Mathematical modeling shows that ball-rolling dung beetles can use dances to avoid competition

Ball-rolling dung beetles shape a portion of dung into a ball and roll it away from the dung pile for later burial and consumption. These beetles perform dances (rotations and pauses) atop their dung balls in order to choose an initial rolling direction and to correct their rolling direction (reorient). Previous mathematical modeling showed that dung beetles can use reorientation to move away from the dung pile more efficiently. In this work, we study whether reorientation can help beetles avoid competition (i.e., avoid having their dung balls captured), and if so, under what circumstances? This is investigated by implementing a model with two different type of beetles, a roller with a dung ball and a searcher which seeks to capture that dung ball. We show that reorientation can help rollers avoid searchers in a wide range of conditions, but that there are some circumstances in which rolling without reorienting can be a beetle’s optimal strategy. We also show that rollers can minimize the probability that their dung ball is captured without making precise measurements of the time interval between dances or the angular deviation for dances.


Introduction
For dung beetles, dung is a highly prized resource. Many dung beetles shape a portion of dung at a dung pile into a ball and roll it away, for burial and later consumption Baird et al. (2012); Bartholomew and Heinrich (1978). These beetles are known as rollers. Some rollers, in order to save the labor of making their own dung balls, attempt to steal others' balls instead Bartholomew and Heinrich (1978); Hanski and Cambefort (1991). Hence, competition for dung balls at or near the dung pile is very common. In the vicinity of large dung piles, as many as thousands of dung beetles and dozens of dung beetle species have been observed. Fights over a dung ball can occur between dung beetle species or within the same species Heinrich and Bartholomew (1979). Naturally, rollers seek to avoid these fights as much as possible, since even rollers that win their fight (and keep their dung ball) may incur a significant energy cost. Hence, a roller's short-term task, prior to burying its dung ball, is to move away from the dung pile as efficiently as possible Baird et al. (2012); Dacke et al. (2013aDacke et al. ( , 2014. Maintaining a straightline rolling path away from the pile is optimal for this purpose, but many factors make this path difficult to achieve. Namely, beetles are facing backwards while pushing dung balls larger than themselves, imperfections in the dung ball can make balls drift off course, and beetles need to roll their balls around obstacles and over uneven ground Baird et al. (2012); Tomkins et al. (1999).
Research has shown that dung beetles use celestial cues, including the sun, the moon, polarized light, and the Milky Way, to navigate Byrne et al. (2003); Dacke et al. (2013aDacke et al. ( , b, 2003. Prior to rolling a dung ball away from the dung pile, most beetles perform a "dance" (a sequence of rotations and pauses) atop the ball. This dance allows beetles to get their bearings and to choose a rolling direction Baird et al. (2012); Byrne et al. (2003). Dances are also observed later in the rolling process, particularly when beetles lose control of dung balls or are driven off their intended course by obstacles or uneven ground Baird et al. (2012). This shows that beetles use dances not only to select an initial rolling direction, but also to help maintain a straightline bearing (i.e., to reorient themselves).
Experiments by Baird et al. (2012) have given us insight into how beetles choose the timing of dances. The authors constructed straight and semicircular tunnels, and directed rollers into these tunnels. Beetles danced significantly more often after emerging from a semicircular tunnel than after a straight tunnel Baird et al. (2012). This indicates that the dung beetle dance is a reorientation behavior. While beetles directed 180 degrees off course usually performed dances, the authors did not observe a consistent angular deviation (i.e., angle between the current path and a straightline path) at which rollers danced in the field Baird et al. (2012). So, perhaps beetles reorient themselves when there is a major disturbance in their path (i.e., a forced u-turn or losing control of a ball).
