Fig. 2. Potential effects of complementarity in Bergmann’s and Allen’s rules.

Bergmann noted that changes in surface and volume typically accrue at different rates such that when body size decreases, the surface-to-volume ratio, SVR—and thereby the ability to dissipate heat—also increases (a). Similarly, Allen noted that appendages like the beak already have high SVRs so that when they become larger, SVR also increases (b). Here we note that by combining small changes in both traits, lineages can achieve comparable changes in their SVR without drastically altering their morphology and, presumably, their ecology (c). Depicted examples were simulated by approximating a bird’s body with two spheres and one cone (cartoon depictions were drawn to exemplify the potential subtlety of these changes). Parameter values: Body size reduction factor in a = 1%; Beak size increase factor in b = 13.7% (volume to volume); Body size reduction factor in c = 0.5%; and beak size increase factor in c = 5.5%.