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The limits to population density in birds and mammals

Abstract We address two fundamental ecological questions: what are the limits to animal population density and what determines those limits? We develop simple alternative models to predict population limits in relation to body mass. A model assuming that within‐species area use increases with the square of daily travel distance broadly predicts the scaling of empirical extremes of minimum density across birds and mammals. Consistent with model predictions, the estimated density range for a given mass, ‘population scope’, is greater for birds than for mammals. However, unlike mammals and carnivorous birds, expected broad relationships between body mass and density extremes are not supported by data on herbivorous and omnivorous birds. Our results suggest that simple constraints on mobility and energy use/supply are major determinants of the scaling of density limits, but further understanding of interactions between dietary constraints and density limits are needed to predict future wildlife population responses to anthropogenic threats.

. Scenarios of resource distribution and home range area. (A) Resource patches (black circles) are randomly scattered in the environment and home range (red circle) is adequate to encompass m patches (filled circles). (B) If the density of patches in the environment declines by a half, home range must double in size to continue to encompass m patches. (C) If, by contrast, the density of patches remains the same but patches are halved in quality (indicated by the reduced fill of patch markers encompassed by the home range), the home range (red circle) must double in size to contain 2m patches. (D) If resources could be located anywhere within the environment, the searcher must cover the entire area with an intensity dictated by its radius of detection (r). One way to do that would be to search systematically, covering all points that are a distance of 2r away from any other visited point. If the frequency of finding food by this method is halved, the forager must double the area searched (from the smaller red circle to the larger red circle). If it continues to search with the same intensity as previously, it will have to visit twice as many points per unit time to find twice as many units of resource.
Our models of the limits to population densities are based on the body-mass scaling of a range of underlying parameters. For both birds and mammals, our models of minimum density based on space use and mobility depended on the scaling of home range area (A, km 2 ) ( In addition, we needed data on the scaling of day range (daily travel distances). Few data are available to suggest the scaling of day range for birds during non-migratory flight periods. To infer plausible day ranges, we determined the scaling of proportion of time active, and combined this with the scaling of travel speed. The scaling of proportion of time active was determined using published data (Table S1). For estimating the upper bound to density, we used data on the scaling of field metabolic rate (FMR, kjd -1 ) in relation to body mass (M). Data were obtained from Nagy et al. (1999), Anderson & Jetz (2005) and Speakman & Król (2010). Duplicates were removed to yield 276 estimates of avian FMRs and 228 estimates of mammalian FMRs. Where multiple data were available for a species, both mass and FMR were averaged across all records for that species. The final data set included FMR estimates   Using these fitted parameters, we were able to determine the scaling (and associated uncertainty) of the predicted limits to density (see Table 1 in the main text). Here, we also present the scaling of cM, which indicates the intensity of habitat use under normal conditions of density (see Here, cM encompasses the relationship between the scaling of home range area and that of daily travel distance, and reflects the intensity of habitat use. Under the targeted-search model, this is highly divergent for small birds and mammals, but converges for large birds and mid-sized mammals (Fig.   S6). Both small birds and mammals, with low values of cM, use their habitat more intensively than do larger species. However, small birds appear to use their habitat much more intensively than do small mammals. This presumably reflects the scale at which species of different sizes and taxa search for resources, with small birds searching very intensively for resources of relatively low detectability, and larger species in both taxa seeking out more widespread but relatively detectable resources. These findings could have implications for other aspects of ecology, such as the scaling of consumer searching (Pawar et al. 2012), or spatial ecology (Jetz et al. 2004). At the smallest body sizes, the greater intensity of space use among birds offsets their greater mobility, resulting in predictions of minimum density similar to those for mammals. With increasing body mass, the intensity of space use converges for birds and mammals, and the greater mobility of birds translates into predictions of lower minimum densities.

C. Bootstrapping to determine parameter uncertainty
Various methods exist to estimate prediction uncertainty for quantile regression (Koenker 2015).
However, we also wanted to estimate uncertainty in derived properties (such as minimum density and population scope; see main manuscript). Consequently, we used bootstrapping. For example, to account  for that uncertainty in evaluating equation 3, which predicts minimum population density on the basis of space use and movement, we drew 2,000 bootstrapped resamples (with replacement) from the relevant data on each of the equation's parameters. To retain structure relative to each original data set, we resampled the raw data from three equally-sized strata with respect to logged body mass. For each bootstrapped sample, we recalculated the scaling parameters for all underlying parameters, and used those to re-evaluate equation 3. For any given logged body mass, confidence intervals were taken as the bounds of the inner 95% of predictions of the lower limit to density, across all 2,000 replicates. The same method was used to determine uncertainty in relationships between empirical density data and body mass, in order to propagate that uncertainty into estimates of population scope.

