Hoverfly (Eristalis tenax) pursuit of artificial targets

The ability to visually detect small moving objects can be essential for survival, as such objects could correspond to predators or prey. Such visual identification of prey is common in vertebrates, including zebrafish larvae (Mearns et al., 2020; Patterson et al., 2013), archerfish (Newport and Schuster, 2020) and birds of prey (Kane et al., 2015). Visually driven, high-performance predatory attacks are also displayed by insects, including robber flies (Wardill et al., 2017), dragonflies (Olberg et al., 2000), killer flies (Wardill et al., 2015) and praying mantises (Nityananda et al., 2018). Killer flies (Wardill et al., 2015), some robber flies (Fabian et al., 2018) and dragonflies (e.g. Olberg et al., 2005, 2000) detect their prey from a perch before launching a high-speed pursuit to catch their prey in mid-air. Killer flies decide whether to attack based on the ratio between the angular size of the target image and its angular speed (Wardill et al., 2015). Libelulid dragonflies also use heuristic cues based on the target's angular size and speed (Lin and Leonardo, 2017).

Non-predatory Eristalis spp. hoverfly males pursue both conspecifics and heterospecifics encountered within their territories (Wellington and Fitzpatrick, 1981), but also other small moving targets presented to them (Fitzpatrick, 1981). In contrast to pursuits by the more widely studied dragonflies (Olberg et al., 2000), killer flies (Wardill et al., 2015) and robber flies (Wardill et al., 2017), Eristalis spp. hoverflies often start their pursuit from a hovering or flying stance (Collett and Land, 1975b; Fitzpatrick, 1981; Wellington and Fitzpatrick, 1981). It has been suggested that hoverflies also use heuristic rules to determine when to initiate pursuit (Collett and Land, 1978). Indeed, based on conspecific size and typical flight speed (Golding et al., 2001; Bartsch, 2009), the male hoverfly can predict the expected angular size and speed at suitable distances (Collett and Land, 1978). In field observations, Eristalis spp. males pursue artificial targets of a similar size to conspecifics (Collett and Land, 1978; Fitzpatrick, 1981; Fitzpatrick and Wellington, 1983; Maier and Waldbauer, 1979; Wellington and Fitzpatrick, 1981) moving at 5–12.5 m s⁻¹ (Collett and Land, 1978). However, female Eristalis spp. often move much slower than this (Thyselius et al., 2018), indicating that there are instances when such heuristic rules may be broken.

Interestingly, Eristalis spp. hoverflies also pursue objects that are much larger than conspecifics, such as leaves (Maier and Waldbauer, 1979), butterflies, hornets and bumblebees (Wellington and Fitzpatrick, 1981), arguing against the use of strict heuristic rules based on angular size. Other fly species also pursue targets that do not appear to be ecologically relevant. Predatory killer flies and non-predatory blowflies may pursue targets that are 3–5 times their own size (Boeddeker et al., 2003; Lyneborg et al., 1975; Solano-Rojas et al., 2017; Wardill et al., 2015). However, when blowflies pursue larger beads, they fly further away, consistent with angular size-based heuristic rules. Whether hoverflies similarly adjust their flight behavior to the target's angular size is currently unknown.

Following the start of the pursuit, killer flies and robber flies intercept the target using proportional navigation by keeping the bearing angle constant (Fabian et al., 2018). In contrast, blowflies and houseflies use smooth pursuit, by correlating their yaw rotation with the target error angle (Boeddeker et al., 2003; Land and Collett, 1974; Wehrhahn et al., 1982). More recent studies show that blowflies have two different pursuit strategies (Varennes et al., 2020). In the horizontal plane, they fly towards the current position of the target, i.e. they use smooth pursuit by aiming to keep the error angle close to anterior, whereas in the vertical plane they use proportional navigation (Varennes et al., 2020). Some hoverflies, such as Syritta pipiens, also use smooth pursuit based on the target error angle (Collett and Land, 1975a), whereas the larger Eristalis and Volucella spp. intercept the target using deviated pursuit, where the pursuer flies towards the predicted future position of the target by keeping the error angle constant (Collett and Land, 1978).

To investigate the nuances of hoverfly target responses with higher resolution and behavioral control, we developed an indoor arena to record conspecific flight chases in the lab. By making the arena large enough to encourage naturalistic behavior, we could reconstruct pursuits of beads of different sizes (6–38.5 mm diameter). We found that E. tenax males pursued these artificial targets in the arena, moving at speeds of up to 2 m s⁻¹, starting their pursuit from the wing either below or above the bead. We found that pursuits were initiated across a large range of angular sizes and speeds, arguing against the use of strict heuristic rules. We show that the hoverflies first accelerate to quickly get close to the artificial target, and then follow the target at a closer distance for several seconds. When the hoverfly was proximal to the target, its translational speed was correlated with the bead speed, but not with its angular size or speed. Furthermore, we found that flight behavior was different when pursuing the 38.5 mm diameter bead than when chasing beads that were more similar in size to conspecifics. Indeed, hoverflies initiated pursuit of the 38.5 mm bead when it moved slower, they spent a longer time distal to it and they interacted with it physically more often.

