Drivers of tree-level 3-D density and abundance
Among subplot-level models, the most parsimonious predictors of tree-level 3-D density were (i) the total abundance within the subplot; (ii) proportion of trees occupied within the subplot; and (iii) the density of trees within the subplot (Table 1). Density of trees had a large and highly significant positive effect on tree-level 3-D density (regression coefficient: 9.664 ± 1.492, p <0.0001), as did the proportion of trees occupied (regression coefficient: 1.067 ± 0.301, p <0.0001) (Table 2). Subplot-level abundance was only a significant contributor when interacting with the proportion of trees occupied, but the effect was small (0.001 ± 0.001, p=0.0492) (Table 2).
Results were comparable when modelled with tree-level abundance, for all variables except density of trees (Table 2). Density of trees had a substantial and significant negative impact on the abundance of bats per tree (regression coefficient: -5.053 ± 0.105, p <0.0001) (Table 2), suggesting that abundance per tree is higher when fewer trees are available for bats to roost in. Bats occupy more of the tree’s vertical space when more bats are present (a pattern consistent across tree crown classes, Appendix S2). The difference between tree-level 3-D density and tree-level abundance indicates that bats change the height range they occupy as total tree abundance increases.
At the roost level, density of trees was a relatively poor predictor of tree-level 3-D density and abundance and was not in the top-ranking model sets (Table 1). All fixed terms in the roost-level model had negligible effects on tree-level 3-D density (roost abundance: -0.084 ± 0.06, p=0.159; roost area: <0.0001, p=0.014; and the interaction term: <0.0001, p=0.021) and tree-level abundance (roost abundance: 0.447 ± 0.005, p=<0.0001; roost area: <0.0001, p<0.0001; and the interaction term: <0.0001, p<0.0001).
Roost-level predictors explained minimal variation in tree-level 3-D density, with the overall most parsimonious GAM only explaining 3.8% of variation (Table 1). Models with subplot-level and tree-level predictors explained slightly more variation (subplot-level: 11.7% of variation; tree-level: 13.6% of variation). The explanatory power of roost-level models with tree-level 3-D density was comparable when species were modelled separately (Appendix S3 in the Supporting Information). Explanatory power and rankings were comparable for models with tree-level abundance as the response variable (7.8% - 11.6% between top ranking models) (Appendix S4 in the Supporting Information).
Neither estimated tree-level 3-D density, nor model outputs, varied substantially under different values realistic for eucalyptus species (Appendix S5). Full model outputs for both response variables are given in Appendix S3 and Appendix S4 in the Supporting Information.

Discussion

We evaluated animal abundance and density at multiple scales to determine what information is relevant for understanding transmission. We used an extensive empirical dataset of roosting Pteropus spp. collected over 13 months and including 2,522 spatially referenced trees across eight roost sites. Measures most commonly used to parameterise models of bat-pathogen interactions (roost-level abundance and area) did not reflect the density of bats at scales where transmission is likely to take place (the abundance or density of bats within trees). Roost-level models explained a little of the variation in these tree-level measures. Density of trees was a better predictor of the likely conditions for transmission than was the population size of bats at the roost, where roosts with low tree density typically had a higher abundance but lower density of bats per individual tree. These results have implications for the structuring of infectious disease models for these species, particularly for pathogens transmitted over small local scales (e.g. within roosting trees), as discussed below.
An important consideration for bat-pathogen interactions should be whether local abundance or density is the more pertinent measure for transmission-relevant contact structure. In subplot-level models the best predictor of tree-level measures (abundance and density) was density of trees within roosts, and this had opposing effects on tree-level bat abundance and tree-level 3-D bat density. Roosts with a lower density of trees typically had more bats per tree, but a lower 3-D density of bats within these trees. This suggests that, while abundance per tree is higher when fewer trees are available for bats to roost in, bats are able to decrease their local density by expanding their occupied tree area (i.e. by spacing themselves out across the tree). Roosts with a sparse tree structure may have larger crown areas or have more foliage height available for roosting. For pathogens transmitted by direct contact, density is likely to be the relevant measure (as per standard mass action principles). If pathogens are transmitted indirectly through contact with liquid urine falling downward, or via contact with aerosolised urine particles, then total abundance within trees may be the more pertinent measure. To help illustrate this point, we provide a visual in Appendix S6 in the Supporting Information. Distinction between these measures will be key to framing ecologically relevant contact structures.