D. Compiling density, body mass and dietary data for birds and mammals
Taking density data from multiple sources is vulnerable to the possibility of duplicated data points. To reduce this risk, we identified and removed identical estimates for a species in the same study site. For birds, we also removed records associated with colonial and semi-colonial species (densities of which are notoriously difficult to determine (Sanderson et al. 2002;Wackernagel et al. 2002;Halpern et al. 2008)), as well as ratites (Order Struthioniformes), and species with a primary habitat affiliation of wetland, marine or aquatic (based on the IUCN level one habitat classifications obtained from (Birdlife International and NatureServe 2015)). Densities of birds are reported in a range of units. Where these were given in individuals per unit area, the conversion was straightforward. For counts of pairs, nests, territories, singing, calling or courting males, we doubled reported densities.
In addition to density data, we compiled, for each species, estimates of body mass and broad dietary classifications. For birds, body mass data came principally from (Dunning Jr. 2008), supplemented with additional data from (Lislevand et al. 2007) and BirdLife International's World Bird Database (http://datazone.birdlife.org/home). Body mass estimates for mammals were largely from the PanTheria data set (Jones et al. 2009), or Damuth (1987. For some mammalian species, mass estimates were taken from species-specific studies (often the same sources as those from which density data were obtained). To avoid inconsistencies, we used a single estimate of mass for each species (taking midpoints where mass ranges were given, or averaging across estimates for species for which multiple mass estimates were available). This neglects intra-specific variation in body mass but error in this variable is likely to be very small compared to error in published density estimates.
For birds, dietary data were taken from the same sources as those used to provide density data, and were grouped into the three broad dietary classifications used in our analyses ("carnivore", "omnivore" and "herbivore"). For mammals, the same classifications were obtained initially from the Animal Diversity Web (Myers et al. 2016). Species with no classification in ADW were classified using PanTheria (Jones et al. 2009) or species-specific studies. A very small number of species were classified based on diets of closely related species.

E. Human population densities
Assessing the scaling of densities and their limits among mammalian guilds provides a context for comparing mammalian population densities with those of human hunter-gatherers, as a result of the shift from more carnivorous diets, to herbivorous and omnivorous diets (Murdock 1967(Murdock , 1981Keeley 1988;Burger et al. 2017). Data on human hunter-gatherer population densities were obtained from (Keeley 1988;Burger et al. 2017). Hunter-gatherers were classified broadly into carnivorous diets based on "simple" hunter-gatherer groups (based on classifications in Keeley (Keeley 1988)) or "fisher/hunter" based on data cited in Burger et al (2017); or omnivorous diets; or largely vegetarian diets (the latter described as "gatherers" in Burger et al). In the case of data from Keeley described as "complex hunter-gatherers", diets were split into omnivores and herbivores by matching the tribal names in Burger et al (2017). Modern human density data were obtained from web sources 1 .
Population densities of hunter-gatherer populations were consistent with those of similarlysized wild mammals ( Fig S7A). Indeed, when population scopes were evaluated for 60kg non-human mammals in the three dietary guilds, the density distributions agreed well with human hunter gatherer societies in the corresponding dietary categories (Fig. S7B). However, modern humans live at much higher densities, even when evaluated at a country-wide scale (Fig. S7). At a global scale, the current human density is as high as the maximum for comparably-sized herbivorous non-human mammals; urban populations reach much higher densities, with the highest densities exceeding those of any nonhuman mammal (Fig. S7A). Distributions are shown for hunter gatherers with carnivorous (red) omnivorous (purple) and predominately vegetarian (green) dietary strategies, and for modern humans evaluated at country level (blue). Bars beneath the polygons show the 95% ranges (between upper and lower quantiles) fitted to non-human carnivorous (red), omnivorous (purple) and herbivorous (green) mammals, evaluated for a body mass of 60kg.