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Recent analyses have found that, despite high and increasing levels of community turnover, there is no clear overall trend in local-scale species richness1,2,3,4. However, it remains unclear how this result translates into functional changes. One of the most fundamental functional traits of a species is its size5,6 and there is an expectation that a warming climate will lead to a shift towards smaller species7,8,9,10,11, drawing upon metabolic theory12 and the observed distributional patterns described by Bergmann’s rule13,14. Temperature-driven shifts towards smaller species have been observed in tundra plant communities15 and some7,9,16, but not all11, aquatic systems. Furthermore, larger species have been more extinction prone during some previous mass extinctions17,18 and are more likely to show strong recent population declines19. Although relationships are threat dependent20,21, larger species tend to be assessed at a higher risk of extinction due to longer generational intervals and increased threat from habitat loss, fragmentation and hunting22.One might therefore expect a detectable signal of shifts in community trait values beneath the apparent underlying consistency in taxonomic diversity. To examine this, we tested whether the size of a species is correlated with the change in abundance through time using the publicly available BioTIME database23. This database is the largest collection of time series of ecological communities and, despite considerable biases that we discuss below, has wide geographic and taxonomic scope23. It consists of ‘studies’ defined by a consistent sampling methodology and taxonomic focus. After cleaning and standardizing the names associated with the records, we linked six fundamental ‘size’ traits from four openly accessible trait databases representing four broad guilds: adult body mass from a database of amniote life history traits24, adult body length and qualitative body size of marine species from the World Register of Marine Species (WoRMS) database25, plant maximum height and seed mass from the TRY database26 and maximum body length of fish from a compilation27 based on data in the FishBase repository28.Observations from single-location studies were combined, whilst widely dispersed studies were separately binned into a global grid of cells, each approximately 10 km wide, and data from each study and cell were treated as discrete assemblages, following previous analyses1,29. Selecting only assemblages with quantitative observations of ≥10 species, over ≥5 years and with ≥40% of the species having records for at least one size trait, we generated 12,956 assemblage time series from 144 studies (Fig. 1). This filtered dataset represented 2,109,593 observations of 10,286 species, of which 7,234 could be linked to at least one size trait (representing 84.02% of observations). Equally weighting studies, the average time series length was 18.2 years (range 5–71.8 years), and the average number of species per included assemblage was 65.4 (range 10–337). The log10 ratio between the largest and smallest species in each study averaged 2.49 (range 0.55–6.73) across the ‘mass’ traits and 1.06 (range 0.3–3.15) across the ‘length’ traits.Fig. 1: Global distribution of studies in our dataset, showing average τ for each study–trait combination and divided into aquatic and terrestrial realms.The aquatic realm is principally marine but includes three freshwater studies. Note that the locations are shown as the centre point of each study, which can cause oceanic studies to be ‘located’ on land. See Extended Data Fig. 1 for full details of study-level results.Full size imageFor each trait and community assemblage time series for which there were sufficient data, we first square-root transformed and standardized each time series following previous approaches3 and calculated βi, the slope of a regression of abundance of species i against time. We then calculated, for each assemblage, τ (the Kendall rank correlation coefficient between the trait in question) and β, across the species for which we had trait data. This gives a non-parametric measure of whether larger species are more or less likely than smaller species to have increased through time and, importantly, can be calculated where trait values for only a fraction of the observed species are available. To weight each study within BioTIME equally, where there were multiple assemblages per study, these were averaged to generate a τ value for each possible study–trait combination. To provide a reference distribution against which to evaluate the statistical significance of this multistage analysis, we repeated the procedure with 10,000 trait randomizations within each assemblage.Certain individual studies showed significant relationships between size traits and population trends (coloured dots in Fig. 2 and Extended Data Fig. 1). However, for five of the six tested size traits, the overall mean τ values did not differ significantly from the null model (Fig. 2). For one trait (amniote body mass, Fig. 2d) we found a marginally significant (unadjusted for multiple comparisons) overall average positive relationship between size and the slope of population trends (β). Alternative population data transformations gave highly concordant results (Extended Data Figs. 2 and 3). Possible confounding factors for the value of τ associated with each study, namely the total span of the time series, the number of sample