In addition to experiments in the field, previous mathematical modeling has addressed the movement of ball-rolling dung beetles. Bijma et al. (2021) numerically simulated dung ball transportation using a quasi-directed random walk. The authors varied both the size of balls and the roughness of terrain, finding that larger balls are transported more effectively on rough terrain than smaller ones. They also found that random-noise activity can actually shorten ball transportation times, and can help beetles escape from deep valleys. Multiple models have also addressed movement with direction resetting. Peleg and Mahadevan (2016), inspired by observations of dung beetle navigation strategies, developed a model in which an agent, moving in the presence of noise, navigates using a random walk interspersed by reorientation events, which reset its direction. The authors show that this mechanism for movement becomes a biased random walk at a long time scale, and also find the optimal frequency of reorientations as a function of noise level. In order to investigate the potential benefits of reorientation for dung beetles specifically, Yin and Zinn-Björkman (2020) formulated a mathematical model of dung ball rolling paths in a circular domain with a similar structure to the model from Peleg and Mahadevan (2016). The authors modeled the rolling process using a persistent (correlated) random walk, in which the rolling direction at each time step was drawn from a wrapped normal distribution centered at the previous rolling direction, the idea being that dung balls would tend to drift off their intended course over time. They also incorporated dancing (reorientation) by redirecting beetles to the "optimal direction" (direction of the shortest path to the boundary of the domain). In addition to being redirected, beetles incurred a "time penalty" from reorientation (i.e., they remained stationary for a given amount of time). The framing question from this work was "How often and under what circumstances should dung beetles reorient in order to minimize the average time to reach the boundary?" Spending as little time as possible in the vicinity of the dung pile seems like an ideal way to reduce competition for dung balls. However, in this work, we study competition more directly by introducing a second type of beetle to the models from Bijma et al. (2021) and Yin and Zinn-Björkman (2020). This beetle, which we call a "searcher," randomly moves in the domain, then moves directly towards the roller when the roller and searcher are sufficiently close (i.e., when the roller has been detected). This model allows us to find out if reorientation can help rollers avoid fights, and if so, under what circumstances.
We begin the next section by recapping previous mathematical modeling in Yin and Zinn-Björkman (2020). In Movement of searching beetle, we describe our model for dung beetle competition, which we refer as the "rollersearcher model." This model incorporates the movement model for rolling beetles from Yin and Zinn-Björkman (2020), while adding an additional and distinct movement model for searchers. In Results for a single searcher, we implement our roller-searcher model and investigate how various parameters affect the percentage of beetles captured by searchers. We study this "percentage captured" for a single roller and searcher, in the presence and absence of reorientation (Choosing the optimal reorientation period and error threshold), when reorientation assessment is "noisy" (Effect of assessment error in reorientation), and as detection radius, searching speed, and dancing time vary (Varying parameters for dancing and searching). We conclude by measuring percentage captured when there are multiple searchers present, and then determining the critical number of searchers for which reorientation is no longer beneficial for the roller (Escaping multiple searchers).

Movement of the rolling beetle: recap of previous mathematical modeling
Yin and Zinn-Björkman (2020) developed a simple mathematical model to describe movement of ball-rolling dung beetles in the time period immediately after forming a dung ball, during which beetles seek to move away from the dung pile as quickly as possible. In this model, beetles were placed (with their dung balls) in the center of a circular arena of radius R (i.e., at the origin (x, y) = (0, 0) ), and each beetle was given a random initial rolling direction. In the absence of "dances," movement of beetles was simulated using a persistent random walk, in which beetles would tend to "drift" away from their current rolling direction (due to the unevenness of the ground beneath them, as well as noise in their motor and sensory systems Khaldy et al. (2019)). This persistent random walk approach is common in modeling movement of foraging animals, including dung beetles Kareiva and Shigesada (1983); Bovet and Benhamou (1988); Turchin (1991); Codling et al. (2008); Byers (2001); Bailey et al. (2018); Peleg and Mahadevan (2016).