Eristalis tenax (Syrphidae) (Linnaeus 1758) hoverflies were reared from farm-collected larvae as described previously (Nicholas et al., 2018). At any one time, we kept 10 male and eight female E. tenax hoverflies in the flight arena (see below), with females present to encourage male competition. Males showed an interest in females about 2 weeks after emerging from the pupae and started pursuing artificial targets about 2 months after emerging.

Hoverflies had constant access to fresh pollen–sugar mix and water. A total of 94 male and 80 female hoverflies lived in the arena until death, or until the time when their physical activity declined noticeably, when they were replaced, to keep the total number constant. Replacement hoverflies came from age matched, or slightly younger, artificial hibernation stock (Nicholas et al., 2018). In addition, all hoverflies were replaced 4 times (Table S1).

We used a Plexiglas arena (1 m³; Fig. 1A; Movies 1–3), custom designed by Akriform Plast AB (Sollentuna, Sweden). The arena was lit from above by two daylight fluorescent lamps (58 W/865, Nova Group AB, Helsingborg, Sweden) and two office fluorescent lamps, giving an average illuminance of 900–1200 lx (LM-120 Light Meter, Amprobe, Everet, WA, USA). The hoverflies were kept on a 12 h:12 h light:dark cycle at room temperature (19–20°C).

A fishing line (0.3 mm diameter) was looped around the arena, entering horizontally through two holes at 0.7 m height (Fig. 1A; Movies 1–3), with a bead attached using a 0.06 mm diameter fishing line. A laptop controlled a stepper motor via a stepper driver (23HS-108 MK.2 stepper motor and ST5-Q-NN DC input stepper driver, Promoco Scandinavia AB, Täby, Sweden), similar to Wardill et al. (2017). We used 10 programs with seven speeds each (0.1–2 m s⁻¹) presented in a randomized order. Which program was used on any given day was randomized using a 10-sided die. The bead started its motion 10 cm from one side of the arena (Fig. 1A), travelled 0.8 m to the other side, paused for 0.5 s, travelled back to its start, paused for 0.5 s and then repeated the motion at a different speed (Movies 1–3). The program was run continuously, pausing 3 s before looping.

We used beads of four sizes (6, 8, 10 and 38.5 mm in diameter), painted glossy black (acrylic paint). The three smaller beads were made of glass (Panduro Hobby AB, Malmö, Sweden) and the 38.5 mm diameter bead was made of polystyrene (Clas Ohlson, Insjön, Sweden). The 6 and 10 mm diameter beads were used for 9 consecutive days each. The 38.5 mm bead was used in two periods of 7 and 2 consecutive days. The 8 mm bead was used for 10 consecutive days and in addition as a control before and after the 6 mm bead experiments, and between the two periods of using the 38.5 mm bead.

We filmed for 39±2 min (mean±s.e.m.) per day using two cameras (120 frames s⁻¹, 640×480 pixels; EXFH25, Casio, Tokyo, Japan). Filming was done continuously, pausing briefly every 10 min to start a new movie, because of saving constraints. The cameras were positioned about 1.5 m from the front of the arena, with a 2.5 m distance between the two. They were placed on tripods (Dörr cybrit medi 4-BA, Dörr GmbH, Neu-Ulm, Germany; SIRUI T-2005X, SIRUI, Verona, NJ, USA) with each camera facing the arena at an angle of about 45 deg, with a 90 deg angle between the two. The cameras had a focal length of 26 mm, with the focus adjusted to the center of the arena, level with the bead track. In the resulting movies, in the center of the arena, 1 pixel corresponds to 3 mm in both the x- and y-plane. As Eristalis hoverflies have an average width of about 4 mm and a length of 1.4 cm (Bartsch, 2009), when far away from the camera, they would sometimes only cover 1 pixel in an individual frame.

The cameras were synchronized to a 1-frame resolution using the torch of a Samsung Galaxy A3 2017 A320 mobile phone (Thyselius et al., 2018; Wardill et al., 2017). The cameras were calibrated as described previously (Thyselius et al., 2018; Wardill et al., 2017) using a 35 mm checkerboard pattern, printed on a polyester sheet and glued to a board (NeverTear, Arkitektkopia AB, Stockholm, Sweden).

