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First published online October 19, 2007
Journal of Experimental Biology 210, 3763-3770 (2007)
Published by The Company of Biologists 2007
doi: 10.1242/jeb.009563
Honeybees perform optimal scale-free searching flights when attempting to locate a food source
1 Rothamsted Research, Harpenden, Hertfordshire, AL5 2JQ, UK
2 Natural Resources Institute, University of Greenwich, Chatham, Kent, ME4
4TB, UK
* Author for correspondence (e-mail: andy.reynolds{at}bbsrc.ac.uk)
Accepted 16 August 2007
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Summary |
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 TOP Summary IntroductionMaterials and methodsResultsDiscussionReferences |
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Key words: optimal search strategy, imprecise Lévy-flight, honeybee, Apis mellifera, harmonic radar tracking
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Introduction |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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) or do
they start searching anew? In the case of honeybees (Apis mellifera
L.), there is also the option of returning to the hive and gathering
information from other bees about resource locations, by means of the waggle
dance (Biesmeijer and Seeley,
2005). In reality, the honeybee responses are likely to be a
combination of the above strategies; Biesmeijer and Seeley showed that
scouting and recruiting behaviour of individual honeybees is flexible such
that one bee is not constrained to be a recruit or a scout but can perform
either role, depending on circumstances
(Biesmeijer and Seeley,
2005). In addition to the resource depletion problem, slight navigational errors might also result in a bee arriving at a location slightly different from the actual location of known forage. In both of these scenarios, we might expect the bee to perform a search strategy to re-find the known food source, or to find a new nearby source.
It is now possible to track individual flying bees as they forage, using
harmonic radar (Riley et al.,
1999; Riley et al.,
2003; Riley et al.,
2005), enabling us to investigate the search strategies used by
bees when a forage source runs out. We hypothesise that when a food source at
a known location ceases to be available, flying bees will spend some time in
an optimised search of the vicinity for the resource in question before they
(eventually) return to the hive (where they may or may not gather information
on other resource localities from nest mates). These local search patterns are
described mathematically, using Lévy-flight theory, in this paper.
Lévy-flights are comprised of random sequences of independent flight
segments whose lengths, l, are drawn from a probability distribution
function having a power-law tail,
P(l)
l–µ, where
1<µ<3 [Bouchaud and Georges
(Bouchaud and Georges, 1990)
and references therein]. When µ
3, the distribution of the total length
of any number of flight segments is Gaussian, by virtue of the central limit
theorem, whilst µ
1 does not correspond to probability distributions
that can be normalised. Lévy-flights have a remarkable statistical
property: namely that distributions of the total length, L, of any
number of flight segments have power-law tails,
P(L)L–µ.
Consequently, Lévy-flights are said to be `scale-free' because their
statistical properties do not depend upon the observational scale. This
absence of a characteristic scale makes Lévy-flights scale-invariant
fractals. Levy-flights may have been observed in the movement patterns of
wandering albatrosses, deer, foraging bumble bees
(Viswanathan et al., 1996;
Viswanathan et al., 1999), a
species of African jackal (Atkinson et al.,
2002), foraging spider monkeys
(Ramos-Fernández et al.,
2004) and Drosophila flying in a small circular arena
(Reynolds and Frye, 2007).
These freely roaming Lévy-flight movement patterns are known
to constitute an optimal searching strategy for the location of randomly and
sparsely distributed targets (Viswanathan
et al., 1999). That is, they minimise the mean distance travelled
before first encountering a target. Sub-optimal Lévy-flight searches
with µ
2 and Gaussian (diffusive) searches (µ=3) can be up to 10
times longer than optimal Lévy-flight searches
(Viswanathan et al., 1999).
Recently, Reynolds et al. reported that honeybees adopt µ=2
Lévy-flight looping patterns when attempting to locate their
hive and when deprived of navigational cues
(Reynolds et al., 2007). In
contrast to their freely roaming counterpart, Lévy looping flights
extend out from and return back to a fixed location, around which the search
is centred. Lévy looping flights with µ=2 are optimal for the
location of a single target location when the most likely location of the
target is known or is presumed known (A.M.R., manuscript submitted). The
equidistant spiral (Archimedian) search would be another alternative optimal
pattern. If the bees were using this then it would be visible in the tracked
flight patterns.