In Yin and Zinn-Björkman (2020), each new rolling direction was drawn from a wrapped normal distribution with standard deviation d , centered at the previous rolling direction (see Yin and Zinn-Björkman (2020), Algorithm 1). This parameter d was called the "drifting tendency" of the beetle. Each simulation ended when the beetle reached the boundary of the circular domain (i.e., when x 2 + y 2 ≥ R 2 ). The authors then incorporated dancing into this "basic drifting" model. Each time a dance was performed, the model beetle was reoriented to the direction of the shortest path to the boundary (designated as the "optimal direction"), while also being given a "time penalty," representing the duration of a dance. The authors considered two strategies for dancing: periodic reorientation, in which the time T between consecutive dances was fixed, and error threshold reorientation, in which beetles danced only when their "error" e (angle between optimal and current rolling directions) eclipsed a certain threshold (see Yin and Zinn-Björkman (2020), Algorithms 2 and 3).
Yin and Zinn-Björkman (2020) compared their simulation results with real data from dung beetles in model arenas Dacke et al. (2013a), predicted the behavior of beetles in varying ground and celestial conditions, and suggested future experiments that would be illuminating. However, a limitation of the authors' model was that they did not address competition between beetles. Instead, each beetle was treated as independent (i.e., beetles were placed in the model domain one at a time), as in experiments by Dacke et al. (2013a). In this work, we use the same mathematical model as Yin and Zinn-Björkman (2020) to simulate a rolling beetle (a "roller"), but we add an additional beetle, which we call a "searcher," which aims to find the roller and steal its dung ball. The goal of the roller is to reach the boundary of the circular domain before its dung ball is stolen (subsequently referred to as "captured") by the searcher. Hence, the roller's aim is to minimize the probability of its dung ball being captured. This is a different optimization problem to the one discussed in Yin and Zinn-Björkman (2020), in which the goal of the model beetle was to minimize its time taken to reach the boundary. In Tables 1 and 2, we list the parameters (along with their default values) and variables used for the rolling beetle in Yin and Zinn-Björkman (2020). We will use the same parameter values as Yin and Zinn-Björkman (2020) throughout this work, unless otherwise specified.

Movement of the searching beetle
As discussed above, movement of our rolling beetle is described by Algorithms 1-3 in Yin and Zinn-Björkman (2020). On the other hand, the searching beetle in our model performs an unbiased random walk. The initial position of searchers is chosen uniformly at random, from 5 cm to 100 cm away from the central dung pile at the origin. Direction from the dung pile is also chosen uniformly at random. We choose these initial conditions with the thought that searchers will want to linger near the pile in order to steal other beetles' dung piles, but not so close that they are at the chaotic center of the pile. Every second (i.e., every 10 time steps in numerical simulations), the searching beetle chooses a new search direction from a uniform distribution on [− , ) . However, if the searcher detects the roller, which occurs when the searcher and roller are within a critical distance D s , the searcher changes its behavior. Instead of performing an unbiased random walk, it now chooses its direction at each time step so as to "beeline" directly towards the roller. Searching beetles move at speed v s throughout simulations. We use reflecting boundary conditions for the

Variable Interpretation
Time needed to first reach the boundary L Length of rolling path to the boundary (t) Direction of rolling beetle at time t (t) Optimal direction to boundary at time t n r Number of reorientations per rolling beetle searcher, so that the searcher never leaves the domain. The simulation ends when one of the following scenarios occurs: -The searcher and roller are within a critical distance R s of one another. The roller's dung ball is considered "captured" in this case. -The roller has moved outside of the domain (i.e., the stopping condition from Yin and Zinn-Björkman (2020)). This is called the "escaped" case.
Note that we consider the roller to have escaped upon reaching the boundary of our domain regardless of whether or not it has been detected by a searcher. Our assumption here is that searchers do not want to stray too far away from the dung pile, and will give up their pursuit once their quarry reaches a critical distance from the pile. For a table of parameters for the searching beetle, see Table 3, and see the supplementary material for an algorithm of our roller-searcher model. We chose a default searching speed of 5 cm/s, with the thought that searching beetles are almost certainly faster than rolling beetles, as they are unburdened by a dung ball.
