERDO et al.
tm =
phase 1
b2 ␶1 + ␶2
4DV2␣2 ␶1␶22
再
⫻ 4 ln共b/a兲 − 3 − 2
冎
phase 2
共V␶2兲2
关ln共␣a兲 + ␥ − ln 2兴 ,
D␶1
y
共98兲
x
␥ being the Euler constant. An approximate criterion to determine if intermittence is useful can be obtained by expanding tm in powers of 1 / ␶1 when ␶1 → ⬁ 共␶1 → ⬁ corresponds to
the absence of intermittence兲 and requiring that the coefficient of the term 1 / ␶1 is negative for all values of ␶2. Using
this criterion, we find that intermittence is useful if
冑2 exp共− 7/4 + ␥兲Vb/D − 4 ln共b/a兲 + 3 ⬎ 0.
b2 4 ln w − 5 + c
,
D w2共4 ln w − 7 + c兲
␶opt
2 =
gain =
共99兲
b 冑4 ln w − 5 + c
,
w
V
共100兲
V
v
l
FIG. 10. 共Color online兲 Ballistic mode in two dimensions.
In this regime, using Eq. 共98兲, the optimization of the
search time leads to
␶opt
1 =
a
tdif f
opt
tm
+
⯝
冑2aV
8D
1
冉
I0关2/冑2 ln共b/a兲 − 1兴
1
4 ln共b/a兲 − 3 I1关2/冑2 ln共b/a兲 − 1兴
2冑2 ln共b/a兲 − 1
冊
−1
.
共105兲
Here, the optimal strategy leads to a significant decrease in
the search time which can be rendered arbitrarily smaller
than the search time in absence of intermittence.
4. Summary
where w is the solution of the implicit equation w
= 2Vbf共w兲 / D with
冑4 ln w − 5 + c
f共w兲
= − 8共ln w兲2 + 关6 + 8 ln共b/a兲兴ln w
− 10 ln共b/a兲 + 11 − c关c/2 + 2 ln共a/b兲 − 3/2兴
共101兲
and c = 4关␥ − ln共2兲兴, ␥ being the Euler constant. A useful approximation for w is given by
w⯝
冉
冊
2Vb
Vb
f
.
D
2D ln共b/a兲
共102兲
We studied the case where the detection phase 1 is modeled by a diffusive mode and obtained an approximation of
the mean first-passage time at the target. We found that intermittence is favorable 共i.e., better than diffusion alone兲 in
the regime of large system size b Ⰷ D / V. The optimal intermittent strategy then follows two subregimes:
共i兲 If a Ⰶ D / V, the best strategy is given by 共100兲. The
search is significantly reduced by intermittence but keeps the
same order of magnitude as in the case of a one-state diffusive search.
共ii兲 If a Ⰷ D / V, the best strategy is given by Eq. 共104兲 and
weakly depends on b. In this regime, intermittence is very
efficient as shown by the large gain obtained for V large.
The gain for this optimal strategy reads
gain =
tdif f
opt
tm
⫻
冉
C. Ballistic mode
1 4 ln b/a − 3 + 4a /b − a /b
2 4 ln b/a − 3 + 2共4 ln w兲ln共b/aw兲
2
⯝
2
1
wD 4 ln w − 7
+
4 ln w − 5 bV 共4 ln w − 5兲3/2
4
冊
4
In this case, the searcher has access to two different
speeds: one 共V兲 is fast but prevents target detection and the
other one 共vl兲 is slower but enables target detection 共Fig. 10兲.
−1
.
共103兲
If intermittence significantly speeds up the search in this regime 共typically by a factor 2兲, it does not change the order of
magnitude of the search time.
3. D Õ V ™ a ™ b: “Universal” regime of intermittence
In the last regime D / V Ⰶ a Ⰶ b, the optimal strategy is
obtained for
␶opt
1 ⯝
ln2共b/a兲
D
,
2V2 2 ln共b/a兲 − 1
and the gain reads
a
1/2
␶opt
2 ⯝ 关ln共b/a兲 − 1/2兴
V
共104兲
1. Simulations
Since an explicit expression of the mean search time is
not available, a numerical study is performed. Exploring the
parameter space numerically enables to identify the regimes
where the mean search time is minimized. Then, for each
regime, approximation schemes are developed to provide
analytical expression of the mean search time.
The numerical results presented in Fig. 11 suggest two
regimes defined according to a threshold value vcl of vl to be
determined later on:
共i兲 for vl ⬎ vcl , tm is minimized for ␶2 → 0;
共ii兲 for vl ⬍ vcl , tm is minimized for ␶1 → 0.
2. Regime without intermittence (␶2 \ 0, ␶1 \ ⴥ)
Qualitatively, it is rather intuitive that for vl large enough
共the precise threshold value vcl will be determined next兲,
031146-12
ROBUSTNESS OF OPTIMAL INTERMITTENT SEARCH …
(a)
PHYSICAL REVIEW E 80, 031146 共2009兲
(b)
(c)
FIG. 11. 共Color online兲 Ballistic mode in two dimensions. ln共tm兲 as a function of ln共␶2兲. Simulations 共symbols兲, diffusive/diffusive
approximation 关Eq. 共94兲兴 with Eq. 共109兲 共colored lines兲, ␶1 → 0 limit 关Eq. 共110兲兴 共black line兲, and ␶1 → ⬁ 共no intermittence兲 关Eq. 共108兲兴
共dotted black line兲. b = 30, a = 1, and V = 1. ␶1 = 0 共black, stars兲, ␶1 = 0.17 共yellow, solid circles兲, ␶1 = 0.92 共green, diamonds兲, ␶1 = 5.0 共blue, X兲,
␶1 = 28 共purple, circles兲, ␶1 = 150 共red, +兲, and ␶1 = 820 共brown, squares兲.
phase 2 is inefficient since it does not allow for target detection. The optimal strategy is therefore ␶2 → 0 in this case. In
this regime, the searcher performs a ballistic motion, which
is randomly reoriented with frequency 1 / ␶1. Along the same
line as in 关2兴 共where however the times between successive
reorientations are Levy distributed兲, it can be shown that the
optimal strategy to find a target 共which is assumed to disappear after the first encounter兲 are to minimize oversampling
and therefore to perform a purely ballistic motion. In our
case this means that in this regime ␶2 → 0, the optimal ␶1 is
given by ␶opt
1 → ⬁.
In this regime, we can propose an estimate of the optimal
search time tbal. The surface scanned during ␦t is 2avl␦t. p共t兲
is the proportion of the total area which has not yet been
scanned at t. If we neglect correlations in the trajectory, one
has
2avl p共t兲
dp
.
=−
dt
␲b2
proceed we approximate the problem by the case of a diffusive mode previously studied 关Eq. 共94兲兴, with an effective
diffusion coefficient,
D=
v2l ␶1
.
2
共109兲
This approximation is very satisfactory in the regime ␶1
→ 0, as shown in Fig. 11.
We can then use the results in 关8,9兴 in the ␶1 → 0 regime
and obtain
共106兲
Then, given that p共t = 0兲 = 1, we find
冉 冊
p共t兲 = exp −
2avl
␲b2
共107兲
and the mean first-passage time at the target in these conditions is
tbal = −
冕
⬁
0
t
dp
␲b2
dt =
.
dt
2avl
共108兲
This expression yields 共Fig. 12兲 a good agreement with numerical simulations. Note in particular that tbal ⬀ v1l .
3. Regime with intermittence ␶1 \ 0
In this regime where vl ⬍ vcl , the numerical study shows
that the search time is minimized for ␶1 → 0 共Fig. 11兲. We
here determine the optimal value of ␶2 in this regime. To
opt
FIG. 12. 共Color online兲 Ballistic mode in two dimension. tm
as
a function of b, logarithmic scale. Regime without intermittence
共␶2 = 0 and ␶1 → ⬁, vl = 1兲, analytical approximation 关Eq. 共108兲兴
共blue, line兲, and numerical simulations 共blue, circles兲. Regime with
intermittence 共with ␶1 = 0, V = 1兲, analytical approximation 关Eq.
共112兲兴 共red, line兲 and numerical simulations 共red, squares兲. a = 1.
031146-13
PHYSICAL REVIEW E 80, 031146 共2009兲
LOVERDO et al.
冉 冊册
冋
冉 冊
a2
tm = ␶2 1 − 2
b
冢
a
b4 − 4a2b2 + a4
b
␶22V2共b2 − a2兲
3 + 4 ln
1
1−
4
冉冑冊
冊 冑
冑 冉
冉 冊
2
b
+
2 −1
V␶2 2 a
a
I0
a 2
␶ 2V
I1
a 2
␶ 2V
冣
共110兲
.
The calculation of ␶opt
2 minimizing tm then gives
␶opt
2 =
a
v
冑冉冊
ln
1
b
− .
2
a
共111兲
关Eq. 共111兲兴 in Eq. 共110兲, we
In turn, replacing ␶2 by ␶opt
2
obtain the minimal mean time of target detection,
opt
=
tm
冉 冊冢
冉 冊
a
2
u冑2V
u
+
1−
a
b2
1 − u2
冋
冉 冊册
a
b4 − 4a2b2 + a4
b
a22共b2 − a2兲
3 + 4 ln
冣
b2
− 1 I0共2u兲
a2
,
I1共2u兲
2b2
aV
冑冉冊
ln
b
.
a
共113兲
Finally the gain reads 关using Eqs. 共108兲 and 共113兲兴
gain =
tbal
opt
tm
⯝
冋 冉 冊册
␲V
b
ln
4vl
a
.