We first identified cruising flights as those where a hoverfly was flying around but clearly not interacting with the bead, another hoverfly or the arena walls (similar to Collett and Land, 1975a). We arbitrarily selected 10 cruising flights from male hoverflies and 10 from females, each 0.83 s (100 frames) long. We next manually identified video recordings where a male hoverfly clearly flew towards the bead and labelled these as visually identified pursuits (Table 1; Movies 1–3).

From each camera frame, we tracked the 2D position of the hoverfly and the bead centroids using custom-written MATLAB scripts (Thyselius et al., 2018; Wardill et al., 2017), modified to facilitate visual control and manual corrections. We first established the 2D position of the hoverfly and the bead in each frame for each camera (as in Thyselius et al., 2018). Because of the limited spatial resolution, only one point was tracked on each. We performed 2D reconstructions (Table 1) of all visually identified pursuits where we could clearly see the hoverfly and the bead throughout. Pursuits where several hoverflies were interacting with each other and/or the bead were excluded.

The 2D data were smoothed using MATLAB's lowess smoothing, which is a locally weighted linear regression method, using 2% of the total number of data points. We translated the 2D positions to 3D (Table 1, Fig. 2A,B), using calibration files obtained from filming the checkerboard, synchronized with both cameras, and previously described methods (Thyselius et al., 2018; Wardill et al., 2017). The 3D data were smoothed again, using 2% of the data points. At this stage, eight pursuits were excluded because of calibration issues and five because the start of the trajectory was not included, the hoverfly hit the arena wall or several hoverflies were interacting.

In all equations below, the x- and y-axes define the two sides and the z-axis the height (Figs 1A and 2A,B); F is used to describe the hoverfly and B the bead. The frame rate is 120 frames s⁻¹ and t subsequently defines the time steps of 8.3 ms.

We used the distance between the hoverfly and the bead to separate each pursuit into ‘distal’ and ‘proximal’. We first determined the distribution of distances across all pursuits, at each time point from pursuit start, for each bead size. We used the lower quartile as a cut-off for the proximal part of the pursuit, i.e. 13.7 cm for the 8 mm bead, 11.9 cm for the 10 mm bead and 8.9 cm for the 38.5 mm bead (see below). If the hoverfly was further away than this cut-off at pursuit start, ‘distal’ was defined as the time between pursuit start and the time when the hoverfly was last outside this cut-off. If the hoverfly was never within the cut-off distance, the minimum distance defined the end of the distal stage. ‘Proximal’ was identified as the time from when the hoverfly was first within the cut-off distance until it left it for more than 200 ms. When calculating the minimum distance d_minimum between the bead and the hoverfly (see below), we subtracted the bead radius from d (Eqn 6).

All error and bearing angles were smoothed using a span of 4% of the total number of data points. The delta error and delta bearing angles were defined as the absolute change between t−2 and t.

Data analysis, statistics and figure preparation were done using Prism 9 (GraphPad Software Inc., San Diego, CA, USA) and MATLAB (R2019b, The MathWorks, Inc., Natick, MA, USA). All data are shown as either individual data points or median±interquartile range, unless otherwise specified. We used a restricted maximum likelihood model to fit mixed-effects models to time-aligned data. For pursuit start quantifications, we calculated the mean from t−2 to t+2, where t=pursuit start. For before-start quantifications, we calculated the mean from t–18 to t−6, where t=pursuit start. As most data were not normally distributed, we used Kruskal–Wallis tests followed by Dunn's multiple comparisons tests, or Mann–Whitney tests. For circular data, we used MATLAB's Circular statistics toolbox (Berens, 2009).

We used the first 1 s of pursuit for all bead sizes to determine the correlation coefficient (using MATLAB's corrcoef) for different time shifts (Fig. S3A,B), from which we extracted the time of the peak correlation. For visualization, we then extracted the data at each time point for each bead size individually, using the peak correlation delay (Fig. 5G,H). The graphs show the linear regression, together with the R² value or Spearman correlation.

To confirm that E. tenax hoverflies behaved naturalistically in our indoor arena (Fig. 1A), we analyzed their flight speed during cruising flights (see Materials and Methods). We found that the top cruising speed was just over 2 m s⁻¹ with a median of 0.37 m s⁻¹ for males and 0.35 m s⁻¹ for females (Fig. 1B), which is similar to field flight speeds (0.32 m s⁻¹: Golding et al., 2001; 0.34 m s⁻¹: Thyselius et al., 2018).