In Fig. 1, we display sample paths for the roller and searcher for each of the scenarios described above. In Fig. 1(a), the searcher detects the roller near the boundary of the domain, beelines toward the roller, and captures its dung ball. In Fig. 1(b), the searcher also detects the roller near the boundary of the domain, but the roller escapes the domain prior to being captured. In this figure, we use the default rolling speed v = 3.83 cm/s and searching speed v s = 5 cm/s. Default parameter values for D s and R s are 20 cm and 5 cm, respectively ( D s will be varied in Varying parameters for dancing and searching).

Results for a single searcher
Choosing the optimal reorientation period and error threshold In their previous study, Yin and Zinn-Björkman (2020) showed that "dancing" (reorientation) improves mean first passage time to the boundary of the domain over "drifting" alone, over a wide range of drifting tendencies d . Reorientation introduces a balancing act: frequent reorientations lead to a very straight path, but the beetle will spend a long time performing dances. The authors found that, as d (which represents unevenness of the ground, along with other factors that may push the beetle off course) increases, it is optimal to increase the frequency of reorientations Yin and Zinn-Björkman (2020).
Our optimization problem is different, though related: our model rolling beetles seek to minimize the probability that they are captured by a searcher before exiting the domain. Intuitively, reducing the amount of time that rollers spend in the domain would decrease their probability of being captured. However, rollers are stationary during reorientations, meaning that they will have no opportunity to escape beelining searchers if detected during a dance. Hence, it is not obvious that reorientation will be beneficial in this case. In order to test the effect of reorientation on capture probability, we simulate 2000 independent roller-searcher pairs for a wide range of drifting tendencies, from d = 3 ∕60 to d = 30 ∕60 , with increments of 3 ∕60 . Then, for each value of d , we calculate the percentage of trials that end with the rolling beetle being captured. We repeat these simulations for three different rolling strategies: no reorientation (the basic drifting model), periodic reorientation, and error threshold reorientation. For the latter two strategies, we find the optimal reorientation period T * and reorientation error threshold * e that minimize the probability of being captured. Results for percentage captured are shown in Fig. 2.
We find that both reorientation techniques substantially decrease percentage captured from the no reorientation case. As drifting tendency d increases, reorientation becomes more and more beneficial. We also see that error threshold reorientation is the superior technique to periodic reorientation for d ≥ ∕5 , but for lower drifting tendencies, there is very little difference in percentage captured. Finally, we see that for all three cases, percentage captured trends upward as d increases. This makes sense since rolling beetles take longer on average to reach the boundary of the domain as d increases Yin and Zinn-Björkman (2020).
In Fig. 3, we display how the optimal reorientation period T * and optimal error threshold * e depend on drifting tendency d . We find that T * generally decreases as d increases, while * e generally increases (although there are several outliers, particularly for error threshold reorientation at lower drifting tendencies). This indicates that, as the ground gets more uneven, beetles using periodic reorientation should reorient more frequently, while beetles using error threshold reorientation should tolerate larger deviations from the optimal path. These results are in agreement with those of Yin and Zinn-Björkman (2020) for mean first passage time. Fig. 2 Percentage of rolling beetles captured (out of 2000 total) for three different rolling strategies: basic drifting (no reorientation), periodic reorientation with optimal period T * , and error threshold reorientation with optimal error threshold * e Fig. 3 Reorientation period (a) and error threshold (b) that minimize percentage of beetles captured, for a broad range of drifting tendencies ( d = 3 ∕60, 6 ∕60, ..., 30 ∕60) How important is it for the rolling beetle to precisely choose the optimal reorientation strategy? To answer this question, we choose a single drifting tendency d = ∕4 and compute the percentage of beetles captured for a broad range of reorientation periods and error thresholds. We find that, for both reorientation strategies, there is a wide range of choices that produces a capture percentage close to the minimum (Fig. 4). For example, any choice of reorientation period between T = 1.4 s and T = 8.5 s or error threshold between e = 34 ∕90 and e = 70 ∕90 , inclusive, gives a percentage captured within 5 percentage points of the optimum. In all subsequent sections, we will implement periodic and error threshold reorientation with optimal reorientation periods and error thresholds, respectively, such that capture percentage is minimized.

