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There were three methods of sampling the populations. For Methods 1 and 2, records were obtained by driving along the road transects, and stopping to score the age groups in herds within some 100 m of the road. There were three road transects, entirely in the administrative boundaries of Serengeti National Park and consistent every year (1962–2018), with records summed over the three for each data entry. Transect 1 was from Seronera (34.823°E, 2.428°S) west to Kirawira (34.208°E, 2.151°S; 120 km), Transect 2 from Seronera to Bologonja (35.173°E, 1.757°S; 115 km), and Transect 3 from Seronera to Olduvai Gorge (35.35°E, 2.993°S; 75 km) (Fig. 1). The first two transects were in similar savanna ecosystems, and comparison of samples from these two showed close similarity.
Fig. 1

Ungulate and ostrich sampling transects in the Serengeti ecosystem.

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The criteria for age classes in each species are given in Online-only Table 1. The sample was the herd within view (such as a group of impalas (Ae. melampus) or hartebeests (Al. buselaphus), which occur in discrete groups), or a subset of it if the herd was very large. One observer, using 8–10 x magnification binoculars, called out the age category while a recorder entered the records on data sheets. These were later entered digitally.
Two exceptions to this were the immense herds of migrant wildebeest (C. taurinus) and zebra (Eq. quagga). Because they were numerous and extensive, herds had to be sampled in a systematic way. A vehicle drove through the herds, stopping every half kilometer, where a 180 degree scan out to 100 m was conducted to count the sample within view. The transects were from the start to the end of the herd, with some being 30 km long through a single, continuous herd. Method 3 used aerial pictures of the herds to score age groups. Although the sampling protocol was different in the three methods (due to different distributions of each species) the same criteria for identifying age classes was used in all methods. All methods used either systematic or random sampling of the populations.
All species were either migrants, if the species shows seasonal variation in habitat, or residents, if the species remains in the same area of the park year-round. A notable exception to this is the wildebeest (C. taurinus). In fact, there were two populations of wildebeest, a large migrant herd and a small resident herd at the far western end of the ecosystem. These two were sampled separately and scored as either migrant or resident.
Method 1
This method was used in all sampling years for impala (Ae. melampus), Coke’s kongoni (Al. buselaphus), topi (D. lunatus), warthog (P. africanus), Defassa waterbuck (K. defassa), and zebra (Eq. quagga). Sampling years 1984–1994 for African buffalo (Sy. caffer), 1965–2012 for giraffe (G. camelopardalus), and 1964–2016 for wildebeest (C. taurinus).
Populations were sampled once or twice a year at specific times, depending on the availability of different age classes in the areas near transects. Because ungulates had different birth seasons samples were collected at two time periods, once in mid-year and once at year-end. Only one time period per year was used for each species. The early age group, “infants”, was sampled usually near the end of the rainy season (March–June) since many species give birth during the rainy season. For some species, there was a second sampling period (August-December) at the end of the dry season, to measure the survival of juveniles during this period of ecological stress. There are a few cases where more than two samples were obtained in a single year, so as to track the survival of the whole cohort throughout a year.
Method 2
This method was used in all sampling years for eland (T. oryx), elephant (L. africana), Grant’s gazelle (N. granti), ostrich (S. camelus), and waterbuck (K. defassa).
These species were sufficiently scarce that an adequate sample could not be obtained at specific times. For these, records were scored whenever the species was seen in a sampling period, and then records for all sampling periods of a single given year were summed. A special case was Thomson’s gazelle (Eu. thomsonii), which, although numerous, was scored only during one short time period (1992–1994) for the months of August and September.
Method 3
This method was used in sample years 1965–1973 for African buffalo (Sy. caffer), and 1926–1933 for giraffe (G. camelopardalus tippelskirchi), wildebeest (C. taurinus), and zebra (Eq. quagga). The area covered was in all cases within the Serengeti ecosystem. Buffalo and giraffe were only found in the savanna, while wildebeest were sampled when they were on the plains. Flights were made systematically over the area, wildebeest was sampled using photographs at regular intervals, buffalo and giraffe were sampled when they were encountered.
The third method, applied only in the very early years, used aerial photographs to identify age classes and females. The same criteria for identifying age classes was used as those for Methods 1 and 2 (Online-only Table 1), with an emphasis on the shape and size of horns for the wildebeest and African buffalo2, and of the relative sizes of young giraffe. The early samples in 1926–1933, were obtained from photographs taken by Martin Johnson. These photos reside in the Martin and Osa Johnson Safari Museum, Chanute, Kansas. Unfortunately, the 1965–1973 photographs of buffalo herds have now all been lost or destroyed. More