共114兲
4. Determination of vcl
It is straightforward than vcl ⬍ V. Indeed, if vl = V, phase 2
is useless since it prevents target detection. Actually, an estimate of vcl can be obtained from Eq. 共114兲 as the value of vl
for which gain= 1,
冋 冉 冊册
−0.5
⬀
5. Summary
We studied the case of ballistic mode in the detection
phase 1 in dimension 2. When vl ⬎ vcl , the optimal strategy is
to remain in phase 1 and to explore the domain in a purely
c
opt
ballistic way. Therefore, ␶opt
2 → 0, ␶1 → ⬁. When vl ⬍ vl , we
b
1
opt
opt a 冑
find on the contrary ␶1 → 0 and ␶2 = V ln共 a 兲 − 2 . The
V
threshold value is given by vcl ⬀ 冑ln共b/a兲
and shows that when
the target density decreases, intermittence is less favorable.
D. Conclusion of the two-dimensional problem
−0.5
Numerical simulations of Fig. 12 show the validity of these
approximations.
␲V
b
ln
vcl ⯝
4
a
reoriented many times and therefore scales as diffusion,
which is less favorable than the nonintermittent ballistic motion.
共112兲
opt 1
⬀ V . Note
with u = 关ln共 ab 兲 − 1兴−1/2. It can be noticed that tm
that if b Ⰷ a this last expression can be greatly simplified,
opt
⯝
tm
FIG. 13. 共Color online兲 Ballistic mode in two dimension. vcl as a
function of ln共b兲 by simulations 共symbols兲, predicted expression
共115兲 共red, dotted line兲, predicted expression multiplied with a fitted
numerical constant 共blue, line兲. V = 1, a = 1.
V
冑ln共b/a兲 .
共115兲
We note that this expression 共Fig. 13兲 gives the correct dependence on b, but it however departs from the value obtained by numerical simulations. This is due to fact that the
opt
with intermittence 关Eq. 共113兲兴 is an under
expression of tm
estimate, while tbal given in Eq. 共108兲 is an upper estimate. It
is noteworthy that intermittence is less favorable with increasing b. This effect is similar to the one-dimensional case
even though it is less important here. It can be understood as
follows: at very large scales the intermittent trajectory is
Remarkably, for the three different modes of detection
共static, diffusive, and ballistic兲, we find a regime where intermittence permits to minimize the search time for one and
b
1
opt a 冑
the same ␶opt
2 , given by ␶2 = V ln共 a 兲 − 2 . As in dimension 1,
this indicates that optimal intermittent strategies are robust
and widely independent of the details of the description of
the detection mechanism.
V. DIMENSION 3
The three-dimensional case is also relevant to biology. At
the microscopic scale, it corresponds, for example, to intracellular traffic in the bulk cytoplasm of cells or at larger
scales to animals living in three dimensions, such as plankton 关40兴 or C.elegans in its natural habitat 关41兴. As it was the
case in dimension 2, different assumptions have to be made
to obtain analytical expressions of the search time. We
checked the validity of our assumptions with numerical
simulations using the same algorithms as in dimension 2.
A. Static mode
We study in this section the case where the detection
phase is modeled by the static mode, for which the searcher
031146-14
ROBUSTNESS OF OPTIMAL INTERMITTENT SEARCH …
冏 冏 冏 冏
dtout
2
dr
phase 1
V
z
PHYSICAL REVIEW E 80, 031146 共2009兲
We find an explicit expression of the mean search time,
a
k
t m = 共 ␶ 1 + ␶ 2兲
FIG. 14. 共Color online兲 Static mode in three dimensions.
does not move during the detection phase and has a finite
reaction rate with the target if it is within a detection radius a
共Fig. 14兲.
冉
⫻ 3
Denoting t1共r兲 the mean first-passage time at the target
starting from a distance r from the target in phase 1 共detection phase兲 and t2,␪,␾共r兲 the mean first-passage time at the
target starting from a distance r from the target in phase 2
共relocation phase兲 with a direction characterized by ␪ and ␾,
we get
1
共t − t
兲 = − 1.
␶2 1 2,␪,␾
Then one has outside the target 共r ⬎ a兲
冉 冕
␲
d␪ sin ␪
0
冕
2␲
冕
␲
0
d␪ sin ␪
冕
2␲
d␾t2,␪,␾ −
0
冉 冊
1
+ k t1 = − 1.
␶1
冊
␤=
t m = 共 ␶ 1 + ␶ 2兲
冉
共119兲
dtout
2
dr
冎冊
共127兲
.
冉
冊
b3共␶2 + ␶1兲 共1 + k␶1兲
6
+ 2 2 .
a
␶1ka2
5 ␶ 2V
␶opt
1 =
冉 冊冑
3
10
1/4
a
,
Vk
共128兲
共129兲
冑 a
␶opt
2 = 1.2 ,
V
共130兲
and the minimum mean search time reads finally
opt
=
tm
共120兲
1 1 b3
冑5 k a 3
冉冑
ak 1/4
24 + 51/4
V
冊
2
.
共131兲
3. Comparisons with simulations
共121兲
We solve these equations for inside and outside the target
using the following boundary conditions:
冏 冏
再
This expression of tm can be minimized for
Making a similar decoupling approximation as in dimension
2, we finally get
1
V 2␶ 2
⌬t2 − 共t1 − t2兲 = − 1.
3
␶2
共126兲
Assuming further that ␣ is small, we use the expansion ␤
⯝ 共b3 / a兲关1 − tanh共␣兲 / ␣兴−1 ⯝ 关b3 / a兴关共3 / ␣2兲 + 共6 / 5兲兴 and rewrite mean search time as
and inside the target 共r ⬍ a兲
冉 冊
− sinh共␣兲a3 + ␣ cosh共␣兲b3
a关− sinh共␣兲 + ␣ cosh共␣兲兴
3a2
9
− b2 + 2
5
␣
With t2 = 41␲ 兰0␲d␪ sin ␪兰20␲d␾t2,␪,␾, we obtain outside the target 共r ⬎ a兲
1
1
t2 −
+ k t1 = − 1.
␶1
␶1
共125兲
,
1
1
− sinh共␣兲a3 + ␣ cosh共␣兲b3
+ 2 2
k ␶ 1 ␶ 2V
a关− sinh共␣兲 + ␣ cosh共␣兲兴
共118兲
1
共t − t 兲 = − 1
␶1 2 1
册冎
and ␣ = 冑3关k␶1 / 共1 + k␶1兲兴共a / V␶2兲.
In the limit b Ⰷ a, this can be simplified to
共116兲
共117兲
冋
1
1
+ 3 2 2 − 2b3共b2 − a2兲 + 共b3 − a3兲
k␶1 b V ␶2
1 5
a2
5
2 + ␤ + 共b − a 兲
5
␣
tm =
and inside the target 共r ⱕ a兲
1 1
␶1 4␲
冊
d␾t2,␪,␾ − t1 = − 1
0
再
with
1. Equations
1 1
␶1 4␲
共124兲
r=a
2. Results
x
ជ t2,␪,␾ +
Vជ . 䉮
r=a
and the condition that tin
2 共0兲 should be finite.
phase 2
y
dtin
2
dr
=
= 0,
共122兲
in
tout
2 共a兲 = t2 共a兲,
共123兲
Data obtained by numerical simulations 共Fig. 15 and additionally in Appendix C兲 are in good agreement with the
analytical expression 关Eq. 共125兲兴. In particular, the position
of the minimum is very well approximated, and the error on
the value of the mean search time at the minimum is close to
10%. Note that the very simple expression 关Eq. 共128兲兴 fits
also rather well the numerical data, except for small ␶2 or
small b.
4. Summary
r=b
In the case of a static detection mode in dimension 3, we
obtained a simple approximate expression of the mean first-
031146-15
PHYSICAL REVIEW E 80, 031146 共2009兲
LOVERDO et al.
(a)
(b)
(c)
(d)
(e)
(f)
FIG. 15. 共Color online兲 Static mode in three dimensions. ln共tm兲 as a function of ln共␶2兲 for different values of ␶1, a, and b / a. Comparison
between simulations 共symbols兲, analytical expression 共125兲 共line兲, expression for b Ⰷ a 关Eq. 共127兲兴 共small dots兲, and simple expression for
冑 aV
冑 aV
冑 aV
b Ⰷ a and ␣ small 关Eq. 共128兲兴 共dotted line兲. ␶1 ⯝ ␶opt
1 ⯝ 0.74 k 关Eq. 共129兲兴 共blue, crosses兲, ␶1 = 0.25 k 共red, circles兲, and ␶1 = 2.5 k 共green,
squares兲. V = 1, k = 1.
passage time at the target tm = 关b3共␶2 + ␶1兲 / a兴(关共1
+ k␶1兲 / ␶1ka2兴 + 6 / 共5␶22V2兲). tm has a single minimum for
3 1/4冑 a
a
opt 冑
␶opt
1 = 共 10 兲
Vk and ␶2 = 1.2 V , and the minimal mean
1 1 3
a
k
search time is 冑5 k 共b / a3兲共冑 V 241/4 + 51/4兲2. With the static detection mode, intermittence is always favorable and leads to
a single optimal intermittent strategy. As in dimensions 1 and
2, the optimal duration of the relocation phase does not depend on k, i.e., on the description of the detection phase. In
addition, this optimal strategy does not depend on the typical
distance between targets b.