points, the species richness, the range of traits in the assemblage, the average size trait completeness, the number of assemblages within the study, the grain of the study and the absolute latitude, did not consistently predict either τ or τ2 (Extended Data Figs. 4 and 5 and Supplementary Tables 1 and 2). Further, the likelihood of an individual species showing either a statistically significant positive or negative population trend was not linked to its relative size trait value within the assemblage (all P  > 0.05; Extended Data Fig. 6 and Supplementary Table 3).Fig. 2: Correlation between six body-size traits and changes in abundance through time (τ).a–f, Distribution of Kendall rank correlation coefficient between body-size traits for body length (a) and qualitative body size (b) of marine species, maximum length of fish (c), adult body mass of amniotes (d) and seed mass (e) and maximum height (f) of plants versus changes in abundance through time. Each dot represents one study, averaging across the constituent assembly time series for studies of large spatial extent. Study-level results are binned into classes 0.05 units of τ wide. Coloured dots highlight studies that were individually identified as showing a significant trend (yellow for negative, blue for positive; see Extended Data Fig. 1 for study-level intervals). The error bar below each plot displays the distribution (central 95% and 66%) of mean τ values over 10,000 permutations of the size trait data, whilst the red line indicates the observed mean τ value within that panel. Displayed P values are calculated from permutation tests. Equivalent results using alternative approaches to transforming the community data are given in Extended Data Fig. 3.Full size imageThese results indicate that there is not yet evidence for widely pervasive within-assemblage trends in a core functional trait, size. Importantly however, this study should not be seen as a refutation or diminishment of the heightened threats faced by the very largest apex species30,31, which constitute only a minor component of the BioTIME database. Rather, against a background of considerable turnover2,3 across whole observed community assemblages, on average, species positions in communities are being taken up by species of comparable size. Our results suggest that previously identified shifts towards smaller species found in some aquatic systems9,16 may not be as universal as currently expected7,11 and align with the divergent changes in global body-size abundance distributions observed between mammal guilds32 and the apparent stability of trait diversity in North American birds despite declines in abundance33.The tendency towards an overall positive association between body-size and population trends across the amniote studies could have a number of drivers that would benefit from further investigation. One putative explanation that has been put forward for positive size trends is that anthropogenic dispersal limitations (generally considered to act more strongly against smaller species) may be having a greater immediate impact than climate change34. There are also indications of differences between terrestrial and marine systems. Previous work with the same datasets1,29 has found greater species richness and abundance changes in marine than terrestrial systems, whilst here we see a signal of greater trait changes in the (largely terrestrial) amniotes.In our dataset, the fish length trait studies displayed a particularly skewed distribution of τ values (Fig. 2c), with a modal peak of studies showing small negative values then a tail of strongly positive relationships. This guild is also the most likely to have experienced sustained anthropogenic pressure35, and many of the ‘fish’ datasets in BioTIME include data from surveys of actively fished and managed areas. Accurately quantifying marine community trends is a challenge36,37, but this pattern could reflect the imposition or relaxation of anthropogenic pressure across marine systems38,39. Positive τ values could represent recoveries from past pressures on larger species, and positive τ values were associated with shorter study durations in the fish studies (Extended Data Fig. 4).Our analysis necessarily sacrifices fine resolution for global scale. Technically, BioTIME studies represent assemblages defined by taxonomy and sampling protocol rather than complete ecological communities. We must implicitly assume that the scope of each study within BioTIME strikes a reasonable balance between the need to include a sufficiently diverse set of species to be able to observe any potential impact of trait differences whilst maintaining meaningful comparability. Limitations to total time series lengths and the limited range of sizes recorded within each dataset inevitably constrain our capacity to detect gradual changes or subtle influences of size. Although the lack of consistent study-level drivers of τ suggests that the results are unlikely to be solely determined by the inevitable spatial and temporal limitations of the BioTIME database, future work should seek to improve the scope and resolution of available data to enable more strongly parametric analyses and examine additional measures of community change.Whilst available trait databases of amniotes and fish are carefully curated, checked and taxonomically tidy, the other databases pose more problems in terms of taxonomic matching and consistency of trait measurements. Without direct correspondence between