We filmed with two cameras (Fig. 1A) and reconstructed the 3D position of hoverflies pursuing artificial targets. We show two example pursuits in Fig. 2A,B (see also Movies 1 and 2), with the bead (8 mm diameter) and male hoverfly locations every 75 ms. As hoverflies initiate pursuit while on the wing (Collett and Land, 1975b), we defined pursuit start as a sharp increase in angular speed (i.e. a turn), followed by a translational speed increase (Fig. 2C–E; see also Collett and Land, 1975b; Varennes et al., 2019). We found no difference in translational speed (Fig. 2F) or angular speed (Fig. 2G) between pursuits of beads of different sizes. The hoverfly acceleration increased after pursuit start (Fig. S1A), but was much lower than the 33 m s⁻² measured in the field (Collett and Land, 1978).

We found that male hoverflies pursued black beads of all four sizes (6, 8, 10 and 38.5 mm diameter; Table 1). However, despite hoverflies often flying in the arena, we only visually identified a few pursuits per hour (Table 1). In addition, pursuits of the 6 mm diameter bead were even more rare than pursuits of the other bead sizes (Table 1; Movie 3) and these data were therefore excluded from further analysis. The hoverfly flight speed 100 ms before pursuit ranged from 0.01 to 1.1 m s⁻¹ (minimum to maximum), with median values of 0.33 m s⁻¹ when pursuing the 8 mm bead, 0.45 m s⁻¹ when pursuing the 10 mm bead and 0.23 m s⁻¹ when pursuing the 38.5 mm bead (Fig. S1B).

The hoverflies started pursuit from a range of distances (from 5.6 cm to 91 cm; Fig. 3A), suggesting that they used the entire arena. The median distance at pursuit start was 43 cm for the 8 mm bead, 48 cm for the 10 mm bead and 44 cm for the 38.5 mm bead (Fig. 3A). The coefficient of variation was 46% for the 8 mm bead and 41% for both the 10 and 38.5 mm beads (Fig. 3A). The angular size of the bead (θ, Fig. 3B, inset) at pursuit start covered a broad range of 0.5–22 deg, with median sizes for the 8, 10 and 38.5 mm bead sizes of 1.1, 1.2 and 5.0 deg (Fig. 3B). Consistent with its larger physical size, the 38.5 mm bead had a significantly larger angular size (Fig. 3B). The angular size coefficients of variation were large: 91%, 66% and 79% (Fig. 3B).

The bead speed at pursuit start ranged from stationary to 2.0 m s⁻¹, with coefficients of variation for the 8, 10 and 38.5 mm bead sizes of 73%, 86% and 138% (Fig. 3C). The median bead speeds at pursuit start were 0.8, 0.4 and 0.2 m s⁻¹, but this trend was not significant (P=0.1). We found that the relative speed between the bead and the hoverfly 100 ms before pursuit start ranged from 0.05 to 2.0 m s⁻¹, with coefficients of variation of 70%, 64% and 71% (Fig. S1C).

The angular speed of the target image (φ, Fig. 3D, inset) at pursuit start was significantly lower for the largest bead size (Fig. 3D), with median speeds of 94, 79 and 36 deg s⁻¹ (Fig. 3D). The angular speed coefficients of variation were large: 153%, 97% and 73% (Fig. 3D).

Taken together, because of the large coefficients of variation, it is unlikely that male E. tenax hoverflies use strict heuristic rules based on the target's angular size or speed (Fig. 3B,D) to determine when to initiate pursuit. Killer flies use the ratio between the angular size and speed to determine which targets to pursue (Wardill et al., 2015). However, we found that the coefficient of variation for this ratio was also large: 136%, 146% and 72% (not shown). Nor did the hoverflies seem to selectively pursue targets from a narrow range of distances, physical sizes or bead speeds (Fig. 3A,C; Fig. S1C).

The data above (Fig. 3) show that male E. tenax hoverflies are unlikely to use strict heuristic rules based on angular size or speed to trigger pursuit initiation. However, visual information from the bead could be used to adjust initial flight behavior. The pursuit start is associated with a translational speed increase (Collett and Land, 1975b; Varennes et al., 2019), here quantified as a speed ratio peak (example trace shown in Fig. 2D). We found that this ratio did not depend on the bead's physical diameter (Fig. S2A) or its angular size (Fig. S2B). Nor did the speed ratio peak depend on the bead's translational speed (Fig. S2C) or its angular speed (Fig. S2D).