One can notice than for the static mode in the three cases
studied 共in one, two, and three dimensions兲, we have the
冑 opt
relation ␶opt
1 = ␶2 / 共2k兲. The optimal durations of the two
phases are related independently of the dimension.
1. Equations
One has outside the target 共r ⬎ a兲
ជ .䉮
ជ t2,␪,␾ +
V
1
共t − t
兲 = − 1,
␶2 1 2,␪,␾
共132兲
z
D
y
phase 2
x
V
phase 1
a
B. Diffusive mode
We now study the case where the detection phase is modeled by a diffusive mode 共Fig. 16兲. During the detection
phase, the searcher diffuses and detects the target as soon as
their respective distance is less than a.
031146-16
FIG. 16. 共Color online兲 Diffusive mode in three dimensions.
ROBUSTNESS OF OPTIMAL INTERMITTENT SEARCH …
PHYSICAL REVIEW E 80, 031146 共2009兲
(a)
(b)
FIG. 17. 共Color online兲 Diffusive mode in three dimension. Comparison between analytical approximations 关Eqs. 共143兲 and 共145兲兴 共black
opt,simulation
simulation
lines兲 and numerical simulations: the symbols are the values of ␶1 and ␶2 for which tm
⬍ 1.05tm
. a = 100, V = 1, D = 1.
D⌬t1 +
冉 冕
1 1
␶1 4␲
␲
d␪ sin ␪
0
冕
2␲
冊
2. Results in the general case
d␾t2,␪,␾ − t1 = − 1
0
共133兲
and inside the target 共r ⱕ a兲
ជ .䉮
ជ t2,␪,␾ −
V
1
t
= − 1,
␶2 2,␪,␾
共134兲
b3␬42共␶1 + ␶2兲
tm =
␬1
共135兲
t1 = 0.
With t2 = 41␲ 兰0␲d␪ sin ␪兰20␲d␾t2,␪,␾, we get outside the target
共r ⬎ a兲
D⌬tout
1 +
Through standard but lengthy calculations we can solve
the above system and get an analytical approximation of tm
共cf. Appendix D 1兲. In the regime b Ⰷ a, we use the assumption 冑共␶1D兲−1 + 3共␶2v兲−2 Ⰶ b and obtain
1 out out
共t − t1 兲 = − 1.
␶1 2
共136兲
The decoupling approximation described in previous sections then yields outside the target,
V2␶2 out 1 out out
⌬t2 + 共t1 − t2 兲 = − 1,
3
␶2
共137兲
␬1
␬2
␬1
␬1␬22␶1Da tanh共␬2a兲 +
− tanh共␬2a兲
␬2
共142兲
␶opt
1 ⬃
V2␶2 int 1 int
⌬t2 − t2 = − 1.
3
␶2
共138兲
tout
2 共a兲
=
=
r=a
tm =
b3冑3
V3␶22
dtint
2
dr
冋冑
共140兲
.
r=a
共141兲
冉 冊册
冑3a
3a
− tanh
V␶2
V␶2
␶opt
2 =
tint
2 共a兲,
共143兲
−1
.
共144兲
In turn, the minimization of this expression leads to
r=b
冏 冏 冏 冏
dtout
2
dr
共139兲
= 0,
6D
.
V2
opt
共Fig.
Note that taking ␶1 = 0 does not change significantly tm
17, left兲 and permits to significantly simplify tm,
These equations are completed by the following boundary
conditions:
冏 冏
册
with ␬1 = 共冑␶22V2 + 3␶1D兲 / 共␶2V冑D␶1兲 and ␬2 = 冑3 / V␶2. As
shown in Fig. 17 left or in the additional Fig. 24 in Appendix
D 2, tm only weakly depends on ␶1, which indicates that this
variable will be less important than ␶2 in the minimization of
the search time. The relevant order of magnitude for ␶opt
1 can
be evaluated by comparing the typical diffusion length Ldif f
= 冑6Dt and the typical ballistic length Lbal = Vt. An estimate
of the optimal time ␶opt
1 can be given by the time scale for
which those lengths are of same order, which gives
and inside the target 共r ⱕ a兲,
dtout
2
dr
冋
tanh共␬2a兲 +
冑3a
Vx
共145兲
with x as solution of
2 tanh共x兲 − 2x + x tanh共x兲2 = 0.
This finally yields
031146-17
共146兲
PHYSICAL REVIEW E 80, 031146 共2009兲
LOVERDO et al.
(a)
(b)
FIG. 18. 共Color online兲 Diffusive mode in three dimension. tm / tdif f 关tdif f given by Eq. 共150兲兴 as a function of ␶2 for different values of
the ratio b / a 共logarithmic scale兲. The full analytical form 关Eq. 共D1兲兴 共plain lines兲 is plotted against the simplified expression 关Eq. 共142兲兴
共dotted lines兲, the simplified expression with ␶1 = 0 关Eq. 共144兲兴 共small dots兲, and numerical simulations 共symbols兲 for the following values of
the parameters 共arbitrary units兲: a = 1 共green, squares兲, a = 5 共blue, stars兲, a = 7 共purple, circles兲, a = 10 共red, +兲, a = 14 共brown, X兲, and a
= 20 共orange, diamonds兲. ␶1 = 6 everywhere except for the small dots, v = 1, D = 1. tm / tdif f presents a minimum only for a ⬎ ac ⯝ 4.
a
␶opt
2 ⯝ 1.078 .
V
共147兲
Importantly this approximate expression is very close to the
冑6 a
expression obtained for the static mode 共␶opt
2 = 5V
a
⯝ 1.095 V 兲 关Eq. 共130兲兴, and there is no dependence with the
typical distance between targets b. The simplified expression
of the minimal tm 关Eq. 共144兲兴 can then be obtained as
3 2
opt
=
tm
bx
冑3a V
2
3
关x − tanh共x兲兴−1 ⯝ 2.18
b
a 2V
共148兲
and the gain reads
gain =
tdif f
opt
tm
⯝ 0.15
aV
.
D
共149兲
tdif f =
1
共5b3a3 + 5b6 − 9b5a − a6兲,
15Dab3
共150兲
which gives in the limit b / a Ⰷ 1
tdif f =
b3
.
3Da
共151兲
5. Criterion for intermittence to be favorable
There is a range of parameters for which intermittence is
favorable, as indicated by Fig. 18. Both the analytical exopt
in the regime without intermittence 关Eq.
pression for tm
共150兲兴 and with intermittence 关Eq. 共148兲兴 scale as b3. However, the dependence on a is different 共cf. Appendix D 4兲. In
the diffusive regime, tm ⬀ a−1, whereas in the intermittent regime, tm ⬀ a−2. This enables us to define a critical ac, such
that when a ⬎ ac, intermittence is favorable: ac ⯝ 6.5 DV is the
value for which the gain 关Eq. 共149兲兴 is 1.
3. Comparison between analytical approximations and
numerical simulations
6. Summary
Numerical simulations reveal that the minimum of tm with
respect to ␶1 is shallow as it was expected 共cf. Fig. 17, left兲.
It approximately ranges from 0 to the theoretical estimate
at the minimum is close to the
关Eq. 共143兲兴. The value ␶opt,sim
2
expected values 关Eq. 共145兲兴 共cf. Fig. 17, right兲, except for
very small b, which is consistent with our assumption b Ⰷ a.
We can then conclude than the position of the optimum in ␶1
and ␶2 is very well described by the analytical approximations even if the value of tm at the minimum is underestimated by our analytical approximation by about 10– 20 %
共Fig. 18兲.
We studied the case where the detection phase 1 is modeled by a diffusive mode and calculated explicitly an approximation of the mean first-passage time at the target. We
found that intermittence is favorable 共i.e., better than diffusion alone兲 when a ⬎ ac ⯝ 6.5 DV :
共i兲 If a ⬍ ac, the best strategy is a one state diffusion,
without intermittence, and the mean first-passage time at the
target is tm ⯝ b3 / 3Da.
共ii兲 If a ⬎ ac, intermittence is favorable. The dependence
on ␶1 is not crucial as long as it is smaller than 6D / V2. The
a
value of ␶2 at the optimum is ␶opt
2 ⯝ 1.08 V . The minimum
opt
3 2
search time is then tm ⯝ 2.18共b / a V兲.