the sources of dynamics and trait data, it is necessary to take traits as fixed values for each species, despite known differences in traits in time8,40,41,42 and space43 that may themselves represent responses to global change. However, in Celtic Sea fish, within-species shifts have been shown to contribute less to community-level size shifts than changes in species composition44. We also note that ‘size’ traits for indeterminately growing plants have a less clear meaning than for animals. However, both seed size and maximum height are linked to environmental variables45,46, plant size is linked to life history47,48 and changes in community height driven by species turnover have been observed in tundra plants15.Many of the criticisms and defences regarding earlier studies using the BioTIME dataset, and indeed other analyses of large collections of time series, also apply to this work49,50. The consistency between the alternative approaches we tested to determine population trends (Extended Data Fig. 3) demonstrates that our conclusions are not dependent on particular data transformation choices. However, a largely non-parametric statistical approach was necessitated by the unevenness of the available data, and it must be noted that it could lack the power and resolution to identify subtle changes. Biases in the underlying BioTIME database towards vertebrate taxa, particular biomes and temperate North American and European sites23 are further exaggerated when crossed with trait data availability (Fig. 1). One particularly concerning gap is the absence of any insect studies in our dataset due to a paucity of usable trait information. Observations suggest that there have been considerable changes in the structure of insect communities34,51,52. Developing comprehensive insect trait datasets, including using proxies and data imputation, will be crucial to address this deficit53,54,55.In conclusion, despite necessary reservations, this global analysis suggests that examples of relative increases of larger species11,34 may in fact be as frequent as shifts towards smaller-sized species16. Community responses appear to be considerably more nuanced and localized than previously considered based on theoretical macro-ecological expectations7. More

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Combining agent mobility patterns and SIR modelTo take into account agent mobility19 in a scenario compatible with a SIR model, we developed the model pictorially illustrated in Fig. 1. As explained in details in the Methods Section, the agents can move on a lattice through jumps processes, modelled using a Lévy walk of jump parameter (beta)36,37,38. When (beta) becomes large, i.e., for (beta rightarrow 2), agents tend to perform a Brownian random walk with very short jumps. As (beta rightarrow 1), agents can travel long distances in just one step. There are no constraints on the number of agents that can occupy a single cell. In each cells, agents can be infected by neighbours according to the SIR rules. Thus, the parameters that control the model are the jump parameter (beta) plus the standard SIR parameters, infection rate (alpha) and removal rate (gamma). The agent-based lattice model considered here reduces to a standard SIR model when the well-mixed population condition is satisfied, i. e. when large jumps dominate the dynamics (Fig. 2).Figure 1Agent-based SIR model on a lattice. (a) Agents of different colors, representing the SIR states, move on a lattice. White cells represent empty sites. Green cells are occupied by susceptible (S) agents, blue cells contain only removed (R) agents. Red cells contain only infected (I) agents. Shaded cells contain agents in a mixture of states. Agents can move among cells performing jumps (black arrows) whose length follows Lévy statistics. The letters i and j, with (i=1,..,N_b) and (j=1,…,N_b) define the location of the cell (i, j). (b,c) Agents in the same cell undergo a SIR dynamics: (b) S become I at a rate (alpha); (c) I become R at rate (gamma). (d) The jump dynamics allows an agent to move from the cell (i, j) to ((i+k,j+l)). The probability to perform a large/small jump is controlled by the parameter (beta in [1.0,1.99]). Large (beta) values correspond to small jumps, i. e., a random walk that gives rise to Brownian motion. Small (beta) values correspond to large jumps.Full size imageFor reproducing the kinetics of real data we made the following assumptions:

In the absence of containing strategies, the infection is characterized by a high infection rate (we take (alpha =0.9)) and a low removal rate ((gamma =0.025) or 0.05). Using as a unit of time the update of all agent positions (see Methods for details), the removal rate introduce a time scale (tau _I = gamma ^{-1}=40) or (20). This characteristic time scale represents the average time an agent remains infected and can thus spread the infection. This condition ensures that we are in an epidemic regime, i. e., the mean-field value is (R_t gg 1). We stress that, since the SIR dynamics with only three sub-populations is a simplification of the real chain of epidemic transmission, the parameters we choose for the epidemic spreading are not strictly related to those of Covid-19. Because we are interested in the effect of mobility restriction on epidemic spreading, we fix the epidemic parameters in a way that, without mobility restrictions, we are sure to stay in the worst-case scenario with an exponentially fast spreading of the infection.