We next looked at the hoverfly's increased angular speed (example in Fig. 2E) associated with a sharp turn at pursuit start and found that it did not depend on the bead size (Fig. S2E), its angular size (Fig. S2F), its translational speed (Fig. S2G) or angular speed (Fig. S2H). In contrast, field work has shown a correlation between the hoverfly's angular speed at pursuit start and the bead's angular speed, which has been interpreted as an effort to put the target image in the frontal visual field (Collett and Land, 1978). In our experiment, the hoverflies' angular speed at pursuit start ranged from 460 to 13,000 deg s⁻¹, with median speeds of 2960 deg s⁻¹ for the 8 mm bead, 4530 deg s⁻¹ for the 10 mm bead and 4230 deg s⁻¹ for the 38.5 mm bead (Fig. S2E), similar to turning speeds measured in the field (Collett and Land, 1978).

We quantified the delay between the hoverfly's peak angular speed (e.g. Fig. 2E, blue dot) and the peak speed ratio (e.g. Fig. 2D, arrow) and found that this did not depend on the physical size of the bead (Fig. S2I) or its angular size (Fig. S2J). Nor did the delay depend on the bead's translational speed (Fig. S2K) or its angular speed (Fig. S2L). The median delays were 25 ms for the 6 mm bead, 17 ms for the 8 mm bead and 21 ms for the 38.5 mm bead (Fig. S2I), similar to delays measured in the field (Collett and Land, 1975b). In summary, hoverflies showed similar pursuit start behavior to that in the field, with a sharp turn followed by a translational speed increase about 20 ms later, suggesting that they behaved naturalistically, but the heuristic rules investigated here could not explain how hoverflies controlled this behavior.

Killer flies pursue artificial beads from above as well as below (Rossoni et al., 2021). We found that this was also the case for hoverflies, across the bead sizes tested (Fig. 4A). We next investigated whether the hoverfly starting position (i.e. above versus below the bead) impacted flight behavior when pursuing the 8 mm bead, as we had most pursuits of this bead size. We found that the starting position did not affect the hoverfly's speed ratio (Fig. 4B), its angular speed at the start of the pursuit (Fig. 4C) or the delay between the angular and translational speed peaks (Fig. 4D). In contrast, pursuits that started from below the bead were initiated when the bead was significantly further away from the hoverfly (Fig. 4E). It is likely that this was due to the larger space available below the bead track, which was located 70 cm above the arena floor (Fig. 1A). Consistent with the distance difference (Fig. 4E), the angular size of the target image also differed (Fig. 4F). We found that when the hoverflies started their pursuit from above, they pursued faster targets (Fig. 4G). Together, the shorter distance to the target and the faster target speed resulted in a significant difference in the target's angular speed when comparing above with below starting positions (Fig. 4H).

We next quantified the hoverfly's behavior during pursuit. As expected, the distance between the male hoverfly and the bead decreased with time (Fig. 5A), which resulted in the target's angular size increasing (Fig. 5B). There was no significant effect of bead size on the distance (Fig. 5A), but the angular size was significantly larger for the 38.5 mm diameter bead (Fig. 5B), consistent with its larger physical size. We also measured the angular speed of the target and found that there was no significant effect of bead size (Fig. 5C).

To investigate the behavior during pursuit further, we first looked into the bead-to-hoverfly distance across all trajectories, at each time point. The bead-to-hoverfly distance range was 0.5–92 cm and the median distance for the 8, 10 and 38.5 mm bead sizes was 25, 22 and 20 cm (Fig. 5D). We also noted that across pursuits, more time points were spent close to the bead compared with further away (Fig. 5D). To investigate whether the behavior was different when closer to the bead compared with further away, we used the lower quartile for each bead size (13.7, 11.9 and 8.9 cm; Fig. 5D,E) to separate each pursuit into a ‘distal’ (far from bead) and a ‘proximal’ (close to bead) stage (see Materials and Methods for details).

We found that the time the hoverfly spent distal to the bead was highly variable, from 83 ms to 3.9 s, with median durations for the 8, 10 and 38.5 mm bead sizes of 0.81, 0.95 and 1.5 s, which was significant (Fig. 5F). The total amount of time that the bead-to-hoverfly distance exceeded the distal cutoff increased linearly with the distance to the bead at pursuit start (R² values of 0.17, 0.25 and 0.19; Fig. 5G), i.e. as may be expected, the further away the fly started from the bead, the longer it spent in the distal phase, but the correlation was not significant for the 38.5 mm bead (P=0.0024 for the 8 mm bead, P=0.0032 for the 10 mm bead and P=0.14 for the 38.5 mm bead; Fig. 5G).

The time the hoverfly was proximal to the bead ranged from 8.3 ms to 2.9 s, with a median duration of 0.39, 0.35 and 0.69 s for the 8, 10 and 38.5 mm bead sizes, but this difference was not significant (Fig. 5H). We hypothesized that the distal stage was optimized to rapidly decrease the distance to the bead (Fig. 5A,D–G), while the proximal stage (Fig. 5D,E,H) was aimed at staying close to the bead. In support of this, the hoverfly translational speed was higher during the distal stage than during the proximal stage (Fig. 5I).