4. Case without intermittence: One state diffusive searcher
C. Ballistic mode
If the searcher always remains in the diffusive mode, it is
straightforward to obtain 共cf. Appendix D 3兲
We now discuss the last case, where the detection phase 1
is modeled by a ballistic mode 共Fig. 19兲. Since an explicit
031146-18
ROBUSTNESS OF OPTIMAL INTERMITTENT SEARCH …
PHYSICAL REVIEW E 80, 031146 共2009兲
1. Numerical study
vl
The numerical analysis puts forward two strategies to
minimize the search time, depending on a critical value vcl to
be determined 共Fig. 20兲:
opt
共i兲 When vl ⬎ vcl , ␶opt
1 → ⬁ and ␶2 → 0. In this regime intermittence is not favorable.
opt
共ii兲 When vl ⬍ vcl , ␶opt
1 → 0 and ␶2 finite. In this regime
the optimal strategy is intermittent.
phase 2
V
z
a
y
phase 1
2. Regime without intermittence (one state ballistic searcher):
␶2 \ 0
x
FIG. 19. 共Color online兲 Ballistic mode in dimension 3.
analytical determination of the search time seems out of
reach, we proceed as in dimension 2 and first explore numerically the parameter space to identify the regimes where
the search time can be minimized. We then develop approximation schemes in each regime to obtain analytical expressions 共more details are given in Appendix E兲.
Following the same argument as in dimension 2, without
intermittence the best strategy is obtained in the limit ␶1
→ ⬁ 共cf. Appendix E 1兲 in order to minimize oversampling
of the search space. Following the derivation of Eq. 共108兲
共see Appendix E 1 for details兲, it is found that the search time
reads
tbal =
(a)
(b)
(c)
(d)
(e)
(f)
4b3
.
3a2vl
共152兲
FIG. 20. 共Color online兲 Ballistic mode in three dimensions. tm as a function of ␶2 in log-log scale. Simulations 共symbols兲. Approximation
vl␶1 ⱕ a 关Eq. 共D1兲with Eq. 共E1兲兴 共colored lines兲, approximation ␶1 = 0 关Eq. 共153兲兴 共black line兲, and approximation ␶1 = 0 and b Ⰷ a 关Eq. 共144兲兴
共dotted black line兲. Ballistic limit 共␶2 → 0 and ␶1 → ⬁兲 共no intermittence兲 Eq. 共152兲 共green, dotted line兲. 共a兲 and 共d兲: vl ⬎ vcl , ␶1,1 = 0.04,
␶1,2 = 0.2, ␶1,3 = 1, ␶1,4 = 5, and ␶1,5 = 25. 共b兲 and 共e兲: vl ⯝ vcl , ␶1,1 = 0.08, ␶1,2 = 0.4, ␶1,3 = 2, ␶1,4 = 10, and ␶1,5 = 50. 共c兲 and 共f兲: vl ⬍ vcl , ␶1,1
= 0.2, ␶1,2 = 1, ␶1,3 = 5, ␶1,4 = 25, and ␶1,5 = 125. V = 1, a = 1. ␶1 = 0 共black, diamond兲, ␶1 = ␶1,1 共brown, +兲, ␶1 = ␶1,2 共red, squares兲, ␶1 = ␶1,3 共pink,
stars兲, ␶1 = ␶1,4 共blue, circles兲, and ␶1 = ␶1,5 共green, X兲.
031146-19
PHYSICAL REVIEW E 80, 031146 共2009兲
LOVERDO et al.
gain =
tbal
opt
tm
V
⯝ 0.61 .
vl
共156兲
As in dimension 2, it is trivial that vcl ⬍ V and the critical
value vcl can be defined as the value of vl such that gain= 1.
This yields
vcl ⯝ 0.6 V.
共157兲
Importantly, vcl neither depends on b nor a. Simulations are
in good agreement with this result 共cf. Appendix E 2兲 except
for a small numerical shift.
5. Summary
opt
FIG. 21. 共Color online兲 Ballistic mode in three dimensions. tm
as a function of b, logarithmic scale. Regime without intermittence
共␶2 = 0 and ␶1 → ⬁, vl = 1兲, analytical approximation 关Eq. 共152兲兴
共blue, line兲, and numerical simulations 共blue, circles兲. Regime with
intermittence 共with ␶1 = 0, V = 1兲, analytical approximation 关Eq.
共155兲兴 共red, line兲, and numerical simulations 共red, squares兲. a = 1.
3. Regime with intermittence
In the regime when intermittence is favorable, the numerical study suggests that the best strategy is realized for ␶1
→ 0 共Fig. 20兲. In this regime ␶1 → 0, phase 1 can be well
approximated by a diffusion with effective diffusion coefficient Def f 关see Eq. 共E1兲兴. We can then make use of the analytical expression tm derived in Eq. 共D1兲. We therefore take
␶1 = 0 in the expression of tm 关Eq. 共D1兲兴, which yields
tm共␶1 = 0兲 ⯝
u
b3aV
冉冑
3
共5b3a2 − 3b5 − 2a5兲
5
共b3 − a3兲2u
+
冑3a关u − tanh共u兲兴
冊
共153兲
,
where u = 冑3a / ␶2V. In the limit b Ⰷ a, this expression can be
further simplified 关see Eq. 共144兲兴 to
tm =
b3冑3
冋
冑3a
V2␶22 V␶2
冉 冊册
− tanh
冑3a
−1
V␶2
,
共154兲
冑3a
and one finds straightforwardly that ␶opt
2 = Vx , where x is solution of x tanh共x兲2 + 2 tanh共x兲 − 2x = 0, that is, x ⯝ 1.606. Using this optimal value of ␶2 in the expression of tm 关Eq.
共144兲兴, we finally get
opt
tm
=
b3
x
b3
⯝
2.18
.
冑3 tanh共x兲2 a2V
a 2V
2
共155兲
These expressions show a good agreement with numerical
simulations 共Figs. 20 and 21兲.
4. Discussion of the critical value vcl
The gain is given by
We studied the case where the detection phase 1 is modeled by a ballistic mode in dimension 3. We have shown by
numerical simulations that there are two possible optimal
regimes, which we have then studied analytically,
共i兲 in the first regime vl ⬎ vcl , the optimal strategy is a one
state ballistic search 共␶1 → ⬁, ␶2 = 0兲 and tm ⯝ 4b3 / 3a2vl and
共ii兲 in the second regime vl ⬍ vcl , the optimal strategy is
intermittent 共␶1 = 0, ␶2 ⯝ 1.1 Va 兲 and tm ⯝ 2.18共b3 / a2V兲 共in the
limit b Ⰷ a兲.
The critical speed is obtained numerically as vcl ⬃ 0.5V
共analytical prediction: vcl ⬃ 0.6V兲. It is noteworthy that when
b Ⰷ a, the values of ␶1 and ␶2 at the optimum and the value of
vcl do not depend on the typical distance between targets b.
D. Conclusion in dimension 3
We found that for the three possible modelings of the
detection mode 共static, diffusive, and ballistic兲 in dimension
3, there is a regime where the optimal strategy is intermittent. Remarkably, and as was the case in dimensions 1 and 2,
the optimal time to spend in the fast nonreactive phase 2 is
independent of the modeling of the detection mode and reads
a
␶opt
2 ⯝ 1.1 V . Additionally, while the mean first-passage time
on the target scales as b3, the optimal values of the durations
of the two phases do not depend on the target density a / b.
VI. DISCUSSION AND CONCLUSION
The starting point of this paper was the observation that
intermittent trajectories are observed in various biological
examples of search behaviors, going from the microscopic
scale, where searchers can be molecules looking for reactants, to the macroscopic scale of foraging animals. We addressed the general question of determining whether such
kind of intermittent trajectories could be favorable from a
purely kinetic point of view, that is, whether they could allow
us to minimize the search time for a target. On very general
grounds, we proposed a minimal model of search strategy
based on intermittence, where the searcher switches between
two phases, one slow where detection is possible and the
other one faster but preventing target detection. We studied
this minimal model in one, two, and three dimensions and
under several modeling hypotheses. We believe that this systematic analysis can be used as a basis to study quantitatively
various real search problems involving intermittent behaviors.
031146-20
l
vl < vlc 0.6V
b a 6D
V
6D
τ1opt V
2
opt
τ2 1.1 Va
b3
topt
m 2.18 V a2
vl >
τ1opt → ∞
τ2opt → 0
opt
4b3
tm 3a
2v
πb2
2avl
vlc
topt
m w, s e e S e c . I 
F o r topt
m , c a n d
PHYSICAL REVIEW E 80, 031146 共2009兲
More precisely, we calculated the mean first-passage time
at the target for an intermittent searcher and minimized this
search time as a function of the mean duration of each of the
two phases. Table II summarizes the results. In particular,
this study shows that for certain ranges of the parameters
which we determined, the optimal search strategy is intermittent. In other words, there is an optimal way for the intermittent searcher to tune the mean time it spends in each of the
two phases. We found that the optimal durations of the two
phases and the gain of intermittent search 共as compared to
one state search兲 do depend on the target density in dimension 1. In particular, the gain can be very high at low target
concentration. Interestingly, this dependence is smaller in dimension 2 and vanishes in dimension 3. The fact that intermittent search is more advantageous in low dimensions 共1
and 2兲 can be understood as follows. At large scale, the intermittent searcher of our model performs effectively a random walk and therefore scans a space of dimension 2. In an
environment of dimension 1 共and critically of dimension 2兲,
the searcher therefore oversamples the space, and it is favorable to perform large jumps to go to previously unexplored
areas. On the contrary, in dimension 3, the random walk is
transient, and the searcher on average always scans previously unexplored areas, which makes large jumps less beneficial.
Additionally, our results show that, for various modeling
choices of the slow reactive phase, there is one and the same
optimal duration of the fast nonreactive phase, which depends only on the space dimension. This further supports the
robustness of optimal intermittent search strategies. Such robustness and efficiency—and optimality—could explain why
intermittent trajectories are observed so often and in various
forms.