The parameter (beta in [1,1.99]) tunes the intensity of mobility restrictions. The higher its value, the stricter the limitations. (beta) is one of the fitting parameters.

Other interventions that mitigate the epidemic spreading tend to increase the removal rate (gamma). We thus assume that (gamma) is another fitting parameter. This is because typical measures, for instance, quarantine, remove infected agents from the system. In this way, we reabsorb the presence of many hidden sub-populations into an effective value of (gamma).

We define the parameter (delta), i. e., the fraction of infected agents at the epidemic peak with respect to the entire population, that provides a quantitative measure of the reduction of the epidemic peak. In other words, the parameter (delta) represents the efficiency of a given containing strategy compared to the uncontrolled situation where all the agents turn out to contract the infection (which is the case of our model for (gamma ll alpha), (alpha =0.9), and (beta =1)).

To detail how mobility restrictions induce deviations from the SIR model, we calculate, via numerical simulations, the epidemic curves as a function of time for different values of (beta) as illustrated in Fig. 2a. Here, the SIR parameters are (alpha =0.9) and (gamma =0.025), i. e., the corresponding SIR model is in the fully blown epidemic regime. For small (beta) the epidemic growth is well captured by the exponential function, indicating that we are in the epidemic regime. As (beta) increases the curve turns out to be flattened and the peak reduces to (80%). Moreover, the growth of the epidemic for the largest (beta) examined is well described by the power law (I(t) sim t^{2}). The value of the exponent is comparable with those measured in different countries during the COVID(-19) epidemic wave23. The model considered here suggests that the crossover from exponential growth to power-law might be related to changes of the mobility patterns that, in our picture, shift from being dominated by large jumps to small ones. This finding is consistent with the observation that a sub-exponential growth in the number of infected people is a consequence of containing strategies23. Moreover, in the microscopic description adopted here, the crossover in the kinetics of I(t) is driven by just one parameter.Figure 2Agent dynamics impacts the epidemic spreading process. (a) The graph shows the dependency of the epidemic curves on (beta =1.20,1.50,1.75,1.80,1.85,1.87,1.90,1.92,1.95,1.97,1.99) (increasing values of (beta) from yellow to violet). As (beta) decreases, the epidemic grows exponentially fast (dotted black curve) and approaches the evolution of SIR model in well-mixed population (dashed red curve). The dash-dot blue curve is a power law (sim t^2). The parameters of the SIR reactions are (alpha =0.9) and (gamma =0.025). (b–g) Typical configurations taken at the same fraction of infected agents (I/N sim 0.25) for increasing values of (beta =1.0,1.2,1.4,1.6,1.8,1.9) (red are infected sites, green the susceptible ones, we keep white the sites populated by removed agents). (h) The probability distribution function of the local density of infected sites. (i) Radius of the cluster of infected agents ((beta =1.99)) as a function of time. The red dashed line is a linear fit.Full size imageThe crossover from exponential to power-law growth reflects the drastic change in the structure of clusters of infected agents, as illustrated in Fig. 2b–g, where typical configurations with the same fraction of infected agents are shown ((I/N=0.25, alpha =0.9, gamma =0.025)). As one can see, in the high mobility region ((beta = 1)), infected agents are spread almost everywhere in the system. As (beta) increases, infected sites tend to form a single cluster. This phenomenology is consistent with the literature of mobile agents undergoing SIR dynamics39,40. This structural change is quantitatively documented by the density distribution of infected sites shown in panel (h) of the same figure (see section Methods for