To investigate how hoverflies control their translational speed during pursuit, we calculated the correlation coefficients between the hoverfly speed and the bead distance, its angular size, the bead speed and its angular speed during the first second of the distal stage (Fig. S3A). We found the strongest correlation between the hoverfly's translational speed and the bead's speed (Fig. S3A, solid line), with a peak at −233 ms (Fig. S3A, dotted vertical line). The graph showing the flight speed at each time point, as a function of the bead speed 233 ms previously (Fig. 5J), highlights that even if they are correlated, there is large variation, with R² values of 0.26, 0.15 and only 0.010 for the 8, 10 and 38.5 mm beads, respectively (Fig. 5J). In addition, 233 ms is slow for typical insect reactions (e.g. Collett and Land, 1978; Mischiati et al., 2015; Varennes et al., 2020; Wehrhahn et al., 1982), so its biological relevance needs to be taken with caution. Together, this suggests that during the distal stage the hoverfly was aiming to rapidly decrease the distance to the bead (Fig. 5A,E,G,I).

We carried out cross-correlations for the proximal stage and found the strongest correlation between the hoverfly's translational speed and the bead's speed (Fig. S3B, solid line) at −150 ms (Fig. S3B, dotted vertical line). We visualized this by plotting the hoverfly speed as a function of the bead speed 150 ms previously, at each time point, from all the pursuits where the hoverfly was proximal to the bead, and found R² values of 0.35, 0.21 and 0.39 for the 8, 10 and 38.5 mm beads, respectively (Fig. 5K).

We found a weaker correlation between the hoverfly's translational speed and the distance to the bead, with a peak at −42 ms (Fig. S3B, dashed line). In the graph showing the hoverfly speed as a function of distance 42 ms previously, for each time point, we found R² values of 0.096, 0.11 and 0.006 for the 8, 10 and 38.5 mm beads, respectively (Fig. S3C).

To test whether the observed reduction in flight speed during the proximal stage (Fig. 5I) was due to the hoverfly being close to the arena wall, we plotted the flight speed at each time point as a function of the horizontal distance to the closest arena wall (Fig. S3D). We found that the hoverflies flew fast even close to the walls, making the wall an unlikely confounding factor. Taken together, our findings suggest that in the proximal phase, the hoverfly matched the target speed (Fig. 5K) because it intended to shadow it, rather than catch it.

If true, this should be reflected in the minimum distance between the bead and the hoverfly. Indeed, we found that pursuits of the 8 and 10 mm beads rarely ended with the hoverfly grabbing or landing on the bead (median minimum distances 5.8 and 6.5 cm; Fig. 6). In contrast, the median distance between the hoverfly and the 38.5 mm bead was 0.99 cm (Fig. 6), suggesting that the hoverflies often landed on the largest bead. This is interesting considering that they initiated pursuit of the 38.5 mm bead when it moved more slowly (Fig. 3C), and they spent a longer time distal to it (Fig. 5F). Maybe the hoverflies categorized it differently to the more conspecific-sized targets.

As we found that E. tenax hoverflies pursue beads from above and below (Fig. 4), we next analyzed whether the initial geometry between the fly and target had an impact on the subsequent pursuit behavior. For this purpose, we calculated the error angle (ε, Fig. 7A,B), defined as the angle between the hoverfly heading (Fig. 7A,B, black arrows) and the LoS to the bead (Fig. 7A,B, dashed line; e.g. Land and Collett, 1974; Rossoni et al., 2021; Varennes et al., 2020). The error angle was quantified in both the azimuth (Fig. 7A) and elevation planes (Fig. 7B). After smoothing the data (see Materials and Methods), we extracted the error angle at five different time points: 100 ms before pursuit start, at the start of the distal stage, 100 ms into the distal stage, at the start of the proximal stage and 100 ms into the proximal stage.

We found that 100 ms before pursuit started, the error angles were evenly distributed, in both the azimuth (Fig. 7Di) and the elevation plane (Fig. 7Ei). This was also the case at the start (0 ms) of the distal stage (azimuth and elevation, Fig. 7Dii,Eii). This is because hoverflies were flying in different directions when the target caught their attention. We found no correlation between the error angle at pursuit start and the hoverfly angular speed for the two smaller bead sizes (8 and 10 mm; Fig. S4), but there was a correlation for the 38.5 mm bead (R²=0.10; Fig. S4).

