Something that has been far less studied is the nature of variation around those mean relationships between population density and body size. However, many plots of observed densities against body mass appear to show quite strong relationships between the extremes of density and body mass. As two examples, consider published data on both mammals and birds (Fig. 1). The data for mammals appear to show upper and lower limits that show similar scaling to that of the mean. By contrast, the data for birds suggest that the upper limit is more steeply negative than the mean, whilst the lower limit is almost mass invariant. Both graphs suggest that the smallest species do not reach densities as high as would be expected from the overall scaling of the upper boundaries. The question arises, what determines these upper and lower limits?

*Fig. 1. Published relationships between population densities (or their proxies) and body mass.*

*Fig. 2. If every individual were a cube with the specific gravity of water, this figure suggests that we would never observe population densities in the triangle. We don't - but, given the scale on the y-axis, this doesn't seem a satisfactory explanation for the location of the upper limit to population density. (From Cotgreave & Harvey 1993 Trends in Ecology & Evolution)*

Prompted by these observations, we decided to identify some simple models that would guide us about where we expected the limits to population density to lie. Specifically, we conjectured that an upper limit to population density might lie where a species accounted for all above-ground net primary productivity (ANPP) in the most productive biomes on earth. With such a varied and coarse diet, assimilation efficiency would be low. Accounting for that, and dividing the density of ANPP by mass-specific individual field metabolic rates identifies that upper bound.

For the lower bound, we conjectured that the area of an individual's home range would increase with the distance they travelled in a day. At the limit to area use, they would be travelling as far as they could each day. If there was no overlap among home ranges but no gaps between them, this would be the limit to individual area use in a spatially-contiguous population. For various reasons, this model assumed that area used would increase in proportion to the square of daily travel distance; we referred to this model as the "targeted search model".

We used published relationships between body mass and key parameters in these models, in order to estimate where the limits to density might lie for both birds and mammals. We compared those predicted limits to estimated population densities for 10,474 populations of birds and mammals and found that, given the simplicity of the models, they did a remarkably good job of describing the empirical extremes of the data (Fig. 3).

*Fig. 3. Empirical estimates of population density for 8,188 bird (left panel) and 2,286 mammal (right panel) populations (open circles), together with fitted quantiles and estimates of where upper (paler solid lines) and lower (paler broken lines) limits should lie.*

*Fig. 4. Left panel: population densities of herbivorous (green), omnivorous (purple) and carnivorous (red) mammals in relation to body mass; open blue circles show human hunter-gatherer populations and filled blue circles are the human densities indicated. Right panel: distributions of hunter-gatherer population densities for societies with relatively carnivorous (red), omnivorous (purple) and herbivorous (green) diets in relation to the quantile spans for the same trophic groups of wild mammals (horizontal lines); the blue distribution shows modern human densities by country.*

Many refinements of these models are doubtless possible. However, an important challenge will be to identify when failures of a given species group to fill expected ranges of population densities are due to sampling biases and when they are due to morphological, physiological or behavioural constraints. More work is required to identify the underlying parameters with greater confidence, to produce unbiased distributions of abundance estimates, and to validate the assumptions of our underlying expectations about the relationship between travel distance and area use. Nevertheless, these models substantially improve on previous estimates of where the limits to density should lie, and provide useful frames of reference against which to evaluate observed distributions of population densities in these and other taxa.

This post relates to work published in

*Ecology Letters*. Read the original paper for free,

**here**.