The nature of the slug movement data collected in our field experiment (i.e. position on or beneath the soil surface observed at discrete moments of time) makes it possible to analyse the slug locomotory track in terms of the discrete-time random movement framework12,16,35,36. Within this framework, a curvilinear movement path is approximated by a broken line (see Fig. 2) and the movement of an individual slug is parameterized by the following frequency distributions:
1.
The distribution of the step sizes along the movement path (i.e. the distance between sequential pairs of recorded positions; Fig. 2) or the corresponding average speed

2.
The distribution of turning angle (the angle between the straight lines drawn between sequential pairs of recorded positions; Fig. 2).

Figure 2

A sketch of animal movement path and its discretization (adapted from36). (a) The original movement path is normally curvilinear. (b) Due to the limitations of the radio-tracking technique, position of the animal is only known at certain discrete moments of time; correspondingly, the curve is approximated by a broken line. (c) The movement path as a broken line is fully described by the sequence of the step sizes (lengths) along the path, i.e. the distances travelled between any two sequential recorded positions, and the sequence of the corresponding turning angles.

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Once all the information is available, it is possible to calculate the mean squared displacement as a function of time12,37. Additionally, in case the movement consists of alternating periods of active movement and immobility (periods with no recorded displacement resulting from feeding or inactivity, hereafter referred to as“resting time”), one should also consider the distribution of the corresponding periods.
Speed, squared displacements and the straightness index
It is apparent from the data that slug movement is intermittent, with periods of locomotion interspersed between periods in which they remain motionless. Tables 1 and 2 show, for the sparse and dense releases respectively, the number of ‘active’ time intervals when the slugs were moving. Periods during which slugs were motionless are marked by the zeros in Tables 1 and 2, but all these individuals resumed their movement during the following hours, confirming that they were alive throughout the assessment period. We therefore retain the zeros in the data for the subsequent analysis.
Table 1 Slug mean speed (averaged over the whole movement path), the mean SSD (see Eqs. (3) and (5), respectively) and the straightness index in the case of sparse release for each of 17 slugs used in the experiment. Here the straightness index is calculated using Eq. (4) where the values of the step size are immediately available from our field data.
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Table 2 Slug mean speed (averaged over the whole movement path), the mean SSD (see Eqs. (3) and (5), respectively) and the straightness index in the case of dense release for each of 11 slugs used in the experiment. Here the straightness index is calculated using Eq. (4) where the values of the step size are immediately available from our field data.
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The baseline discrete-time framework considers animal position at equidistant moments of time. However, in the field experiment (as described in the previous section), time taken to locate slugs at each assessment resulted in the time interval varying between measurements (sparse release treatment: 27–87 mins; dense release treatment: 20–103 mins). The step size, i.e. the displacement during one time interval, depends in part on the duration of that interval, hence risking bias in the results. We address this issue by scaling the step size by the duration of the corresponding time interval, i.e. by considering the average speed during the step:

$$begin{aligned} v_k(i)=, & {} frac{|Delta {mathbf{r}|_k(i)}}{Delta {t}_k(i)}, quad i=1,2,ldots ,N, end{aligned}$$
(1)

where

$$begin{aligned} |Delta {mathbf{r}|_k(i)}=, & {} |mathbf{r}_k(t_i)-mathbf{r}_k(t_{i-1})|, end{aligned}$$
(2)

is the displacement of the kth slug during the ith time interval, i.e. the distance between the two sequential positions in the field. Here N is the total number of steps made by the given slug during the full period of the experiment (in our field data, for all slugs (N=10)).
For each individual slug, we then calculate the mean speed over all steps along the movement path:

$$begin{aligned} _k=, & {} frac{1}{N} sum ^{N}_{i=1} v_k(i). end{aligned}$$
(3)