Effect of assessment error in reorientation
In the previous section, we showed that a broad range of choices for reorientation periods or error thresholds yield similar success in avoiding searchers. This suggests that rolling beetles do not need to be extremely precise in the timing of reorientations in order to achieve close to optimal results. In this section, we explore this idea further by adding a noise component (i.e., an error in the assessment of when to reorient) to periodic and error threshold reorientation.
For "noisy" periodic reorientation, rolling beetles choose a reorientation period T, but cannot measure this time precisely. Instead, the time T a between successive reorientations is chosen as follows: Here, the random variable Z is sampled from a normal distribution with mean 0 and standard deviation p . We refer to p as the assessment error for periodic reorientation. Similarly, for "noisy" error threshold reorientation, rolling beetles cannot precisely choose the same error threshold e each time they reorient. Instead, the error threshold a between each reorientation is where t is called the assessment error for error threshold reorientation.
To test whether noise has a detrimental effect on capture probability, we begin by choosing a single drifting tendency d = ∕4 . We then implement ten different levels of detection error (noise): p = 0.1, 0.2, ..., 1 s for periodic reorientation and t = ∕90, 2 ∕90, ..., 10 ∕90 for error threshold reorientation. For each noise level, we find the optimal reorientation period T * and error threshold * e , then find the percentage of rolling beetles captured for these optimal values. We then compare these capture percentages to those for the zero noise case. Results are shown in Fig. 5. We find that noise does not have a significant effect on percentage captured for d = ∕4.
Does noise have more influence for other drifting tendencies? To address this question, we choose a single value for detection error ( dt = 1 s for periodic reorientation, de = 10 ∕90 for error threshold reorientation), and find the optimal percentage captured for values of d from 3 ∕60 to 30 ∕60, with increments of 3 ∕60. We then compare these capture percentages to those from the no-noise case (see Fig. 2, blue and green curves). Results are shown in supplementary Fig. S1. We find that, again, noise has no discernible effect on percentage captured. The results from this section reinforce that our model beetles do not need to make precise measurements in order to achieve close to optimal results. It seems that it is the average time between reorientations and average error threshold that influence percentage captured, and not the precision of these measurements prior to each dance.

Varying parameters for dancing and searching
In this section, we vary several of our rolling and searching parameters and measure their impact on capture percentage. When one of these parameters is varied, all other parameters are set to their default values (see Tables 1 and 3). In Choosing the optimal reorientation period and error threshold and Effect of assessment error in reorientation, we used a "dance duration" (time penalty) t pen = 5 s in accordance with experimental results from Baird et al. (2012). However, in field experiments, this duration was highly variable from beetle to beetle and between subsequent dances Baird et al. (2012). That being said, how beneficial is it to the rolling beetle to complete dances quickly? To investigate, we run 2000 simulations of our roller-searcher model for each time penalty t pen = 1, 2, ..., 10 s and for both reorientation methods. We then compute the percentage of beetles captured for each value of t pen . Surprisingly, increasing t pen only minimally increases the probability that rolling beetles are captured. In fact, for both periodic and error threshold reorientation, a tenfold increase in time penalty results in only about a 15% increase in capture percentage (Fig. 6), and capture percentage for t pen = 10 s is still much lower with reorientation than in its absence. This indicates that reorientation can remain a valuable strategy for avoiding competition, even for beetles that take a relatively long time to perform dances.