In this paper, we show how Lévy-flight looping patterns underlie the flight paths of honeybee foragers searching a local area for a known food source. These flight patterns can be associated explicitly with the adoption by the bees of an optimal scale-free searching strategy for the location of a single target. The theory of random Lévy-flight searching is advanced by the formal demonstration that these searches will remain optimal despite errors due to imperfections in the bees' navigation system.
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Materials and methods |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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; Giurfa and Menzel,
1997). There was very little natural forage on the experimental
site, and no honeybees were seen foraging in the vicinity prior to the start
of our experiments. The honeybees to be tracked were foragers from a small
colony of 5000 workers, housed in a brood box mounted on a stand. The
hive was fitted with a transparent Perspex® entrance tunnel with removable
doors and roofs, enabling bees to be easily captured or returned to the hive
or the hive to be completely closed. The front of the hive stand and the hive
roof were painted white to aid visibility.
Bees were trained to a feeder, a glass jar containing 70% sucrose solution,
inverted onto a grooved Perspex plate. The feeder was placed on a small wooden
board 46 cm2 in the grass. A visual resolution of 1–2° is
possible for the bee compound eye (Giurfa
and Menzel, 1997) so the board would have been visible to the bee
from a maximum distance of 18.6 m; and probably a lot less (7 m) since Giurfa
et al. (Giurfa et al., 1996)
suggest that 5° is a more realistic approximation of visual resolution
once spectral contrast is taken into account. In the experiments, the feeder
was removed in each of the test situations so that the bees were not
responding to visual or olfactory cues from the feeder itself.
The feeder was positioned 210 m from the hive. In order to minimise the effect of navigational cues, such as any distant features, each evening after the end of the experimental work, the hive, and hence the entire experimental set-up, was moved 30 m to the east or west. This did not appear to affect the bees' ability to find the feeder or to find their way home. After moving the hive, no orientation flights were observed, and no bees were seen searching around the previous site of the hive or feeder the following morning.
The observations were carried out on marked bees that had been observed
several times at the feeder and were therefore regular foragers. When one of
these bees was about to leave the hive, she was captured and fitted with a
harmonic radar transponder, as described by Capaldi et al.
(Capaldi et al., 2000). The
feeder (and the bees feeding on it) was removed from the experimental arena,
and the bee with transponder was then released from the hive entrance. Her
flight route was tracked using scanning harmonic radar
(Riley et al., 1996;
Riley and Smith, 2002). Only
one such regular forager was tracked at a time and the transponder was removed
when the bee returned to the hive. The errors in fixing positions of the bees
were approximately ±3 m in range and ±1.3 m in azimuth (at a
range of 300 m). Bees can be detected by the harmonic radar when flying below
about 10 m (depending on range and terrain). We are unable to determine the
precise height of flight but there is no evidence in this study or in others
we have done (Reynolds et al.,
2007) to suggest that bees fly higher when searching. Rather, the
height of flight may be related to the maintenance of a certain optic flow
(Riley et al., 1999).
Some bees were trained to the feeder when a landmark (a white vehicle approximately 4 m long x 2 m high) was positioned broadside-on either 150 m from the hive, but offset from the direct line between the hive and feeder by 12°, or 160 m from the hive on the direct line between the hive and feeder. In some cases, the landmark was removed when the feeder was removed.
Each of the 39 tracks used in this study comprised a `vector' flight
(Riley et al., 2003) from the
hive to the vicinity of the former position of the feeder (`virtual feeder'),
the ensuing searching flights and a return flight. A further 12 flights were
excluded from the analysis either because the radar recordings were incomplete
or because the bees flew around the hive rather than making an immediate
vector flight towards the virtual feeder.
Analysis method for the honeybee search flights
After release, most bees flew immediately to the vicinity of the virtual
feeder before engaging in long, looping flights indicative of searching. After
a period of searching, most bees returned to the hive (see the example shown
in Fig. 1). We carried out an
analysis of the bees' flight tracks to determine how they compared with what
is known about optimal search strategies. To do this, we represented the
flight paths as sequences of straight-line segments between the points at
which significant changes in direction occur (see
Fig. 1). A `searching' flight
is defined to be the entire flight pattern that arises after the first
significant change in flight orientation and before the last significant
change in flight orientation.
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The representations of the searching flights (see
Fig. 1) were analysed in detail
by using `random walk methods', which can detect the presence of long-term
correlations. Our first analysis is based on the fact that the number of
turning points occurring within the time intervals t to
t+
t defines a time series, u(t),
and an associated net `displacement':
 |
,
where 
=
and where the angular brackets denote an ensemble
average over all flights in the data set
(Peng et al., 1995).