At the roost level, the associations between tree density, and bat density and abundance were diminished, as density of trees was a relatively poor predictor of tree-level abundance and 3-D density. This is likely an artefact of scale, and heterogeneity of tree density across larger areas (see similar issues of spatial heterogeneity and scale originally discussed in Krebs (1999)). Individual roosts in our dataset varied substantially in their density of trees across space. As a result, the mean density of trees (as used in these roost-level models) may not be a meaningful measure of density in roosts with a heterogenous tree structure. In other words, the variation in tree density is important at localised scales (i.e. patches within the roost), but not if averaged over the roost.
Measures of density also varied greatly by scale. This reflects the highly aggregative nature of bat distribution which is captured to different extents across the scales. Estimated mean density ofPteropus at the roost level was 13-fold lower than the subplot-level mean estimate that accounted for heterogenous distributions of bats (0.38 bats/m2 with an interquartile range of 0.21-0.47, vs 5.13 bats/m2 with an interquartile range of 2.71-6.09). At the roost-level, the total roost area can encapsulate substantial unoccupied space, if trees are sparsely distributed or not occupied, as the perimeter of the roost boundary captures the maximum extent of inhabited roosting habitat, but not trees that are occupied and unoccupied within this boundary. This contrasts with other scales of density estimate in this study, like subplot-level kernel density, which more effectively delineate unoccupied and occupied space, and so generate higher estimates of density. The latter estimates were more consistent with previous estimates of Pteropus density, which have ranged between 0-8.7 bats/m2 (average 2.1 bats/m2) for tree-level visual approximations (Welbergen 2005). The finding that spatial distributions are a function of scale is not new (e.g. see discussions of spatial distribution and scale in Krebs (1999)), but highlights the need to consider which scale (or scales) are ecologically relevant when considering the nature of density-transmission scaling in host-pathogen interactions.
Mean estimated tree-level 3-D density was 0.34 bats/m3(0.03-0.32). The low level of variation explained by roost-level and subplot-level models (minimum 3.8% and maximum 13.6% of variation across top ranking models) likely reflects the highly heterogenous spatial structuring of Pteropus bats, and indicates that neither roost-level measures nor subplot-level measures adequately capture heterogeneity in these finer, tree-level estimates. In Pteropusbats, ecological processes operate in complex ways to influence animal density across different scales - at the roost level, a population can expand in area in response to increasing total abundance (and so remain constant in density), or remain stable in area occupied (and increase in density). If a roost does not expand its roost area in response to increasing total abundance (e.g. due to restrictions on space), bats may either fill more trees within the perimeter of the roost (and increase the density of bats at an intermediate subplot level by increasing the proportion of trees occupied but not the density within individual trees), or fill already occupied trees (and so increase both intermediate subplot density and local tree-level abundance). Whether tree-level 3-D density increases will be determined by how much bats increase their utilisation of tree space, which will be driven by the height and crown area available for roosting. The implication is that the relationship between total roost abundance and density at any scale may be unpredictable, and critically, that roost and subplot measures may not provide adequate approximations for population density at scales relevant for transmission. Tree density within roosts may provide a better standard of comparison across roosts when reflecting the conditions for transmission, but only when considered in local scales, and in context of whether local abundance or density is the more pertinent measure for transmission-relevant contact structure.
We would note here that our estimates of tree-level 3-D density and 2-D density were based on overall estimated crown area, not occupied crown area, and so may be underestimates of true density. This is an acknowledged limitation of our approximation of crown area by Dirichlet-Voronoi tessellation. True estimates of tree-level density would require empirical estimation of occupied crown area in the field. However, crown area can be difficult to measure accurately (Vermaet al. 2014) and measurement of occupied area may not be practical. Our Dirichlet-Voronoi tessellation approach allowed us to estimate crown area for a large number of trees which would not have been feasible with field methods. While this approach could be influenced by the choice of maximum crown area set for edge trees and trees in open areas, we show that neither estimated tree-level 3-D density, nor model outputs, varied substantially under different values realistic for eucalyptus species (Verma et al. 2014) (Appendix S5).