We found that 100 ms into the distal stage, the mean error angle was directed anteriorly (Fig. 7Diii,Eiii, red line). At the start of the proximal stage, the mean error angle was even more strongly anterior (Fig. 7Diii,Eiv, red line), but this strong directional preference had decreased after 100 ms (Fig. 7Dv,Ev, red line). Taken together, our data suggest the hoverflies attempted to adjust their flight direction to keep the target anterior relative to its direction of flight, but there was a large variation (Fig. 7D,E).

Our data above suggested that during the distal stage the hoverflies attempted to rapidly decrease the distance to the target (Fig. 5E–G), potentially by intercepting its future position (Fig. 2A,B), as suggested in field work (Collett and Land, 1978). In contrast, during the proximal stage they appeared to follow the speed of the bead more closely (Fig. 5H,I,K), by keeping the target image anterior (Fig. 7Div,Eiv). Target interception can be achieved by keeping either the error angle (Fig. 7A,B, also referred to as deviated pursuit) or the bearing angle (Fig. 7C, also referred to as proportional navigation) constant. We therefore investigated whether the error or bearing angles in 3D space were kept constant, by quantifying how much they changed from one time point to another, two frames later (16.7 ms). The closer to 0 the delta angle is, the more constant the angle is. We quantified the delta error and delta bearing angles in the 1 s preceding pursuit start, during the distal and proximal stages.

We found that the delta error angle was not significantly different between the three stages, for any of the bead sizes (Fig. 7F–H). In addition, during the distal and proximal stages, there was a large variation of delta error angles, suggesting that the hoverflies did not attempt to keep the error angle constant. In contrast, the bearing angle (Fig. 7C) was held much more constant (delta bearing angle close to 0) before the pursuit and during the distal stage, compared with that during the proximal stage (Fig. 7I–K). This suggests that the hoverflies could use proportional navigation (Fig. 7C) to intercept the bead during the distal stage of the pursuit, whereas they may use smooth pursuit (keeping the error angle close to 0; Fig. 7Div,Eiv) during the proximal stage. Future modelling endeavors will help elucidate this.

We show that E. tenax males pursue artificial targets ranging from 6 to 38.5 mm in diameter (Table 1) in an indoor flight arena (Figs 1A and 2; Movies 1–3). We show that male E. tenax pursue targets from above as well as below (Figs 4 and 7), with pursuits lasting several seconds (Fig. 5). At the start of the pursuit, the hoverflies fly fast to decrease the distance to the bead, whereas they adjust their translational speed to the bead speed when they are proximal to it (Fig. 5; Fig. S3), only rarely physically interacting with it (Fig. 6). We found that male E. tenax are unlikely to use strict heuristic rules based on angular size or speed (Fig. 3; Figs S2–S4), and that pursuits of the largest bead (38 mm; Figs 3D, 5F and 6) differed, suggesting possible categorization.

The pursuit flight speed in our indoor arena (Figs 2F and 5I) was lower than the 10 m s⁻¹ recorded in the field, and the acceleration (Fig. S1A) was also lower than the 33 m s⁻² measured in the field (Collett and Land, 1978). Therefore, while hoverflies pursued targets in the arena, they were not flying as fast as they do in the field. However, the high angular speeds associated with pursuit start (Fig. 2E,G; Fig. S2E) were similar to field measurements (Collett and Land, 1978), suggesting that turning behaviors were naturalistic.

We found it unlikely that hoverflies use strict matched filters, also referred to as heuristic rules, to trigger pursuit start, as the angular size and speed covered a large range of values (Fig. 3B,D). Nor did they seem to adjust their saccade-like turn followed by a translational speed increase (Fig. 2C–G; Movies 1–3) at pursuit start to the angular size or speed of the target (Figs S2 and S4), as previously suggested (Collett and Land, 1978). It is possible that being indoors affected territoriality, and thus reduced the saliency of cues that might be important in the field. Indeed, the pursuit ratio was relatively low (Table 1) compared with field behavior (e.g. Wellington and Fitzpatrick, 1981). Furthermore, having many hoverflies in the arena simultaneously might have added competition, which could affect underlying heuristic rules. Indeed, Drosophila fly more erratically when density is low (Combes et al., 2012), suggesting that group dynamics affect flight behavior. From our data it is therefore unclear what cues triggered pursuit start. As all our experiments used a black bead moving against a brighter background (Fig. 1A; Movies 1–3), it would be interesting to determine whether this dark contrast is an important driver.