Financial support from ANR grant Dyoptri is acknowledged.
a 6D
V
τ1opt → ∞
τ2opt → 0
opt
b3
tm 3Da
ACKNOWLEDGMENT
APPENDIX A: DIFFUSIVE MODE IN DIMENSION 1
1. Exact results (cf. Sec. III B 2)
topt
m
a lw a y s in t e r m it t e n c e
` 3 ´1 p a
4
τ1opt 10
Vk
τ2opt 1.1 Va
«2
„q
1
1
b3
ak
4 + 54
√5ka
3
V 24
Mean first-passage time at the target exactly reads:
tm =
1 共 ␶ 1 + ␶ 2兲 N
,
3 ␤ 2b F
共A1兲
with
N = ␣1 + ␣2 + ␣3 + ␣4 + ␣5 + ␣6 + ␣7 ,
共A2兲
F = ␥1 + ␥2 + ␥3 + ␥4 ,
共A3兲
␣1 = L32兵关3L22共L21 − L22兲 + 2h2␤兴h冑␤S + 3L1L2共L42 − 2h2␤兲C其,
共A4兲
␣2 = − L1hL52共2␤ + 3L22兲RC,
3 D
τ2opt → 0
`b´
b2
topt
m 2D ln a
τ1opt → ∞
b<
D
V
b
2 D
topt
m
a lw a y s in t e r m it t e n c e
´1
p a ` `b´
opt
1 4
τ1 2V k ln a − 2
q ` ´
b
τ2opt Va ln a
− 12
„
q «2
√ ` ` b ´´ 1
b2
V
topt
2 ln a 4 +
m aV
ak
b2
3D
topt
m τ2opt → 0
τ1opt → ∞
1 D
D
V a
2
opt
b
τ1 D w24(4lnlnw−5+c
w−7+c)
√
τ2opt Vb 4 ln w−5+c
w
B
2
topt
m
q ` ´
2
b
2b
ln a
aV
τ2opt τ1opt τ1opt → 0
opt
τ2 1.1 Va
3
opt
tm 2.18 Vba2
1
2
τ2opt → 0
τ1opt → 0
q ` ´
b
τ2opt Va ln a
−
q
`
´
2
opt
b
tm 2b
ln a
aV
τ1opt → ∞
vl > vlc
D
V
“ ””2
ln b
D
“a”
2
2V
2 ln b −1
q ` a´
a
b
ln a
− 12
V
ba
“
b
vl
topt
m b
a
topt
m √2b
3V
q
a
V
τ2opt b
3a
τ2opt → 0
τ1opt → 0
q
b
τ2opt Va
3a
q
2b
b
topt
m V
3a
`
`
´´− 1
b
2
vl < vlc πV
ln a
4
τ1opt → ∞
q
Db
48V 2 a
τ1opt 3a
b
q
V
2
vl < vlc B a llis t ic m o d e
vl > vlc
b D
a V
q
a
D
V ,
b>
D i u s ivq
e m o d e
b D
a
a V
“ 2 ”1
3
b D
τ1opt 36V
4
“ 2 ”1
D 3
τ2opt 2b
9V 4
“ 5 4 ”1
3
3 b
topt
m 24 DV 2
D
V ,
b>
D
V
b<
S t a t ic m o d e
a lw a y s in t e r m it t e n c e
p a ` b ´1
4
τ1opt Vk
12a
q
opt
a
b
τ2 V
3a
„q
«
q
` 3a ´1/4 2
b
b
2ak
ak
+
3a
V
b
TABLE II. Recapitulation of main results: strategies minimizing the mean first-passage time on the target. In each cell, validity of the regime, optimal ␶1, optimal ␶2, and minimal tm
opt
共tm with ␶i = ␶opt
i 兲. Yellow background highlights the value of ␶2 independent from the description of the slow detection phase 1. Results are given in the limit b Ⰷ a.
ROBUSTNESS OF OPTIMAL INTERMITTENT SEARCH …
031146-21
共A5兲
PHYSICAL REVIEW E 80, 031146 共2009兲
LOVERDO et al.
␣3 = L1共2h4␤2 − 3L82兲BC,
共A6兲
␣4 = h2冑␤关6L62 + h2␤共␤ + L21兲兴RS,
共A7兲
␣5 = 冑␤hL32关4h2␤ + 3L22共L22 − L21兲兴BS,
共A8兲
␣6 = L1L32兵3共2h2L2␤ + L52兲B + h关3L22共␤ + L22兲 + 2h2␤兴R其,
part of the system: b / L1 Ⰷ 1. Alternatively, having a ballistic
phase of the size of the system is a waste of time, thus close
−2
to the optimum b / L2 Ⰷ 1. Consequently 2共b − a兲冑L−2
1 + L2
Ⰷ 1.
We use the numerical results 共Table III兲 to make assumpopt
tions on the dependence of ␶opt
1 and ␶2 with the parameters.
We define k1 and k2,
␶1 = 共k1兲−1
共A9兲
␣7 = − L1共3L82 + 2h4␤2兲,
共A10兲
␥1 = L32L1R共C − 1兲,
共A11兲
␥2 = 冑␤h共2L21 + L22兲RS,
共A12兲
␥3 = 冑␤L32共B − 1兲S,
共A13兲
␥4 = 2h␤L1共BC − 1兲,
共A14兲
冉 冊
B = cosh
2a
,
L2
−2
C = cosh共2h冑L−2
1 + L2 兲,
冉 冊
共A15兲
␶2 = 共k2兲−1
冉 冊
冉 冊
b 2D
V4
1/3
b 2D
V4
1/3
,
共A23兲
.
共A24兲
We make a development of tm for b Ⰷ a. We suppose k1 and
k2 do not depend on b / a,
tm =
冉 冊
1 D bV
3 V2 D
4/3
k1 + k2 2
共k + 3冑k1兲.
k 1k 2 2
共A25兲
We checked that this expression gives a good approximation
of tm in this regime, in particular around the optimum 共Fig. 4
in Sec. III B 4兲.
Derivatives of Eq. 共43兲 as a function of k1 and k2 must be
equal to 0 at the optimum. It leads to
共A16兲
3
冑
− 3k3/2
1 + 3k2 + 3 k1k2 = 0,
共A26兲
3
2
3k3/2
1 − 2k2 − k1k2 = 0.
共A27兲
2a
,
R = sinh
L2
共A17兲
−2
S = sinh共2h冑L−2
1 + L2 兲,
共A18兲
␤ = L21 + L22 ,
共A19兲
L1 = 冑D␶1 ,
共A20兲
L2 = V␶2 ,
共A21兲
4. Details of the optimization of the universal intermittent
regime bD2 Õ a3V2 ™ 1 (cf. Sec. III B 5)
h = b − a.
共A22兲
We start from the exact expression of tm 关Eq. 共A1兲兴. We
have to make an assumption on the dependency of ␶opt
2 with
a
b
b and a. We define f by ␶2 = 1f V 冑 3a and we suppose that f is
independent from a / b. We make a development of a / b → 0.
The first two terms give
2. Numerical study (cf. Sec. III B 2)
We studied numerically the optimum of the exact tm expression 关Eq. 共A1兲兴 共Table III兲. We could distinguish three
regimes: one with no intermittence and two with favorable
intermittence but with different scalings. Intermittence is favorable when b ⬎ DV . The demarcation line between the two
intermittent regimes is bD2 / a3V2 = 1.
3. Details of the optimization of the regime where
intermittence is favorable, with bD2 Õ a3V2 š 1 (cf. Sec. III B 4)
We suppose that target density is low: ba Ⰶ 1.
We are interested in the regime where intermittence is
−2
favorable. We have both 2共b − a兲冑L−2
1 + L2 ⬎ 2关共b − a兲 / L1兴
−2
−2
and 2共b − a兲冑L1 + L2 ⬎ 2关共b − a兲 / L2兴. In a regime of intermittence, one diffusion phase does not explore a significant
On four pairs of solutions, only one is strictly positive,
␶opt
1 =
冑
冑
1 3 2b2D
,
2
9V4
3
␶opt
2 =
tm ⯝
b
a
冉冑
2b2D
.
9V4
冊
a + af 2 + 冑D␶1 f 2
ab 1
+ ␶1
.
3 Vf
a + 冑D ␶ 1
共A28兲
共A29兲
共A30兲
This expression gives a very good approximation of tm in the
bD2 / 共a3v2兲 Ⰶ 1 regime, especially close to the optimum 共Fig.
5 in Sec. III B 5兲.
