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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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Definitions of terms
GPP is defined here as ‘the sum of gross carbon fixation (carboxylation minus photorespiration) by autotrophic carbon-fixing tissues per unit area and time54. GPP is expressed as mass of organic carbon produced per unit area and time, over at least one year. NPP consists of all organic carbon that is fixed, but not respired over a given time period54:

$${mathrm{NPP}} = {mathrm{GPP}}-R_{mathrm{a}} = {Delta}B + L + F + H + O = {mathrm{BP}} + O$$
(3)

with all terms expressed in unit of mass of carbon per unit area and time. Ra is autotrophic respiration (composed of growth and maintenance respiration components); ΔB is the annual change in standing biomass carbon; litter production (roots, leaves and woody debris) is L; fruit production is F; the loss to herbivores is H, which was not accounted here because of the very limited number of observations available. BP is biomass production4. Symbol O represents occult, carbon flows, i.e. all other allocations of assimilated carbon, including changes in the nonstructural carbohydrate pool, root exudates, carbon subsidies to symbiotic fungi (mycorrhizae) or bacteria (e.g. nitrogen fixers), and BVOCs emissions (Supplementary Fig. 1). These ‘occult’ components are often ignored or unaccounted when estimating NPP, hence this bias is necessarily propagated into the Ra estimate when Ra is calculated as the difference between GPP and NPP55.
Estimation methods
We grouped the ‘methods’ into four categories:

biometric: direct tree stock measurements, or proxy data together with biomass expansion factors, allometric equations and the stock change as a BP component. If not otherwise stated, we assumed that the values included both above- and below-ground plant parts (n = 13 for GPP; n = 200 for NPP or BP).

micrometeorological: micrometeorological flux measurements using the eddy-covariance technique to measure CO2 flux and partitioning methods to estimate ecosystem respiration and GPP (n = 98 for GPP; n = 4 for NPP or BP).

model: model applications ranging from single mathematical equations (for canopy photosynthesis and whole-tree respiration) to more complex mechanistic process-based models to estimate GPP and Ra, with NPP as the net difference between them (n = 53 for GPP; n = 24 for NPP or BP).

scaling: upscaling of chamber-based measurements of assimilation and respiration (GPP and Ra) fluxes at the organ scale, or the entire stand (n = 73 for GPP; n = 9 for NPP or BP).

The difference between ‘scaling’ and ‘modelling’ lies in the data used. In the case of ‘scaling’ the data were derived from measurements at the site. ‘Model’ means that a dynamic process-based model was used, but with parameters calibrated and optimized at the site, based on either biometric or micrometeorological measurements.
Data selection
The data were obtained from more than 300 peer-reviewed articles (see also ref. 5), adding, merging and extending published works worldwide on CUE or BPE4,9,11,23,25,56,57. Data were extracted from the text, Tables or directly from Figures using the Unix software g3data (version 1.5.2, Jonas Frantz). In most studies, NPP, BP and GPP were estimated for the tree stand only. However, GPP estimated from CO2 flux by micrometeorological methods applies to the entire stand including ground vegetation. We therefore included only those micrometeorological studies where the forest stand was the dominant primary producer. The database contains 244 records (197 for BPE and 47 for CUE) from >100 forest sites (including planted, managed, recently burned, N-fertilized, irrigated and artificially CO2-fertilized forests; Supplementary Information, Supplementary Fig. 3 and online Materials; https://doi.org/10.5281/zenodo.3953478), representing 89 different tree species. Globally, 170 records out of the total data are from temperate sites, 51 from boreal, and 23 for tropical sites, corresponding to 79 deciduous broad-leaf (DBF), 14 evergreen broad-leaf (EBF), 132 evergreen needle-leaf (ENF) and 19 mixed-forests records (MX). The majority of the data (∼93%) cover the time-span from 1995 to 2015. We assume that when productivity data came from biometric measurements the reported NPP would have to be considered as BP because ‘occult’, nonstructural and secondary carbon compounds (e.g. BVOCs or exudates) are not included. In some cases, multiple datasets from the same site were included, covering different years or published by different authors. We considered only those values where either NPP (or BP) and GPP referred to the same year. From studies where data were available from more than 1 year, mean values across years were calculated. When the same reference for data was found in different papers or collected in different databases, where possible, we used data from the original source. When different authors described the same values for the same site, one single reference (and value) was used (in principle the oldest one). By using only commonly available environmental drivers to analyse the spatial variability in CUE and BPE, we were able to include almost all of the data that we found in the literature. We examined as potential predictors site-level effects of: average stand age (n = 204; range from 5 to ∼500 years), mean annual temperature (MAT; n = 230; range −6.5 to 27.1 °C) and total annual precipitation (TAP; n = 232; range from ∼125 to ∼3500 mm yr−1), method of determination (n = 237), geographic location (latitude and longitude; n = 241, 64°07′N to −42°52′S and 155°70′W to −173°28′E), elevation (n = 217; 5–2800 m, above sea level), leaf area index (LAI, n = 117; range from 0.4 to 13 m2 m−2), treatment (e.g.: ambient or artificially increased atmospheric CO2 concentration; n = 34), disturbance type (e.g.: fire n = 6; management n = 55), and the International Geosphere-Biosphere Programme (IGBP) vegetation classification and biomes (n = 244), as reported in the published articles (online Materials). The methods by which GPP, NPP, BP (and Ra) were determined were included as random effects in a number of possible mixed-effects linear regression models (Supplementary Table 4).
We excluded from statistical analysis all data where GPP and NPP were determined based on assumptions (e.g. data obtained using fixed fractions of NPP or Ra of GPP). In just one case GPP was estimated as the sum of upscaled Ra and NPP58; however, this study was excluded from the statistical analysis. NPP or Ra estimates obtained by process-based models (n = 23) were also not included in the statistical analysis. No information was available on prior natural disturbance events (biotic and abiotic, e.g. insect herbivore and pathogen outbreaks, and drought) that could in principle modify production efficiency, apart from fire. The occurrence of fire was reported by only a few studies59,60,61. These data were included in the database but fire, as an explanatory factor, was not considered due to the small number of samples in which it was reported (n = 6).
Data uncertainty
Uncertainties of GPP, NPP and BP data were all computed following the method based on expert judgment as described in Luyssaert et al.55. First, ‘gross’ uncertainty in GPP (gC m−2 yr−1) was calculated as 500 + 7.1 × (70−|lat|) gC m−2 yr−1 and gross uncertainties in NPP and BP (gC m−2 yr−1) were calculated as 350 + 2.9 × (70−|lat|). The absolute value of uncertainty thus decreases linearly with increasing latitude for GPP and for NPP and BP, because we assumed that the uncertainty is relative to the magnitude of the flux, which also decreases with increasing |lat|. Subsequently, as in Luyssaert et al.55, uncertainty was further reduced considering the methodology used to obtain each variable, by a method-specific factor (from 0 to 1, final uncertainty (δ) = gross uncertainty × method-specific factor). Luyssaert et al.55 reported for GPP-Micromet a method-specific factor of 0.3 (i.e. gross uncertainty is reduced by 70% for micrometeorological measurements); and for GPP-Model, 0.6. GPP-Scaling and GPP-Biometric were not explicitly considered in ref. 55 for GPP. We we used values of 0.8 and 0.3, respectively. For BP-Biometric and NPP-Micromet we used a reduction factor of 0.3; for NPP-Model, 0.6; and for NPP-Scaling (as obtained from chamber-based Ra measurements), 0.8. When GPP and/or NPP or BP methods were not known (n = 7), a factor of 1 (i.e. no reduction of uncertainty for methods used, hence maximum uncertainty) was used. The absolute uncertainties on CUE (δCUE) and BPE (δBPE) were considered as the weighted means62 by error propagation of each single variable (δNPP or δBP and δGPP) as follows:

$$delta {mathrm{CUE}} = sqrt {left( {frac{{delta {mathrm{NPP}}}}{{{mathrm{GPP}}}}} right)^2 + left( {delta {mathrm{GPP}}frac{{{mathrm{NPP}}}}{{{mathrm{GPP}}^2}}} right)^2}$$
(4)

and similarly for δBPE, by substituting NPP with BP and CUE with BPE.
Data and model selection
The CUE and BPE data were combined into a single variable, as sites for which both types of estimates existed did not show any significant differences between these entities (Supplementary Fig. 2). CUE values based on modelling were excluded (in our database we do not have BPE data from modelling). Tests showed that the CUE value was systematically higher when GPP was estimated with micrometeorological methods, compared to values based on biometric or scaling methods. Only data with complete information on CUE, MAT, age, TAP, and latitude were used. Altogether, 142 observations were selected.
In order to use the most complete information possible, a full additive model was constructed first (Eq. (1)). The method used for estimation of GPP (GPPmeth) was specified as a random effect on the intercept, as visual inspection suggested that CUE values were smaller where ‘scaling’ was used to estimate GPP compared to cases where ‘micromet’ was used to estimate GPP.
In Eq. (1) the variable ‘age’ represents the development status of the vegetation, i.e. either average age of the canopy forming trees or the period since the last major disturbance. The other three parameters represent different aspects of the climate. The absolute latitude, |lat|, was chosen as a proxy of radiation climate, i.e. day length and the seasonality of daily radiation. The term ηZGPPMeth represents the random effect on the intercept due to the different methods of estimating GPP.
These variables were not independent (Supplementary Table 1). If the different driver variables contain information that is not included in any of the other driver variables, multiple linear regression is nonetheless able to separate the individual effects. If, on the contrary, two variables exert essentially the same effect on the response variable (CUE) this can be seen in an ANOVA based model comparison. These considerations led us to the selection procedure in which we started with the full model (Eq. (1)) and compared it with all possible reduced models (Supplementary Table 2). The result of this analysis is the model with the smallest number of parameters that does not significantly differ from the full model.
We also examined, whether there were any significant interactions of predictor variables. There were not.
We used the R function lmer from the R-package lme463 to fit the mixed and ordinary multiple linear models to the data. We checked for potential problems of multicollinearity using the variance inflation factor (VIF)64. All predictors had VIF  More