details). As one can appreciate, the distribution becomes double-peaked as (beta) increases. The first peak around zero indicates the presence of an extended region of susceptible agents. The peak at high values is due to the growing cluster of infected agents. As highlighted in panel (i), the cluster grows linearly in time and thus the number of infected grows with (t^2).Another interesting aspect to understand with this model is the trade off between mobility restrictions and and other kind of interventions that have the effect of increasing the removal rate. In particular in Asian countries41, NPIs applied during the COVID-19 waves have relied mostly on contact tracing and/or preventive quarantine, with little mobility reduction, leading to effective and durable control of epidemic spreading, as reviewed by Ref.21. To understand if there is an optimal balance between containing strategies (characterized by (beta)) and efficiency in removing infected agents (denoted by (gamma)), we calculate the fraction of infected population at the epidemic peak (the maximum of I(t)) as a function of the jump parameter (beta) and of the removal rate (gamma). As above, the initial occupation number of each site is, on average, one. The infection rate is (alpha =0.9). The resulting phase diagram is shown in Fig. 3. The color indicates the fraction of infected population: in the violet region, this fraction goes to zero (epidemic is suppressed) while in the yellow region such a value goes to one, indicating an epidemic regime. The phase diagram fully recapitulates the effectiveness of the two strategies used to mitigate the infection spread, a strong lockdown with limited contact tracing, or an efficient contact tracing a moderate reduction of the mobility.Figure 3Effect of different containment strategies. The phase diagram is obtained considering as control parameters (beta), that represents mobility restrictions, and (gamma), the efficiency in removing infected agents. The color scale represents the fraction of the initial susceptible population that becomes infected, ranging between 0 (epidemic suppression, violet region) and 1 (fully-blown epidemic, yellow region). Containment is achieved as (beta) increases (corresponding to increasing mobility restrictions) even with low removal rate, or increasing (gamma) (effective removal of infected agents), even with limited mobility restrictions.Full size imageHowever, even under the strictest lockdown, several activities could not be stopped (hospitals, food supply chain, …), meaning that a single mobility parameter cannot fully describe this varied situation. To understand what could be the impact of heterogeneous motility patterns on the evolution of the epidemic, we introduce in the model some regions characterized by a high mobility (jump parameter, (beta _2)), while the majority of the the cells have restricted mobility, with a jump parameter (beta _1=1.99) (see Methods for more details). By varying (beta _2) and the density of more mobile cells (parameter (rho)) we are able to draw the phase diagram shown in Fig. 4.Figure 4Sites of different mobility affect epidemic spreading. (a) Each cell labelled by (i, j) is characterized by its own mobility parameter (beta _{ij}). We consider the special case of a binary mixture ((beta _{ij} = beta _{1,2})) of high and low mobility regions. Changing the density (rho) of (beta _2) sites and the value of (beta _2), we obtain the the phase diagram presented in panel (b), obtained for (beta _1=1.99), (alpha =0.9), and (gamma =0.05), conditions that grant contained epidemic spreading thanks to the low-mobility group. A small amount of sites with small values of (beta _2) can trigger the epidemic spreading.Full size imageAs in the previous case, in the violet area the epidemic spreading is stopped, while in the yellow area the epidemic peak reaches the entire population. Epidemic spreading takes place above a critical curve: for a given value of mobility (beta _2 More

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