We then minimize tm 关Eq. 共A30兲兴 as a function of f and
␶1. We introduce w defined as
w=
aV
D
冑
a
.
b
共A31兲
We make an assumption on the dependency of ␶opt
1 with a / b,
inferred via the numerical results
031146-22
ROBUSTNESS OF OPTIMAL INTERMITTENT SEARCH …
PHYSICAL REVIEW E 80, 031146 共2009兲
TABLE III. Diffusive mode in one dimension. Optimization of tm as a function of ␶1 and ␶2 for different sets of parameters 共D = 1, V
= 1兲. For each 共a , b兲, numerical values for the exact analytical function 关Eq. 共A1兲兴 are given with the values expected in the regimes where
opt
intermittence is favorable, either with bD2 / a3V2 Ⰷ 1 共th , 1兲 or with bD2 / a3V2 Ⰶ 1 共th , 2兲. gainth,1 关Eq. 共47兲兴, gain= tm
/ tdif f , and gain2,th 关Eq.
th,1
th,2
th,1
th,2
opt
opt
共52兲兴. ␶1 关Eq. 共44兲兴, ␶1 , and ␶1 关Eq. 共49兲兴. ␶2 关Eq. 共45兲兴, ␶2 , and ␶2 关Eq. 共50兲兴. Colored backgrounds indicate the regime: dark red
when intermittence is not favorable, medium green in the bD2 / a3V2 Ⰷ 1 regime, and light blue in the bD2 / a3V2 Ⰶ 1 regime.
gainth,1
gain
gainth,2
th,1
τ1
opt
τ1
th,2
τ1
th,1
τ2
opt
τ2
th,2
τ2
100
103
104
105
106
107
0.085
1
0.014
0.39
1
0.046
1.8
2.1
0.14
8.5
8.7
0.46
39
40
1.4
180
180
4.6
0.19
0.89
4.1
19
89
410
∞
∞
6.1
21
90
410
2.1
21
210
2100
21000
2.1.105
0.38
1.8
8.2
38
180
820
0
0
8.4
38
180
820
0.029
0.091
0.29
0.91
2.9
9.1
gainth,1
gain
gainth,2
th,1
τ1
opt
τ1
th,2
τ1
th,1
τ2
opt
τ2
th,2
τ2
1.8
2.4
1.4
8.5
9.4
4.6
39
41
14
180
190
46
850
850
140
4000
4000
460
4.1
19
89
410
1900
8900
3.5
15
78
390
1900
8800
2.1
21
210
2100
21000
2.1.105
8.2
38
180
820
3800
18000
7.6
36
170
810
3800
18000
2.9
9.1
29
91
290
910
gainth,1
gain
gainth,2
th,1
τ1
opt
τ1
th,2
τ1
th,1
τ2
opt
τ2
th,2
τ2
39
150
140
180
470
460
850
1500
1400
3900
5500
4600
18000
21000
14000
89
410
1900
8900
41000
85000
91000
46000
1.9.105
2.2
22
230
2500
21000
2.1
21
210
2100
21000
180
820
3800
18000
82000
290
980
3500
15000
72000
3.8.105
3.6.105
290
910
2900
9100
29000
91000
gainth,1
gain
gainth,2
th,1
τ1
opt
τ1
th,2
τ1
th,1
τ2
opt
τ2
th,2
τ2
850
15000
14000
3900
46000
46000
18000
1.4.105
1.4.105
1900
8900
41000
85000
4.6.105
4.6.105
1.9.105
3.9.105
1.5.105
1.4.106
8.9.105
1.8.106
4.7.106
4.6.106
4.1.106
2.2
21
210
2100
21000
2.1
21
210
2100
3.8.105
21000
1.8.106
b/a
a =
0 . 0 0 5
a =
0 . 5
a =
5 0
a =
5 0 0 0
3800
1800
29000
91000
82000
2.9.105
29000
91000
2.9.105
031146-23
9.2.105
9.1.105
2.9.106
2.9.106
1.4.105
2.1.105
2.2.105
2.1.105
8.2.106
9.8.106
9.1.106
PHYSICAL REVIEW E 80, 031146 共2009兲
LOVERDO et al.
s=
1 Db
.
␶1 V2 a
␶opt
1 =
共A32兲
We write Eq. 共A30兲 with these quantities. Its derivatives with
f and s should be equal to zero at the optimum. It leads to
opt
⯝
tm
共A33兲
冑3s3/2w2共f 2 − 1兲 = 0.
共A35兲
Consequently f = 1. We incorporate this result to Eq. 共A34兲,
12s3/2w2 − w2s2冑3 + 15ws + 6冑s = 0.
The relevant solution is
冉
1 冑u 5冑3 + 16w 4
ssol =
+
冑3 u + 冑3
3 w
3
冊
gain ⯝
冑
b a
=
3a V
冑
b
,
3a
b
.
a
共A42兲
共B1兲
h2共h + 3L1兲
,
3␣
共B2兲
␥1 =
␥2 =
When w → ⬁, ssol = 48. As we made the assumption
bD2 / a3V2 = w−2 Ⰶ 1, difference from the asymptote will be
small 共Fig. 22兲.
It leads to
1 a
= opt
f V
共A41兲
␶1 + ␶2
共␥1 + ␥2 + ␥3兲,
b
共A37兲
with
␶opt
2
冑
.
Mean first-passage time at the target exactly reads:
2
共A38兲
1 aV
D
2 冑3
3/2
APPENDIX B: BALLISTIC MODE IN ONE DIMENSION:
EXACT RESULT (cf. SEC. III C 2)
共A36兲
u = 共270w + 27冑3 + 192w2冑3 + 9冑55冑3w + 84w2 + 27兲w.
b
V 3 a
It is in very good agreement with numerical data 共Table III兲.
Gain can be very large if the density is low.
tm =
,
冑 冉 冊
2a
We get
共A34兲
We take Eq. 共A33兲 and here we need to make the assumption
than a Ⰶ b Ⰶ a3V2 / D2. We get
共A40兲
It corresponds to numerical results 共Table III兲.
We use Eqs. 共41兲 and 共48兲 to calculate the gain,
− 冑3s3/2w2 + 冑3s3/2w2 f 2 + 冑3swf 2 + 6冑swf 3 + 6f 3 = 0,
6s3/2w2 f 3 + 6s3/2 fw2 − w2s2冑3 + 3wsf + 12wsf 3 + 6冑sf 3 = 0.
Db
Db
=
.
s V a 48V2a
opt 2
L2共h + L1兲
冑
关g4共− 1兲e−2a/L2 − g4共1兲兴关g3共− 1兲e2共 ␣a/L1L2兲
␣3/2F
+ g3共1兲e2共
冑␣b/L1L2兲
F = g1共1兲e−2关a共L1−
兴,
共B3兲
冑␣兲/L1L2兴
+ g2共1兲e−2关共aL1−
+ g1共− 1兲e2共
冑␣b兲/L1L2兴
共A39兲
␥3 = −
冑␣b/L1L2兲
+ g2共− 1兲e2共
冑␣a/L1L2兲
, 共B4兲
L 2 N 1N 2
,
4␣3/2 F1F2
共B5兲
N1 = f 1共1兲 + f 1共− 1兲e−2a/L2 + ␴1 + ␴2 + ␴3 + ␴4 ,
共B6兲
f 1共⑀兲 = 2关␣g4共⑀兲共h + L1兲 + L42共⑀L2 − L1兲兴,
共B7兲
␴1 = 关f 2共− 1兲 + f 4共1,1兲 + f 3共1兲兴e
冑␣2h/L1L2
␴2 = 关f 2共1兲 + f 4共1,− 1兲 + f 3共− 1兲兴e−
冑␣2h/L1L2
␴3 = 关f 4共− 1,1兲 + f 5共1兲 + f 6共1兲兴e2关共−aL1+
共B8兲
,
共B9兲
,
冑␣h兲/L1L2兴
,
共B10兲
␴4 = 关f 4共− 1,− 1兲 + f 5共− 1兲 + f 6共− 1兲兴e−2关共aL1+
冑␣h兲/L1L2兴
,
共B11兲
FIG. 22. 共Color online兲 Diffusive mode in one dimension. ssol
关Eq. 共A37兲兴 共red, line兲 as a function of ln共w兲, with the asymptote
共blue, dotted line兲.
031146-24
f 2共⑀兲 = 共冑␣ + ⑀L2兲L2共L1 − L2兲g3共⑀兲,
共B12兲
f 3共⑀兲 = − L22L1冑␣共h + L1兲共冑␣ + ⑀L2 + ⑀L1兲,
共B13兲
ROBUSTNESS OF OPTIMAL INTERMITTENT SEARCH …
PHYSICAL REVIEW E 80, 031146 共2009兲
冉 冊 冉 冑冊
冉 冊 冉 冑冊
f 4共⑀1, ⑀2兲 = h␣共h + L1兲关共2L2 + ⑀1L1兲共⑀2冑␣ + L2兲 + L21兴,
␰6 = − L2␣共2h + L1兲sinh
共B14兲
f 5共⑀兲 = − ⑀h冑␣L2共L1 + L2兲关2共⑀冑␣ + L2兲L2 + L21兴,
2a
2h ␣
cosh
,
L2
L 1L 2
␰7 = − 冑␣关共2h + L1兲␣ + L31兴sinh
共B15兲
2a
2h ␣
sinh
,
L2
L 1L 2
共B29兲
f 6共⑀兲 = L22L1关L2共⑀冑␣ + L2兲共L1 + L2兲
F2 = g1共1兲e−2关a共L1−
− 冑␣共h + L1兲共冑␣ + ⑀L2 − ⑀L1兲兴,
共B28兲
共B16兲
冑␣兲/L1L2兴
+ g2共1兲e−2关共aL1−
+ g1共− 1兲e2共
冑␣b兲/L1L2兴
冑␣b/L1L2兲
+ g2共− 1兲e2共
冑␣a/L1L2兲
,
共B30兲
N2 = ␵1 + ␵2 − g4共1兲e
− g4共− 1兲e
2a/L2
−2a/L2
关f 7共1兲 + f 8共1兲兴
关f 7共− 1兲 + f 8共− 1兲兴,
g1共⑀兲 = L2共L1 + L2兲共冑␣ − ⑀L2兲,
共B17兲
g2共⑀兲 = 冑␣共2h + L1兲共− ⑀冑␣ − L2 + ⑀L1兲 + ⑀␣3/2 + L32 − ⑀L31 ,
冑
冑
␵1 = 2冑␣关共L22 − h2兲␣ − L31h兴关e2共 ␣a/L1L2兲 + e2共 ␣b/L1L2兲兴,
共B32兲
共B18兲
␵2 = 2L2关h共h + L1兲␣ −
冑
L42兴关e2共 ␣a/L1L2兲
−e
2共冑␣b/L1L2兲
兴,
共B19兲
f 7共⑀兲 = 关␣ + 共L1 + ⑀L2兲h兴冑␣关e2共
冑␣a/L1L2兲
+ e2共
冑␣b/L1L2兲
g3共⑀兲 = h冑␣共− ⑀冑␣ − L2兲 + ⑀2L22L1 ,
共B33兲
g4共⑀兲 = 关共⑀L2 − L1兲h + L22兴,
共B34兲
h = b − a,
共B35兲
␣ = L21 + L22 ,
共B36兲
L 1 = v l␶ 1 ,
共B37兲
L2 = V␶2 .