Short-term correlations in the data may cause the initial slope of a plot of
log (F)/log(t) to differ from , although it will
still approach at longer times. Long-term power-law correlations,
however, will generate values . Our subsequent analyses
consist of a determination of the fractal dimension of the represented
honeybee flight patterns and an examination of the lengths and durations of
the straight-line segments in those representations. |
Results |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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). The
vector flights to the virtual feeder made by three groups of bees (i.e.
flights with no landmark present, and with the landmark in one of the two
locations) are statistically indistinguishable. This suggests that the bee's
`vector navigation' system was accurate over these scales and was unaffected
by removal of the landmark during our experiment. The mean (± s.d.)
duration of the searching flights was 249±164 s.
Fig. 2D shows that some, but
not all, of the `return' flights to the hive were beeline vector flights that
emanated from the vicinity of the virtual feeder and approximately retraced
the initial outward beeline vector flight. The figure also shows that some of
the return flights over- or undershot the hive location and that these bees
arrived at the hive location only after first having executed local looping
flight manoeuvres.
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, of flight segments in the searching flights
were, however, uniformly distributed between 0° and 360°
(Fig. 4).
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Fig. 5 shows that for our
data the index is equal to 0.85, and, as explained above, this implies
that long-term power-law correlations exist in the data, or, in other words,
the bee flight patterns were similar on all temporal scales. The presence of
this scale-free characteristic is confirmed by the fractal scaling property of
the `represented' honeybee flights shown in
Fig. 6. Scaling properties of
the first and second halves of the searching flights are statistically
indistinguishable. This suggests that the search pattern does not change with
time. The territory covered does, however, tend to increase with time.
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The observed scale invariance of the representations of the honeybee
flights can be understood within the context of a Lévy-flight
(scale-free) model. In this model, the orientations, , of independent
straight-line flights are, in accordance with observations
(Fig. 4), drawn at random from
a uniform distribution P()=1/2
for
0<2. Flight lengths, l, are drawn at random from a
Levy-distribution
P=(µ–1)l0µ–1l–µ
for ll0 and P(l)=0 for
l<l0 where 1<µ<3.
Fig. 6 shows that the observed
fractal scaling D=1.3 is close to the value of 1.2 predicted by the
model when µ=2 (Reynolds et al.,
2007). Note that the fractal dimension of these finitely long
Lévy-flights differs from the fractal dimension of infinitely long
Lévy-flights, D=µ–1
(Reynolds et al., 2007).
Viswanathan et al. showed that such a model also reproduces the observed
power-law scaling (=0.85) of the root-mean-square displacement,
F (Viswanathan et al.,
1996). The lengths of the straight-line flights in the
representations of the honeybee flights are seen in
Fig. 7 to be distributed
according to a Lévy stable distribution (Cauchy distribution). The tail
of this distribution obeys an inverse-square law. This corresponds to µ=2,
the optimal value for the location of a single target (A.M.R., manuscript
submitted). A Cauchy distribution of flight-segment lengths would arise if
each of the Lévy flight-segments resolved by the analysis were
actually comprised of many shorter unresolved Lévy
flight-segments (Gnedenko and Kolmogorov,
1954). A Cauchy distribution may also arise if movement patterns
were exclusively associated with the adoption of an optimal scale-free
searching strategy. This is because a Cauchy distribution constitutes a
least-biased choice for a distribution with an inverse-square-law tail
(Alemany and Zanette,
1994).
|
Fig. 8 shows that relatively
short flight segments tend to be associated with relatively slow speeds whilst
longer flight segments tend to be associated with faster speeds. This
correlation between flight-segment length and speed cannot be attributed to
the effects of wind speed on flight speed because short and long flight
segments are not executed along distinctly different directions. As a
consequence of this correlation, bees spend more time searching in the
location where the feeder is expected to be. If the search were to continue
indefinitely then eventually it becomes advantageous to refrain from looping
back to the origin of the search and instead adopt a freely roaming
Lévy-flight searching pattern (A.M.R., manuscript submitted). The slow,
shorter flight segments can then be associated with an `active local searching
phase' whilst the faster, longer flight segments can be associated with a
`relocation' phase where the bee moves to a new centre-of-search. Such
intermittent searching has been observed in a diverse range of species (e.g.
ground foraging birds, crickets, sea birds, octopi, planktivorous fish)
(Kramer and McLaughlin, 2001).
