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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

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Honeybees perform optimal scale-free searching flights when attempting to locate a food source

Andrew M. Reynolds^1,*, Alan D. Smith¹, Don R. Reynolds², Norman L. Carreck¹ and Juliet L. Osborne¹

¹ Rothamsted Research, Harpenden, Hertfordshire, AL5 2JQ, UK
² Natural Resources Institute, University of Greenwich, Chatham, Kent, ME4 4TB, UK

View larger version (15K): [in this window] [in a new window]   Fig. 1. (A) A typical flight pattern of a honeybee trained to an artificial feeder, which was then removed, resulting in a localized search around the former position of the feeder. The location of the honeybee was recorded every 3 s unless the radar failed to detect the radar transponder. The hive was located at (x, y)=(210 m, 0 m) and the feeder was located at (x, y)=(0 m, 0 m) (approximately). The flight begins and ends in the vicinity of the hive (marked with an `H'). The locations where there are significant changes in flight orientation are indicated (•). A significant change in orientation is taken to arise when the angle between the current flight segment (joining two successive recorded positions) and the flight segment immediately following the last change in orientation is less than 90°, i.e. when the current non-local flight orientation differs from proceeding flight orientation by more than 90° (Reynolds et al., 2006). (B) Representation of the honeybee flight in terms of straight-line flights and changes in flight orientation. The statistical properties of these representations do not differ significantly from representations in which local abrupt changes in orientation are taken to arise when the angle between two successive flight segments (i.e. between three successive recorded positions) is less than 90°. The close correspondence between these two [local (C) and non-local (D)] representations indicates that most changes in flight orientation occur abruptly rather than through the accumulation of small changes.

View larger version (16K): [in this window] [in a new window]   Fig. 2. (A) The recorded locations of the honeybee flights in an x–y coordinate system in which the virtual feeder (F) is located at the origin (0 m, 0 m), and the hive (H) is located at (210 m, 0 m). (B) The locations where the flights changed orientation. (C) The first significant change in flight orientation. It can be seen that this generally occurs in the vicinity of the virtual feeder. (D) The last significant change in flight orientation.

View larger version (6K): [in this window] [in a new window]   Fig. 3. The number of changes in flight orientation in an x, y coordinate system, showing that the search flights are centred on the location of the virtual feeder. The virtual feeder is located at the origin (0 m, 0 m), and the hive is located at (210 m, 0 m).

View larger version (4K): [in this window] [in a new window]   Fig. 4. Distribution of the directions, , of the flight-segments in the representation of the searching flights.

View larger version (3K): [in this window] [in a new window]   Fig. 5. Power-law scaling of the root-mean-square, F, net displacement indicative of the presence of long-range correlations, demonstrating that the honeybee flights are scale-free. The averaging is over the `searching phases' of 39 bees. Approximate power-law scaling behaviour with =0.85 extending over about one decade is indicated (straight line). This scaling and others presented later are not sensitively dependent upon the presence or absence of four non-returning bee tracks within the averaging. The scaling exponent, , does not change significantly when the angle (90°) between successive flight segments used to define a change in flight orientation is changed by ±30°.

View larger version (4K): [in this window] [in a new window]   Fig. 6. The mean number, nbox, of boxes of size lbox required to cover the representations of the honeybee flights. Power-law scaling, nboxlbox-D is indicative of a scale-free characteristic with fractal dimension D. Power-law scaling with D=1.3 obtained from a linear least-squares fit (r2=0.99) is indicated. The insert shows the nbox of boxes of size lbox required to cover simulated Lévy-flights with µ=2. The number of straight-line flights within the simulated Lévy-flights is equal to the mean number of straight-line flights in the representations of the honeybee flights. The model predicts that D=1.2. A similar level of correspondence between the fractal dimension of the representations of bee flights and simulated Lévy flights with µ=2 is also attained when fractal dimensions are calculated using the method of dividers rather than the box-counting method.

View larger version (5K): [in this window] [in a new window]   Fig. 7. Number of straight-line flights, nl, of length l in the representations of the honeybee flights (histogram) on linear–linear scales (A) and on log–log scales (B). A Lévy stable distribution (Cauchy distribution), P(l)=(1/)/(l2+2), with scale parameter, , chosen to minimize the sum of the squared differences between the Lévy stable distribution and the data, nl (solid line). The inverse-square law scaling of the high tail (l>10 m) of the distribution, obtained from a linear least-squares fit of the distribution (r2=0.94), is indicated (broken line).

View larger version (5K): [in this window] [in a new window]   Fig. 8. The distribution of speeds associated with (A) short (L<10 m) and (B) long (L>10 m) flight-segments. The mean speeds are 1.6 m s–1 and 3.2 m s–1. Distinctly different distributions are also obtained when the length scale L is increased or decreased by a factor of 2.

View larger version (5K): [in this window] [in a new window]   Fig. 9. Mean times in the local active searching phase, ts, comprising short flight-segments having length L<10 m, and in the relocation phase, tr, comprising long flight-segments having length L>10 m (X). Mean times do not change significantly when the length scale L is increased or decreased by a factor of 2. Mean times for a diverse range of intermittent foragers (•) [copepod nauplius, phorid fly, cricket, octopus, Arctic grayling fish (foraging for large and small prey) and freely flying Drosophila fruit flies] (Kramer and McLaughlin, 2001; Bénichou et al., 2005; Reynolds and Frye, 2007). The scaling relation trts, predicted by the Levy-flight model of optimal searching (Reynolds, 2006), is shown (solid line).