Searching speed v s is another potentially interesting parameter to vary in our model. Experimental data on searching speeds is lacking, and speeds are likely highly Fig. 5 Percentage of beetles captured for various levels of assessment error (i.e., noise in reorientation), for periodic reorientation (a) and error threshold reorientation (b) variable in time and between dung beetle species. Hence, it is important to test our model on a broad range of searching speeds. We use rolling speed v = 3.83 cm/s as a lower bound for searching speed, and then measure percentage of rolling beetles captured for searching speeds v s = 1, 1.25, 1.5, ..., and 3 times v. We run these simulations for each rolling strategy (basic drifting, as well as periodic and error threshold reorientation). Unsurprisingly, we find that percentage captured increases with v s , though not dramatically (Fig. 7). This increase appears to be roughly linear for all three rolling models, and reorientation decreases percentage captured over drifting alone for all values of v s we tested.
In the previous two sections, we used detection radius D s = 20 cm for searchers. Ground conditions could potentially influence this radius. For example, uneven ground with obstacles could obscure searchers' vision, while even, unobstructed ground could allow for detection of rollers over a longer distance. We now vary D s from 5 cm to 50 cm, with increments of 5 cm. D s = 5 cm indicates no detection ability for searchers, since D s equals the capture radius R s in this case. For each value of D s , we run 2000 trials for each rolling strategy. Results for percentage captured are shown in Fig. 8. We find that percentage captured increases roughly linearly with D s . Reorientation provides a fairly consistent amount of benefit over the drifting model over the range of values for D s , with error threshold reorientation remaining the slightly superior method to error threshold reorientation.

Escaping multiple searchers
In the wild, of course, each roller is not necessarily pursued by a single searcher -it is very possible that a roller will need to evade several searchers during its journey with a Fig. 6 Percentage of rolling beetles captured over a range of time penalties (dancing times), for periodic and error threshold reorientation Fig. 7 Percentage of rolling beetles captured over a range of speeds for the searching beetle, in the presence and absence of reorientation dung ball Heinrich and Bartholomew (1979); Hanski and Cambefort (1991). In this section, we adjust our rollersearcher model to incorporate multiple searchers. Each searcher starts from a random location in the domain (as before, between 5 and 100 cm from the dung pile), and has the same movement behavior (unbiased random walk with beelining) as in the single searcher model (see Movement of the searching beetle). Our simulations end when the roller escapes or when it is captured by one of the searchers.
We now simulate our roller-searcher model for number of searchers n s = 1, 2, 3, ..., 10 , 12, and 15, with drifting tendency d = ∕4 and detection radius D s = 20 cm. For each value of n s , we compute the percentage of rolling beetles captured for the basic drifting model, optimal periodic reorientation, and optimal error threshold reorientation. Results are shown in Fig. 9. Interestingly, we find that reorienting to escape searchers is no longer beneficial for n s ≥ 4. We also see that, as usual, error threshold reorientation is the slightly superior technique to periodic reorientation. We find that there is no discernible trend for optimal period and error threshold as number of searchers increases (data not shown).
How does the critical number of searchers at which reorientation is no longer beneficial depend on the parameters in our model? We choose to vary searching speed and time penalty, as these parameters directly affect the performance of the searcher and roller, respectively. Our set of searching speeds v s is 1, 1.5, 2, 2.5, and 3 times v, and our set of time penalties t pen is 1, 3, 5, 7, and 9 s. For each pair of values for searching speed and time penalty, we compute percentage captured for each reorientation method as number of searchers n s increases. The simulation stops when we find a value of n s for which percentage captured with reorientation exceeds percentage captured for the basic drifting model. We call this value of n s the critical searcher count, denoted by Fig. 8 Percentage of rolling beetles captured for the basic drifting model, periodic reorientation, and error threshold reorientation, for a range of detection radii D s for searchers Fig. 9 Percentage of rolling beetles captured for three different rolling strategies and for various number of searchers n * s . Values of n * s for each searching speed and time penalty are shown in Fig. 10. Note that n * s generally decreases as t pen increases, since reorientation becomes less effective for eluding capture when dances take longer. However, there is no obvious trend in n * s as v s increases. This is likely because faster searchers are more effective at catching rollers, regardless of whether or not rollers reorient. Finally, we see that values of n * s are generally higher for error threshold reorientation ( Fig. 10(a)) than for periodic reorientation (Fig. 10(b)), further reinforcing that error threshold reorientation is the superior reorientation technique for avoiding capture.