Lévy-flight models of intermittent searching predict that searching is
optimal when µ=2 and that, when the searching is optimal, the mean times
within the searching and relocation phases obey the scaling relation
tr
ts
(Reynolds, 2006).
Fig. 9 shows that the honeybee
flight patterns are consistent with this prediction and with observational
data for many other species.
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Pl(l)Ps(s)
[l*–l(1+s)]dlds,
where Pl(l) is the distribution of intended
flight lengths and Ps(s) is the distribution of
proportional errors, s. If a bee seeks to adopt a scale-free
searching strategy, then the distribution of intended flight lengths must have
a power-law tail, l–µ. The tail of the
distribution of realised flight lengths is then determined by
Pl*(l*)=l–µ
(1+s)–µPs(s)dsl–µ.
This tail coincides with the tail of the distribution of intended flights. It
follows that the realised scale-free searching strategy will be optimal when
the intended, perfectly executed, searching strategy is optimal.
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Discussion |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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).
Some of the vector flights took place in strong (5 m s–1) and
blustery cross-winds, which displaced the bees from a direct `bee-line' flight
path between hive and feeder. Nonetheless, the bees usually managed to get
`back on track' as they approached the hive or feeder location, demonstrating
the impressive accuracy of the bees' path-integration system, at least on the
landscape scale used in the present experiment. Cumulative navigational errors
are inevitable, however, and they could become large in long-distance foraging
excursions. So, despite the use of other navigational aids, particularly
landmarks (Collett, 1996;
Collett et al., 2002;
Chittka et al., 1995), it is
quite conceivable that a foraging bee might initially arrive at a marginally
incorrect location. The search strategies studied here would facilitate the
relocation of a floral patch if the error was slight and the resource was
close by. Moreover, if in the mean time the flowers in the patch had dropped
or ceased to produce nectar and pollen, then the described strategy might also
facilitate the finding of other resources in the vicinity. As with our tracked
bees, foragers will give up searching after a time and return to the hive.
Under normal circumstances, the bee may then start to gather information from
other foragers dancing on the comb: Biesmeijer and Seeley showed that bees
were more likely to follow dances after they had returned from a failed trip
(without nectar and pollen) (Biesmeijer and
Seeley, 2005). Local searching may also be required when honeybees
are directed by the waggle dance to a new food source, because the dance
communication system will not specify a food location with pinpoint accuracy
(Riley et al., 2005).
We turn now to the precise nature of the search strategy as revealed by our
experiments. At the end of its vector flight, the bee adopted a stereotypical
flight pattern comprised of loops of ever-increasing size that start and end
at the origin of its search and point in different azimuthal directions. This
strategy ensures that the central area where the feeder is most likely to be
is searched most extensively. We have shown that the looping flight patterns
made by honeybees are consistent with their having adopted an optimal
Lévy-looping searching strategy for the location of a single target
when the most likely location of the target is known or is presumed known
(A.M.R., manuscript submitted). This was done using a `random walk analysis'
of the turning points in the honeybee flight patterns, an examination of the
fractal scaling properties of the flight patterns and a determination of the
distribution of flight-segment lengths (Figs
5,
6,
7). The results of these
independent analyses yield a consistent picture; namely that µ=2
Lévy-flights underlie our recorded honeybee flight patterns. Some other
reported instances of Lévy-flight animal movement patterns, most
notably that of the wandering albatross
(Viswanathan et al., 1999),
are perhaps less secure because they are founded solely on the results of a
single analysis. We also showed that the strategy remains optimal when the
execution of Lévy-flights is imprecise due to the accumulation of
navigational errors and unpredictable displacements by gusts of wind, i.e.
when the execution of an optimal isotropic scale-free searching pattern is
being compromised by flight errors (see section on Imprecise
Lévy-flight searching strategies). In a Lévy-looping search, a
searcher travels out from the origin of its search along a randomly orientated
straight line whose length is drawn at random from a distribution with an
inverse-square power-law tail. If the target is detected, the search ends
– otherwise the searcher returns to the origin and then randomly chooses
a new direction and distance before travelling out again. As a search
progresses without success, the probability of finding the target at the
origin decreases. Eventually it will become more profitable to desist from
repeated returns to the original location and instead adopt a freely roaming
Lévy-flight searching pattern. Such a strategy is not only optimal for
the location of the original single target (i.e. the hive or nest or nectaring
plants), it is optimal for the location of sparsely and randomly distributed
targets (i.e. food sources) that, once visited, are not depleted but instead
remain targets for future searches
(Viswanathan et al., 1999).