When blowflies pursue a bead, they sometimes follow it for a long time, during which they keep a fixed distance to the bead, by controlling their forward speed based on the target's angular size (Boeddeker et al., 2003), so that physically smaller beads are followed at a closer distance. However, we did not see a similar relationship between bead size and distance (Fig. 5A,D) or a correlation between hoverfly flight speed and the target's angular size (Fig. S3A,B, dotted lines). In contrast to blowflies, Syritta pipiens hoverflies control their forward speed based on the distance to the target (Collett and Land, 1975a), as do houseflies (Wehrhahn et al., 1982). We found only a weak correlation between hoverfly flight speed and distance to the bead (Fig. S3A–C).

It is unlikely that the lack of correlation was caused by technical limitations, such as our relatively low recording rate of 120 frames s⁻¹. Indeed, behavioral delays during target pursuit are often much longer than the 8.3 ms temporal resolution provided in our set-up. For example, when filmed at 1000 frames s⁻¹, predator steering changes have delays of 28 ms in the robber fly Holcocephala, 18 ms in the killer fly Coenosia and 47 ms in the dragonfly Plathemis (Fabian et al., 2018; Mischiati et al., 2015). Furthermore, Lucilia blowflies display behavioral delays of between 10 and 32 ms, when recorded at 190 frames s⁻¹ (Varennes et al., 2020), which is close to the temporal resolution we used.

Previous work suggested that E. tenax males pursue targets traveling at female flight speeds (Collett and Land, 1978). However, in the field, Eristalis spp. males pursue artificial targets moving at 5–12.5 m s⁻¹ (Collett and Land, 1978), which is faster than typical female Eristalis spp. flight speeds (Thyselius et al., 2018). We showed here that hoverflies also pursue beads moving much slower than this, and even stationary targets (Fig. 3C). This is important as male E. tenax often wait for females to land before trying to mate with them (Fitzpatrick, 1981). Male E. tenax are capable of flying very fast, up to 10 m s⁻¹ (Collett and Land, 1978). Indeed, even in our limited physical space, we found pursuit speeds at individual time points of up to 3.3 m s⁻¹ (Fig. 5J). This could suggest that the males perceived fast-moving beads as a male competitor rather than a cruising female. Furthermore, escaping female E. tenax can fly at up to 1.5 m s⁻¹ (Thyselius et al., 2018), so the faster beads might have been perceived as escaping females.

Dragonfly and robber fly eyes often have areas with improved spatial and temporal resolution, so-called acute zones. They attempt to keep the target image in this acute zone during pursuit (Olberg et al., 2007; Wardill et al., 2017), a strategy shared with non-predatory dipterans, such as houseflies (Wagner, 1986; Wehrhahn et al., 1982) and the hoverfly S. pipiens (Collett and Land, 1975a). We found that male E. tenax pursue targets from above as well as from below (Figs 4 and 7). Male E. tenax harbor a dorso-frontal bright zone (Straw et al., 2006). Even if we did not reconstruct the head movements, the target image is unlikely to fall within the dorsal visual field when the hoverfly is flying above the bead (Figs 4 and 7E). However, the hoverflies attempted to keep the bead anterior relative to the flight direction, especially at the start of the proximal stage (Fig. 7Div,Eiv). The anterior visual field has higher resolution than the lateral visual field (Straw et al., 2006).

We broke down each pursuit into two stages, where the distal stage appeared to be optimized to rapidly decrease the distance to the target, and the proximal stage to staying close to the target (Fig. 5). Indeed, the distal stage could use proportional navigation based on the bearing angle, whereas the proximal stage did not (Fig. 7I–K). For E. tenax males, the goal may not be to catch a target (Fig. 6), but to either chase it out of its territory if it is an intruder, or stay close until it lands if it is a potential mate. Similar shadowing behavior has been described in dragonflies, previously referred to as motion camouflage (Mizutani et al., 2003). Indeed, staying close to the target allows the hoverfly to gather more information. In the field, Eristalis spp. males often chase intruders out of their territories without contact (Fitzpatrick, 1981). The males also often follow females, waiting for them to settle before mating, rather than grasping them in the air (Fitzpatrick, 1981). Indeed, we found that E. tenax males followed the artificial target for up to 3 s (Fig. 5H), and that when proximal, the hoverfly's translational speed was correlated with the bead speed (Fig. 5K), probably to stay in close proximity, even if it was well within its capacity to speed up (e.g. Fig. 5I) and catch the target. Indeed, they rarely got close enough to the 8 or 10 mm beads to suggest physical contact (Fig. 6). It might be beneficial for hoverflies to keep a greater distance to the target to avoid physical and potentially lethal contact. Could it thus be that E. tenax have developed a strategy that will take them close to but rarely in contact with their targets?

We thank Annika Olsén, Mats Thyselius, Moa Thyselius, AB Cederholms Lantbruk, Louise Gustafsson, and current and past lab members for valuable feedback during the many stages of this work.