The results for the sparse and dense releases are shown in Tables 1 and 2, respectively; see also Fig. 3a.
The mean speed of slug movement, although being an important factor for slug dispersal, does not provide enough information about the rate at which the slug increases its linear distance from the point of release, because it does not provide information on the frequency of turning or the turning angle. In order to take that into account, we calculate the straightness index35, i.e. the ratio of the total displacement (distance between the point of release and the final position at the end of the experiment) to the total distance travelled along the path:

$$begin{aligned} s_k= & {} |mathbf{r}_k(t_N)-mathbf{r}_k(t_0)|/left( sum ^{N}_{i=1}|Delta {mathbf{r}}|_k(i) right) , end{aligned}$$
(4)

where (t_0) is the time of slug release and (t_N) is the time of the final observation. The actual distance travelled is approximated by the length of the corresponding broken line (see the dark solid line in Fig. 2).
Figure 3

(a) Slug mean spead and (b) slug mean SSD, black diamonds for the sparse release and red circles for the dense release.

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The straightness index quantifies the amount of turning (a combination of the frequency and angles of turns) along the whole movement path, i.e. over the whole observation time, but it says nothing about the rate of turning on the shorter time scale of a single ‘step’ along the movement path. To account for this, along with the mean speed we calculate the mean scaled squared displacement (SSD):

$$begin{aligned} langle sigma ^2 rangle _k=, & {} frac{1}{N} sum ^{N}_{i=1} sigma ^2_k(i) qquad text{ where }qquad sigma ^2_k(i)~=~frac{|Delta {mathbf{r}|^2_k(i)}}{Delta {t}_k(i)}, end{aligned}$$
(5)

see Tables 1 and 2 and Fig. 3b. For the same value of mean speed, a larger value of the SSD corresponds to a straighter movement on the timescale of a single step, with a smaller turning rate.
An immediate observation from visual analysis of the data shown in Fig. 3 is that both slug speed and the SSD are smaller in the case of dense release than in the sparse release. Therefore, a preliminary conclusion can be drawn that average slug movement is slower in the dense release compared to the sparse release treatment.
Turning angles
Figure 4

Frequency distribution of the turning angle in the case of (a) sparse and (b) dense releases of slugs. In calculating the turning angle, the periods of no movement were disregarded. The red curve shows the best-fitting of the data with the exponential function; see details in the text.

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We now proceed to analyse the distribution of turning angles. The histogram of different values of the angle is shown in Fig. 4. Let us consider first the case of sparse release (see Fig. 4a). We readily observe that the distribution is roughly symmetrical and has a clear maximum at (theta _T=0). The latter indicates that, on this timescale, slug movement is better described as the CRW than the standard diffusion1,16. Indeed, the standard diffusion (also known as the simple random walk) assumes that there is no bias in the movement direction, in particular there is no correlation in the movement direction in the intervals before and after the recorded position, which means that the turning angle is uniformly distributed over the whole circle. On the contrary, in the case where a correlation between the movement directions exists (hence resulting in the CRW), the distribution of the turning angle becomes hump-shaped. This is in agreement with the results of previous studies on animal movement (in particular, invertebrates12,38) as well as a general theoretical argument13.
In order to provide a more quantitative insight, we look for a functional description of the turning angle distribution using several distributions that are commonly used in movement ecology. The results are shown in Table 3. We establish that the turning angle data are best described by the exponential distribution. Somewhat unexpectedly, it outperforms the Von Mises distribution, although the latter is often regarded as a benchmark and its use has some theoretical justification1. However, the exponential distribution of the turning angle has previously been observed in movement data on some other species, e.g. on swimming invertebrates39.
Table 3 The (r^2) values for the turning angle movement data (in case of sparsely released slugs) described by different standard frequency distributions. The corresponding data are shown in Fig. 4a.
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The distribution of turning angle obtained in the case of dense release exhibit different features; see Fig. 4b. However, in this case, the distribution is not symmetric and has a clear bias towards positive values: the mean turning angle corresponding to the data shown in Fig. 4b is (langle theta _T rangle = 0.772approx pi /4). Since the slugs used in the dense release are from the same cohort as those used in the sparse release, we consider this bias as an effect of the slug density: the movement pattern of an individual slug is affected by the presence of con-specifics. We discuss possible specific mechanisms for the responsiveness to this factor in the Discussion.
An attempt to describe the turning angle data from the dense release by a symmetric distribution returns low values of (r^2) (see Suppl. Appendix A.1). However, the accuracy of data fitting comparable with the sparse release can be achieved by using an asymmetric distribution, i.e. where the corresponding function has different parameters for the positive and negative values of the angle. The results are shown in Table 4.
Table 4 The (r^2) values for the turning angle movement data (in case of densely released slugs) described by asymmetric frequency distributions. In calculating the turning angle, the periods of no movement were disregarded; the corresponding data are shown in Fig. 4b.
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The turning angle data shown in Fig. 4 were obtained using all active steps along the movement paths. However, since periods of slug movement alternate with periods of resting, it may raise the question of the relevance of the turning angle at the locations where slugs remained motionless for some time. In order to check the robustness of our results, we now repeat the analysis to calculate the turning angle differently by omitting the segments adjoined with the rest position. The results are shown in Fig. 5. In this case, a reliable fit may not be possible due to there being insufficient data. However, a visual inspection of the corresponding histograms suggests that the main properties of the turning angle distribution agree with those observed above for the bigger data set. Namely, in both cases the distribution has a clear maximum at (theta _T=0) (this is seen particularly well in the case of sparse release). In the case of sparse release the distribution is approximately symmetric, while in the case of dense release there is a clear bias towards positive values. We therefore conclude that the properties of the turning angle distribution are robust with regard to the details of its definition.
Figure 5