共B38兲
兴,
共B20兲
f 8共⑀兲 = 共⑀h␣ + L32 + ⑀L31兲共− e2共
冑␣a/L1L2兲
+ e2共
冑␣b/L1L2兲
兲,
共B21兲
F1 = 冑␣共␰1 + ␰2 + ␰3 + ␰4 + ␰5 + ␰6 + ␰7兲,
共B22兲
␰1 = 2L1共h + L1兲␣ ,
共B23兲
冋
冉 冊册
冉 冑冊
␰2 = L2冑␣ 共␣ + L22兲sinh
2h ␣
2h冑␣
+ 2L2冑␣ cosh
L 1L 2
L 1L 2
,
共B24兲
冋 冉 冊
␰3 = L1L2 ␣ sinh
冉 冊册
2a
2a
− 2L1L2 cosh
L2
L2
, 共B25兲
This result have been checked by numerical simulations and
by comparison with known limits.
APPENDIX C: STATIC MODE IN THREE DIMENSIONS:
MORE COMPARISONS BETWEEN THE ANALYTICAL
EXPRESSIONS AND THE SIMULATIONS
(cf. SEC. V A 3)
The numerical study of the minimum mean search time
共Fig. 23兲 shows that the analytical values give the good position of the minimum in ␶1 and ␶2 as soon as b / a is not too
small. However, the value of the minimum is underestimated
by about 10%.
APPENDIX D: DIFFUSIVE MODE IN THREE
DIMENSIONS
冉 冊 冉 冑冊
␰4 = − L2冑␣共␣ + 2L1h + L22兲cosh
2a
2h ␣
sinh
,
L2
L 1L 2
1. Full analytical expression of tm (cf. Sec. V B 2)
共B26兲
The approximation of the mean first-passage time at the
target reads:
冉 冊 冉 冑冊
␰5 = − 2关L1共h + L1兲␣ + L42兴cosh
共B31兲
2a
2h ␣
cosh
,
L2
L 1L 2
tm =
共B27兲
with
031146-25
1
共X + Y + Z兲,
b3␣4dpD
共D1兲
PHYSICAL REVIEW E 80, 031146 共2009兲
LOVERDO et al.
(a)
(b)
FIG. 23. 共Color online兲 Static mode in three dimensions. Study of the minimum: 共a兲 its location in the ␶1 , ␶2 space and 共b兲 its value. sim
means values obtained through numerical simulations; th means analytical values. Value expected if there was a prefect agreement between
theory and simulations 共black line兲 and values taking into account the simulations noise 共dotted black lines兲 共we performed 10 000 walks for
each point兲. a = 0.01 共brown, squares兲, a = 0.1 共red, crosses兲, a = 1 共purple, circles兲, a = 10 共blue, stars兲, and a = 100 共green, diamonds兲. V = 1,
k = 1.
2
共␶−1
1 + ␣ dp兲
X=
冉
␣2共b3 − a3兲
− 3S
a
冉
␶1 共␶−1
1
+␣
2
冊冉
1/3
dp兲␶−1
2 R␣
␶−1共␣aR + 1兲 ␣dp共− 1 + TT兲
共b3 − a3兲共␣2dp − ␶−1
2 兲
+ 2
+
a
␣2
␣22
−1
−1
TT␣2dp共␶−1
共− ␣2dp + ␶−1
2 兲␶1
1 + ␶2 兲
+
+
a
a
,
共D2兲
␶−1
1 aS
,
␣2
共D3兲
2
1 共− b + a兲3␣2共a3 + 3ba2 + 6b2a + 5b3兲共␶−1
1 + ␣ dp兲
,
15
a
共D4兲
Y=3
Z=−
冊
冊
␣ = 冑共␶1D兲−1 + 共␶2D2兲−1 ,
共D5兲
D2 = 31 V2␶2 ,
共D6兲
DD2
,
D − D2
共D7兲
␣2 = 共␶2D2兲−1 ,
共D8兲
dp =
␣b tanh关␣共b − a兲兴 − 1
,
␣b − tanh关␣共b − a兲兴
共D9兲
共␣2ba − 1兲tanh关␣共b − a兲兴 + ␣共b − a兲
,
␣b − tanh关␣共b − a兲兴
共D10兲
R=
S=
031146-26
ROBUSTNESS OF OPTIMAL INTERMITTENT SEARCH …
PHYSICAL REVIEW E 80, 031146 共2009兲
TT =
␣ 2a
.
tanh共␣2a兲
2. Dependence of tm with ␶1
APPENDIX E: BALLISTIC MODE IN THREE
DIMENSIONS
The mean detection time is very weakly dependent on ␶1
as long as ␶1 ⬍ 6D / v2 共Fig. 24兲.
3. tm in the regime of diffusion alone (cf. Sec. V B 4)
We take a diffusive random walk starting from r = r0 in a
sphere with reflexive boundaries at r = b and absorbing
boundaries at r = a, we get the following equation for t共r0兲
the mean time of absorption:
Def f
1
r20
再 冋 册冎
d 2 dt共r0兲
r
dr0 0 dr0
t共r0兲 =
冉
1. Without intermittence (cf. Sec. V C 2)
In the regime without intermittence, ␶1 is not necessarily
0. We calculate tm in two limits: ␶1 small or ␶1 large.
a. Limit ␶2 \ 0, vl␶1 ⱕ a
In the limit vl␶1 ⱕ a, we can consider phase 1 as diffusive,
with
共D12兲
= − 1.
With the boundary conditions, the solution is
冊
1
2b3
2b3
.
+ a2 − r20 −
6Def f a
r0
共D13兲
D = 31 v2l ␶1 .
1
共5b3a3 + 5b6 − 9b5a − a6兲.
15Dab3
共D14兲
tm =
b3
.
3Da
1
共5b
5v2l ␶1ab3
a + 5b6 − 9b5a − a6兲.
3 3
共E2兲
And in the limit b Ⰷ a,
tm =
In the limit b / a Ⰷ 1,
tdif f =
共E1兲
We use the approached expression of tm obtained in the diffusive mode 关Eq. 共150兲兴 with this effective diffusive coefficient,
Then we average on r0, because the searcher can start from
any point of the sphere with the same probability,
tdif f =
共D11兲
共D15兲
b3
.
v2l ␶1a
共E3兲
b. Limit ␶2 \ 0, ␶1 \ ⴥ
We name Vol the volume of the sphere. g共t兲 is the volume
explored by the searcher after a time t. The volume explored
during dt is ␲vla2dt. If we consider that the probability to
4. Criterion for intermittence: Additional figure (cf. Sec.
V B 5)
opt
Figure 25 shows the dependence of tm
with a.
FIG. 24. 共Color online兲 Diffusive mode in three dimension. tm
from Eq. 共D1兲; tm共b Ⰷ a , ␶1 → 0兲 from Eq. 共144兲. ␶2 = ␶opt,th
关Eq.
2
共145兲兴, D = 1, V = 1, a = 10 共dotted lines兲, a = 100 共lines兲, a = 1000
共symbols兲, b / a = 10 共light blue, circles兲, and b / a = 100 共red,
squares兲.
FIG. 25. 共Color online兲 Diffusive mode in three dimensions.
Simulations: b / a = 2.5 共red, crosses兲, b / a = 5 共blue, squares兲, b / a
= 10 共green, circles兲, and b / a = 20 共brown, diamonds兲. Analytical
expressions in the low target density approximation 共b / a Ⰷ 1兲: ␶1
= 0 关Eq. 共144兲兴 关with ␶2 = ␶opt,th
; Eq. 共145兲兴 共dotted line兲; diffusion
2
alone 关Eq. 共150兲兴 共continuous line兲. V = 1, D = 1.
031146-27
PHYSICAL REVIEW E 80, 031146 共2009兲
LOVERDO et al.