Discussion
In this work, we formulated and simulated a simple competition model for ball-rolling dung beetles. The competition occurs between a "roller," which has formed a dung ball and seeks to roll it away from the main dung pile, and a "searcher," which aims to discover the roller and steal its dung ball to save the labor of creating one itself. We do not include the battle for the dung ball, instead assuming that the roller's ball is "captured" as soon as it comes into close proximity with the searcher. As far as we are aware, this is the first mathematical model to address competition for dung balls.
That being said, many previous studies have modeled the searching process for foraging organisms Namboodiri et al. (2016); Viswanathan et al. (1999); Ross et al. (2018);da Luz et al. (2016); Viswanathan et al. (2011);Turchin (1991). Namboodiri et al. (2016) used an agent-based mathematical model to explain why path lengths for animals that learn about their environments are power-law distributed de Jager et al. (2011);Sims et al. (2008); Raichlen et al. (2014). The foraging model used by Viswanathan et al. (1999) is similar to our model for searchers: in Viswanathan et al. (1999), if a target is sufficiently close to the forager, the forager beelines toward it, and if no targets are sufficiently close, the forager chooses a new direction at random and a distance from a power-law distribution. The primary difference between our model and foraging models such as Viswanathan et al. (1999) is that our searchers use a constant step length, rather than a power-law distributed step length. Another notable difference is that our "targets" (ball-rolling dung beetles) are in motion, whereas foraging targets are typically modeled as stationary (though there are exceptions, including the model in Bartumeus et al. (2002)). Both Namboodiri et al. (2016) and Viswanathan et al. (1999) demonstrate that Lévy flights, random walks in which the step length is drawn from a heavy-tailed distribution, optimize foraging efficiency and reproduce similar path length distributions to those observed in experiments. By tracking positions of searching beetles in the field and fitting this position data, one could perhaps determine whether these beetles use Lévy flights, and mathematical modeling could address whether Lévy flights result in a higher capture percentage for searchers than an unbiased random walk.
Our model is also closely related to previous modeling of predator-prey systems Abe and Kasada (2020); Oshanin et al. (2009);Domenici et al. (2008Domenici et al. ( , 2011; Weihs and Webb (1984); Krapivsky and Redner (1996): the "prey" in our context being the roller, which seeks to avoid capture, and the "predator" being the searcher. Abe and Kasada (2020) showed that Lévy flights, in addition to optimizing search efficiency, also benefit prey in escaping from predator's attacks. Could Lévy flights also benefit the rollers in our model? This could be an interesting question for future work. In Oshanin et al. (2009), the authors study survival of a prey animal hunted by N predators. Prey and predators move on a lattice, with prey considered caught when it occupies the same grid point as a predator. Predators perform random walks, while prey are "lazy," only moving when a predator Fig. 10 Critical searcher counts n * s , at which reorientation is no longer a beneficial strategy for rollers, as time penalty t pen and searching speed v s vary. Counts are shown for rollers employing periodic reorientation (a) and error threshold reorientation (b) is within their sighting range. This work demonstrates that short-sighting prey undergoes a superdiffusive motion, while far-sighting prey undergoes diffusive motion. In our work, we briefly explored sightedness of the predator by varying its detection radius (Varying parameters for dancing and searching), but we did not allow our prey to detect predators, since we assumed that their only goal was to maintain a straightline path away from the dung pile. An interesting extension of our work would be for rollers to use dances not only to beeline away from the dung pile, but also to detect nearby searchers. In the case that an encroaching searcher was detected during a dance, the roller could choose its new rolling direction by a weighted-sum approach, with certain weights being given both to the boundary direction and to the direction away from a searcher.