Frequency distribution of the turning angle in case of (a) sparse release, (b) dense release. The turning angle is only calculated for consecutive movements, i.e. if a slug does not move during a time step then its previous angle of movement is not used.

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Movement and resting times
Figure 6

Distribution of the proportion of the total time spent in movement in case of (a) sparse release and (b) dense release. The red curve shows the best-fit of the data with the normal distribution; see details in the text.

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Our field data shows that, while foraging, slugs do not move continuously but alternate periods of movement and rest; see the second column in Tables 1 and 2. Such behaviour is typical of many animal species38,40. In this section, we analyse the proportion of time that slugs spend moving, in particular to reveal the differences, if any, between the sparse and dense release.
Figure 6 shows the corresponding data where for the convenience of analysis the slugs are renumbered in a hierarchical order, so that slug 1 spends the highest proportion of time moving, slug 2 has the second highest, etc. We readily observe that the sparse release slugs tend to move more frequently than those from the dense release treatment: slugs that move for more than half of the total observation time constitute about 50% of the group in the case of sparse release but less than 30% in the case of dense release.
Figure 7

Distribution of the movement frequencies in case of (a) sparse release and (b) dense release.

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In order to make a more quantitative insight, we endeavour to describe the data using several standard distributions; see Tables 5 and 6. We find that the normal distribution performs better than others both in sparse and dense release treatments. Importantly, however, the parameters of the distribution are significantly different between the two cases; in particular, the standard deviation appears to be approximately twice as large in the case of sparse release. Arguably, it confirms the above conclusion that slugs move more frequently or for longer in the case of sparse release. Slugs released as a group tend to spend considerably more time at rest compared to the slugs released individually.
Table 5 The (r^2) values for the proportion of movement time described by different standard frequency distributions in the case of sparse release.
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To avoid a possible bias due to the different group size (17 slugs in the sparse release and 11 in the dense release), we now rearrange the data in terms of the proportion of the group that moves with a given frequency. The results are shown in Fig. 7. Although the amount of data in this case does not allow us to describe them using a particular function, the two cases clearly exhibit distributions with different properties. In particular, the average movement frequency is 0.467 for the sparse release and 0.264 for the dense release, and the corresponding variances are 0.090 and 0.065, respectively.
Table 6 The (r^2) values for the proportion of movement time described by different standard frequency distributions in the case of dense release.
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To further quantify the differences, Fig. 8 shows the number of slugs moving in each observation interval. Once again, we observe that the graph exhibits essentially different properties between the two releases. In particular, over the first interval, the majority of slugs (14 out of 17) move in the case of sparse release but none of the slugs move in the case of dense release. In the second half of the observation time (intervals 6–10) on average about 50% of slugs (8 out of 17) move in the case of sparse release but only about 25% of slugs (2–3 out of 11) move in the case of dense release.
Based on the differences between the two releases, we conclude that the presence of con-specifics is the factor that affects the distribution of slug movement time. Thus, along with the results of the previous sections, it suggests that slug movement is density dependent.
Figure 8

The number of moving slugs at each observation moment in case of (a) sparse release and (b) dense release.

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