FIG. 26. 共Color online兲 Ballistic mode in three dimensions. Regime without intermittence 共␶2 = 0兲. ln共tm / b3兲 as a function of
ln共␶1兲; simulations for b = 5 共green, squares兲 and b = 20 共blue,
circles兲. Ballistic limit 共␶1 → ⬁兲 共no intermittence兲 关Eq. 共152兲兴 共red
horizontal line兲. Diffusive limit 共v␶1 ⬍ a兲 关Eq. 共E2兲兴 with b = 5
共green, dotted line兲, b = 20 共blue ,small dots兲, and b Ⰷ a limit 关Eq.
共E3兲兴 共black line兲. a = 1, vl = 1.
encounter a unexplored space is uniform, which is wrong at
short times but close to the reality at long times, the average
of first explored volume at time t during dt is 共关Vol
− g共t兲兴 / Vol兲␲vla2dt. Then in this hypothesis, g共t兲 is solution
of
g共t兲 =
冕
t
0
Vol − g共u兲
␲vla2du.
Vol
冕
target is not yet found at time r is the solution of
dp
= − 关1 − f共u兲兴.
dr
tbal =
共E5兲
4b3
.
3a2vl
共E7兲
c. Numerical study
These expressions give a very good approximation of the
values obtained through simulations 共Figs. 21 and 26兲. In the
regime without intermittence, tm is minimized for ␶1 → ⬁.
2. Numerical vcl (cf. Sec. V C 4)
r
关1 − f共w兲兴dw.
共E6兲
As p共0兲 = 1, the result is p共r兲 = e−r. Then the renormalized
mean detection time of the target is 1, which in real time
reads
共E4兲
This equation can be simplified taking a renormalized time r
as r = 共␲vla2 / Vol兲t and f = g / Vol,
f共r兲 =
FIG. 27. 共Color online兲 Ballistic mode in three dimensions. vcl as
a function of ln共b兲 through simulations. a = 1, V = 1.
Then, as f共0兲 = 0 共nothing has been explored at time 0兲,
f共r兲 = 1 − e−r. The probability to encounter the target at time t
during dt 共and not before兲 is the newly explored volume at
time t divided by the whole volume Vol if we make the
mean-field approximation. Then the probability p共r兲 that the
In simulations 共Fig. 27兲, when b is small vcl decreases, but
it stabilizes for larger b, which is coherent with the fact that
this value is obtained through a development in b. The value
of vl for large b is different 共even if close兲 to the expected
value. The main explanation of this discrepancy is that in the
intermittence regime, the approached value of tm is about
20% away from the value obtained through simulations.
关1兴 J. R. Frost and L. D. Stone, Review of search theory: Advances
and applications to search and rescue decision support, 2001,
http.//www.rdc.uscg.gov/reports/2001/cgd1501dpexsum.pdf
关2兴 G. M. Viswanathan, S. V. Buldyrev, S. Havlin, M. G. E. Da
Luz, E. P. Raposo, and H. E. Stanley, Nature 共London兲 401,
911 共1999兲.
关3兴 A. M. Edwards, R. A. Phillips, N. W. Watkins, M. P. Freeman,
E. J. Murphy, V. Afanasyev, S. V. Buldyrev, M. G. E. Da Luz,
E. P. Raposo, H. E. Stanley, and G. M. Viswanathan, Nature
共London兲 449, 1044 共2007兲.
关4兴 G. M. Viswanathan, E. P. Raposo, and M. G. E. da Luz, Phys.
Life. Rev. 5, 133 共2008兲.
关5兴 M. F. Shlesinger, Nature 共London兲 443, 281 共2006兲.
关6兴 O. Bénichou, M. Coppey, M. Moreau, P.-H. Suet, and R. Voituriez, Phys. Rev. Lett. 94, 198101 共2005兲.
关7兴 O. Bénichou, M. Coppey, M. Moreau, P. H. Suet, and R. Voituriez, J. Phys.: Condens. Matter 17, S4275 共2005兲.
关8兴 O. Bénichou, C. Loverdo, M. Moreau, and R. Voituriez, Phys.
Rev. E 74, 020102共R兲 共2006兲.
关9兴 O. Benichou, C. Loverdo, M. Moreau, and R. Voituriez, J.
Phys.: Condens. Matter 19, 065141 共2007兲.
关10兴 O. Benichou, C. Loverdo, M. Moreau, and R. Voituriez, Phys.
0
031146-28
ROBUSTNESS OF OPTIMAL INTERMITTENT SEARCH …
PHYSICAL REVIEW E 80, 031146 共2009兲
Chem. Chem. Phys. 10, 7059 共2008兲.
关11兴 M. A. Lomholt, T. Koren, R. Metzler, and J. Klafter, Proc.
Natl. Acad. Sci. U.S.A. 105, 11055 共2008兲.
关12兴 M. Slutsky and L. A. Mirny, Biophys. J. 87, 4021 共2004兲.
关13兴 M. Coppey, O. Bénichou, R. Voituriez, and M. Moreau, Biophys. J. 87, 1640 共2004兲.
关14兴 I. Eliazar, T. Koren, and J. Klafter, J. Phys.: Condens. Matter
19, 065140 共2007兲.
关15兴 M. A. Lomholt, T. Ambjornsson, and R. Metzler, Phys. Rev.
Lett. 95, 260603 共2005兲.
关16兴 I. Bonnet, A. Biebricher, P.-L. Porte, C. Loverdo, O. Benichou,
R. Voituriez, C. Escude, W. Wende, A. Pingoud, and P. Desbiolles, Nucleic Acids Res. 36, 4118 共2008兲.
关17兴 O. Bénichou, C. Loverdo, and R. Voituriez, EPL 84, 38003
共2008兲.
关18兴 B. van den Broek, M. A. Lomholt, S. M. J. Kalisch, R. Metzler, and G. J. L. Wuite, Proc. Natl. Acad. Sci. U.S.A. 105,
15738 共2008兲.
关19兴 C. Loverdo, O. Benichou, R. Voituriez, A. Biebricher, I. Bonnet, and P. Desbiolles, Phys. Rev. Lett. 102, 188101 共2009兲.
关20兴 W. J. Bell, Searching Behaviour: The Behavioural Ecology of
Finding Resources 共Chapman and Hall, London, 1991兲.
关21兴 W. J. O’Brien, H. I. Browman, and B. I. Evans, Am. Sci. 78,
152 共1990兲.
关22兴 D. L. Kramer and R. L. McLaughlin, Am. Zool. 41, 137
共2001兲.
关23兴 J. T. Pierce-Shimonura, T. M. Morse, and S. R. Lockery, J.
Neurosci. 19, 9557 共1999兲.
关24兴 C. Loverdo, O. Benichou, M. Moreau, and R. Voituriez, Nat.
Phys. 4, 134 共2008兲.
关25兴 B. Alberts, Molecular Biology of the Cell 共Garland, New York,
2002兲.
关26兴 S. Huet, E. Karatekin, V. S. Tran, I. Fanget, S. Cribier, and J.-P.
Henry, Biophys. J. 91, 3542 共2006兲.
关27兴 O. Benichou, M. Moreau, P.-H. Suet, and R. Voituriez, J.
Chem. Phys. 126, 234109 共2007兲.
关28兴 G. Oshanin, H. S. Wio, K. Lindenberg, and S. F. Burlatsky, J.
Phys. Condens. Matter 19, 065142 共2007兲.
关29兴 F. Rojo, C. E. Budde, and H. S. Wio, J. Phys. A: Math. Theor.
42,125002 共2009兲.
关30兴 P. Levitz, M. Zinsmeister, P. Davidson, D. Constantin, and O.
Poncelet, Phys. Rev. E 78, 030102共R兲 共2008兲.
关31兴 A. James, M. J. Planck, and R. Brown, Phys. Rev. E 78,
051128 共2008兲.
关32兴 M. Moreau, G. Oshanin, O. Benichou, and M. Coppey, Phys.
Rev. E 67, 045104 共2003兲.
关33兴 M. Moreau, G. Oshanin, O. Benichou, and M. Coppey, Phys.
Rev. E 69, 046101 共2004兲.
关34兴 M. Moreau, O. Bénichou, C. Loverdo, and R. Voituriez, EPL
77, 20006 共2007兲.
关35兴 M. Moreau, O. Bénichou, C. Loverdo, and R. Voituriez, J.
Phys. A: Math. Theor. 共to be published兲.
关36兴 C. W. Gardiner, Handbook of Stochastic Methods: For Physics, Chemistry and the Natural Sciences 共Springer, New York,
1996兲.
关37兴 A. Dussutour, J. L. Deneubourg, and V. Fourcassie, Proc. R.
Soc. London, Ser. B 272, 705 共2005兲.
关38兴 A. M. Berezhkovskii, V. Zaloj, and N. Agmon, Phys. Rev. E
57, 3937 共1998兲.
关39兴 S. Redner, A Guide to First Passage Time Processes 共Cambridge University Press, Cambridge, 2001兲.
关40兴 F. Bartumeus, F. Peters, S. Pueyo, C. Marrase, and J. Catalan,
Proc. Natl. Acad. Sci. U.S.A. 100, 12771 共2003兲.
关41兴 K. Kiontke and W. Sudhaus, in Ecology of Caenorhabditis
species, edited by The C. elegans Research Community 共http://
www.wormbook.org, 2005兲.
031146-29