The effect of step size on straight-line orientation

Abstract

Moving along a straight path is a surprisingly difficult task. This is because, with each ensuing step, noise is generated in the motor and sensory systems, causing the animal to deviate from its intended route. When relying solely on internal sensory information to correct for this noise, the directional error generated with each stride accumulates, ultimately leading to a curved path. In contrast, external compass cues effectively allow the animal to correct for errors in its bearing. Here, we studied straight-line orientation in two different sized dung beetles. This allowed us to characterize and model the size of the directional error generated with each step, in the absence of external visual compass cues (motor error) as well as in the presence of these cues (compass and motor errors). In addition, we model how dung beetles balance the influence of internal and external orientation cues as they orient along straight paths under the open sky. We conclude that the directional error that unavoidably accumulates as the beetle travels is inversely proportional to the step size of the insect, and that both beetle species weigh the two sources of directional information in a similar fashion.

Keywords: compass; dung beetle; navigation; orientation; random walk; step size.

Figure 1.

Description of the experimental arenas and beetles. Individuals of Scarabaeus ambiguus (left) and S. lamarcki (right) are depicted side-by-side for size comparison. Photo: Christopher Collingridge (a). For all treatments, a beetle was placed alongside a ball in the centre of a circular, sand-coated arena (b) and filmed with an overhead camera (c). The beetle was allowed to roll its ball to the perimeter of the arena, where the exit angle was noted (0° = magnetic North). Depending on the experimental treatment, the beetle either rolled only once, or was repeatedly placed back in the arena centre beside its ball until it had exited the arena 20 times. The ball rolled was either a natural dung ball or a standard putty ball (shown in a). For the experiment where the sun was mirrored, a 75 cm radius arena was used (not depicted in the figure). For all other experiments conducted, three differently sized arenas were used depending on the species of beetle (b): 50 cm (S. ambiguus and S. lamarcki, black solid line), 33 cm (S. ambiguus, red dotted inner circle), 52 cm (S. lamarcki, red solid outer circle) radius. (Online version in colour.)

Figure 2.

The role of the sun in the celestial compass system of two beetle species. The response to a mirrored sun while rolling outdoors, under a clear sky, was tested in two closely related, but differently sized beetle species. A schematic diagram of the experiment is presented in (a). Forty-five individuals of Scarabaeus ambiguus (b) and S. lamarcki (c), respectively, were individually placed under the natural sky, alongside a dung ball in the centre of a 75 cm radius, circular sand-covered arena. The beetles were allowed to roll to the perimeter of the arena where their exit angles were noted. From here, the beetles were placed back in the centre to exit a second time, either under (i) the same natural sky as during the previous roll (control, grey circles), or (ii) a manipulated sky where the apparent position of the sun is changed by 180° by the use of a mirror (test, yellow circles). The difference between two exit angles was calculated and used to define the mean change in bearing (control, dotted grey lines; test, solid red lines). Error bars represent one circular standard deviation. When allowed to roll twice under the sun, individuals of both species showed no significant change in bearing between consecutive rolls (dotted grey line). Under the mirrored sun, both species responded to this treatment by a change in exit bearing approaching 180° (solid red lines). (Online version in colour.)

Figure 3.

Rolling trajectories in the absence of external visual cues. The two closely related, but differently sized beetles, were allowed to roll a dung ball from the centre of a flat, sand-coated arena, in complete darkness. The full trajectories of 10 randomly chosen beetles of each species are shown. On the 50 cm radius arena (black perimeter) S. ambiguus (a), obtained a significantly lower straightness index (higher tortuosity) compared to the larger S. lamarcki (p = 0.02, N = 15) (b). When analysed over a radial distance corresponding to 20 steps for each species respectively (32 cm for S. ambiguus and 52 cm for S. lamarcki) (a, inner red perimeter; b, outer red perimeter) no significant difference in straightness was recorded (p = 0.08, N = 15). (Online version in colour.)

Figure 4.

Estimation of motor errors, compass errors, and their balance. A model of a beetle performing a random walk where θ_i is the direction of movement of the previous step and ΔX_i, ΔY_i are the distance travelled in step i along the x and y directions, respectively (defined in equations (2.1) and (2.2)) (a). A flow diagram describing the process of estimating pairs of w and θ^*BRW for two beetle species that differ in size (b). Step 1: θ^*CRW is estimated from the width of the angular errors of a beetle orienting in the absence of visual cues. Step 2: Two sets of BCRW trajectories are illustrated; one at the limit of pure CRW (w = 0) and one at the limit of pure BRW (w = 1). To generate these trajectory examples, we chose arbitrarily θ^*BRW = θ^*CRW = 5° (these values were arbitrarily chosen for the purpose of illustrating the model). Each trajectory is shown in a different colour. Step 3: Mean vector length (R) for each species is generated from the simulation and compared to the experimentally measured values (shown as red dotted line on the colour bar and on the heat-map). Step 4: The extracted w and θ^*BRW for each species is shown. (Online version in colour.)

Figure 5.

Measuring orientation performance in the presence of external visual cues. As a measure of orientation performance (a), the mean vector length for each beetle was calculated from 20 tracks over a radius equivalent of 50 cm, as well as of a radius equivalent of 20 step lengths of the corresponding species (32 cm for Scarabaeus ambiguus and 52 cm for S. lamarcki) (white circle, mean value for S. ambiguus; black circle, mean value for S. lamarcki; red solid line, median value for S. ambiguus and S. lamarcki). An R-value of 1 indicates that the beetles maintained the same direction over 20 rolls. When rolling over a radius of 50 cm, the smaller species, S. ambiguus (N = 20), showed a significantly shorter resultant vector length compared to the larger species (N = 20) (R_{S. ambiguus}: 0.88 ± 0.02; R_{S. lamarcki}: 0.92 ± 0.01, p = 0.028, N = 20). However, no significant difference was seen when both species rolled over a distance equivalent to 20 steps (R_{S. ambiguus}: 0.91 ± 0.015; R_{S. lamarcki}: 0.91 ± 0.02, p = 0.42, N = 20). Paths travelled by four individuals for each species and radial distance (b) are shown (from left: S. ambiguus (50 cm); S. lamarcki (50 cm); S. ambiguus (32 cm); S. lamarcki (52 cm)). Each colour represents 20 trajectories of one individual. * = p < 0.05, n.s. = p > 0.05. There was no difference in the straightness of the 20th exit path compared to the 1st exit path performed by the same individual in any of the conditions (p_{50 cm(S. ambiguus)} = 0.16, p_{50 cm(S. lamarcki)} = 0.16; p_{step length(S. ambiguus)} = 0.30, p_{step length(S. lamarcki)} = 0.16, N = 20). (Online version in colour.)

Figure 6.

Orientation performance in a natural environment. The smaller Scarabaeus ambiguus and the larger S. lamarcki were allowed to form a dung ball and roll it away from a dung pat (marked with a star) in their natural environment (N, north; E, east; S, south; W, west) (a). Their trajectories (blue lines, S. ambiguus; orange lines, S. lamarcki), were recorded until they started to bury their balls (blue circles, S. ambiguus; orange circles, S. lamarcki (b)). This marked the end of the trial and the radial distance from the dung pat to the site of burial was measured. The dotted concentric circles indicate radial distances from the dung pat in 5 m increments. In total, trajectories of 10 individuals per species were recorded. The smaller species, S. ambiguus, rolled a significantly shorter radial distance from the pat before burying its ball when compared to the larger species (S. ambiguus: 7.56 m ± 1.05 m, S. lamarcki: 12.45 m ± 1.28 m, N = 10) (p = 0.001, N = 10). (Online version in colour.)