Previous work by Yin and Zinn-Björkman (2020) was motivated by experiments from Baird et al. (2012) and Dacke et al. (2013a), who studied the behavior of isolated ball-rolling beetles. Hence, the mathematical model formulated in Yin and Zinn-Björkman (2020) focused on a single roller, in an effort to reasonably simulate ball-rolling paths. Many of the findings from Yin and Zinn-Björkman (2020) focused on the role of "dancing" (reorientation) in reducing time to reach the boundary of a circular domain. We were interested in the benefits that reorientation can provide rollers in avoiding battles for their dung ball.
In our model (as in Yin and Zinn-Björkman (2020)), reorientation is implemented either at fixed time intervals (for periodic reorientation) or at a specific angular deviation (for error threshold reorientation). Both methods would require unreasonably precise measurements from rollers in order to minimize the probability of being captured by searchers. However, we found that a broad range of values for T (time period between reorientations) and e (angular deviation from the optimal path) produce close to optimum results, and that reorientation with noise is equally effective to "perfect" reorientation. These are significant findings, as they indicate that reorientation can be an effective strategy for avoiding searchers even when timing of reorientations is unpredictable. Indeed, results from Baird et al. (2012) have demonstrated that there is not a fixed angular deviation from a straight path at which rolling beetles dance. It would be interesting to also record the timing of dung beetle dances in the field, to test whether rollers might reorient periodically, and if they do, how well they can regulate the time between consecutive reorientations.
As there are often hundreds or even thousands of beetles at a dung pile, it is unreasonable to assume that each roller is pursued by a single searcher Heinrich and Bartholomew (1979). We therefore devoted Escaping multiple searchers to study how adding additional searchers might change a roller's optimal strategy. Interestingly, we found that there is a critical number of searchers at which reorientation is no longer a beneficial strategy (i.e., it is optimal to roll the dung ball without any course correction), and that this critical number of searchers depends on speed of searchers and the duration of roller's dances. As number of searchers increases, the time penalty associated with reorientation becomes more and more costly. Each time a roller remains stationary in our model, it gives searchers more opportunity to "beeline" towards it and capture its dung ball. Hence, rolling "blindly" (without reorientation) becomes an optimal method, as beetles using this strategy are always moving. This result could potentially be tested experimentally, either in the field or in a model arena, by observing whether or not a roller's behavior changes as its number of competitors increases.
A limitation of our model is that it does not directly account for obstacles in the way of the roller or the searcher. Incorporating obstacles into the roller-searcher model (by using a jump-diffusion model or similar) could be an interesting direction to pursue. Obstacles in the rolling path would seemingly be of greater detriment to rollers than to searchers, as rollers are facing backwards with a large dung ball blocking its view of the path ahead, while searchers are unencumbered and could potentially fly over obstacles Tomkins et al. (1999). Movement of our searching beetle (random walk with beelining) is also very basic. If searchers' positions could be tracked in the field (as Dacke et al. (2013a) did for rollers), we could perhaps formulate a more accurate model to describe movement of searchers.
In future, we hope to investigate dung beetle competition more directly by including the fight over dung balls in our model, in addition to the pursuit of dung balls by searchers. For example, a roller could potentially fight off a searcher, or win back its dung ball after having it stolen. One potential approach could be to use an agentbased model, in which the probability of a beetle winning a "battle" for a dung ball depends on a variety of factors, including size, body temperature, as well the number and recency of prior battles. In this agent-based approach, rollers and searchers could switch roles following fights (i.e., a searcher that wins a fight would become a roller, while the roller would now be a searcher). We could use such a model to answer interesting questions such as "How does the size of a roller affect the probability that it escapes with its dung ball?" and "How many times do we expect a dung ball to change hands before it is buried?"
Author contributions ZY wrote code, ran simulations, and composed figures and tables. LZ conceived of the study and helped to draft the manuscript. All authors read and approved the final manuscript.

Funding
The authors did not receive support from any organization for the submitted work.
Data availability Additional data provided in Supplementary Material.

Declarations
Ethics approval Not applicable.

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Conflicts of interest
The authors have no relevant financial or nonfinancial interests to disclose.