This strategy minimizes the mean distance travelled, and so presumably the
mean energy expenditure, before first encountering a target. Freely roaming
Lévy-flight movement patterns may have been observed in a diverse range
of organisms that includes the wandering albatross, an African jackal, spider
monkeys and Drosophila fruit flies
(Viswanathan et al., 1996;
Viswanathan et al., 1999;
Atkinson et al., 2002;
Bartumeus et al., 2003;
Reynolds and Frye, 2007).
The random Lévy-looping searching strategy is clearly less efficient
than an equidistant spiral search pattern. A spiral search could, however,
work only if the bees' navigation were precise enough and their visual
detection ability reliable enough to ensure that all areas are explored and
that no intervening regions escape scrutiny. Should the objective be missed
there would be no possibility of encountering it a second time because the
flight path is an ever-expanding spiral. Relying on a spiral search pattern
when attempting to locate the hive would therefore be disastrous where
navigational systems are less than ideal; even then, this method could be used
only for short searches before the inevitable cumulative navigational error
became too large to allow a true spiral to be maintained. Switching from
spiral to random looping search paths has been observed in the desert isopod
Hemilepistus reaumuri when it gets lost after an excursion from its
burrow (Hoffman, 1983), in
male ladybird beetles (Adalia bipunctata) after they encounter a
conspecific female (Hemptinne et al.,
1996) and in desert ants (Cataglyphis) returning to their
nest after foraging beyond the range of their known landmark map
(Wehner and Srinivasan, 1981)
and if they are displaced by strong winds that have blown them off the ground
(Wehner et al., 2002).
The case of Cataglyphis ants returning from foraging trips is
interesting because it mirrors closely that of the honeybees searching at the
end of their vector flights. In the Cataglyphis case, foragers that
have moved beyond the range of their landmark map return to their nest using a
path-integration (dead-reckoning) strategy
(Wehner and Srinivasan, 1981).
When adopting this strategy, an ant must continuously monitor its motion
during foraging so that the mean vector pointing from its current position to
its nest can be computed. Even small inaccuracies in this mode of navigation
can result in large discrepancies between the end of the homing vector and the
actual location of the nest. If a homing ant gets lost, it adopts a
stereotypical search strategy that is comprised of loops of ever-increasing
size that start and end at the origin of its search and point in different
azimuthal directions (Wehner and
Srinivasan, 1981). This strategy ensures that the central area
where the nest is most likely to be located is searched most extensively.
Müller and Wehner suggested that underlying the ant's searching strategy
is a spiral search programme that gets transformed into the observed pattern
of loops by the ant's idiosyncratic path integration system
(Müller and Wehner,
1994). This search programme was described as a sequence of
ever-expanding spiral movement patterns interspersed with reset episodes
during which the ant returns to the origin. Our results for honeybees suggest
that the searching patterns of homing desert ants are, in fact, consistent
with their having adopted an optimal `Lévy-loop' searching strategy.
Lévy-loop searches may also be adopted by other insects, including, for
example, Formica schaufussi worker ants. Upon returning to a site at
which food had previously been found, F. schaufussi workers adopt a
fractal (i.e. Lévy-like) searching pattern and repeatedly return to the
origin of their search (Fourcassié
et al., 1992;
Fourcassié and Traniello,
1994). Their searching patterns, like those of desert ants, become
progressively more expansive. F. schaufussi show a greater tendency
to return to, and search at, a site of a prior food find when offered a source
of carbohydrate (sucrose solution) than when offered a source of protein
(insect prey). That is, they search more persistently for resources that are
renewed at a more or less regular rate (honeydew produced by homopterans),
whereas they do not return as frequently to a rewarding site or give up their
search rapidly when exploiting resources (e.g. dead arthropods) that have a
high unpredictability in space and time
(Fourcassié and Traniello,
1994). Our results with honeybees call for a re-examination of the
movement patterns of these and other species as part of the development of a
unifying theory of foraging patterns in animals. Indeed, our analysis
(Fig. 9) suggests that this
unification may extend well beyond insects and embrace a large class of
animals. This is because the results presented in
Fig. 9 support the conjecture
that scale-free and intermittent behaviours are not manifestations of two
distinctly different kinds of searching strategy but rather are constituent
parts of a single, complex, widely adopted searching strategy
(Reynolds, 2006;
Reynolds and Frye, 2007).
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Acknowledgments |
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