The mathematics of thermal sub-optimality: Nonlinear regression characterization of thermal performance of reptile metabolic rates.

Although several approaches have been suggested, there is no broadly accepted single approach for quantitative characterization of thermal performance in ectotherms. I sought to identify the most appropriate non-linear function with which to represent thermal performance of ectothermic metabolic rate, and to interrogate the biological relevance of the thermal parameters of this function. I used published data for exercise-induced metabolic rates of eight species of reptile from a broad phylogenetic base and global distribution. Using an Akaike Information Criterion, I compared 12 different models proposed to characterize thermal performance adapted from a broad range of disciplines, finding that a beta-distribution model described the reptile metabolic rate data most parsimoniously. Using the beta-distribution model, unique functions were parameterized for each species. Four parameters were extracted from each species-specific fit: the temperature coincident with the peak of the thermal performance curve, Topt; the point at which the function intersected the x-axis, CTmax; and two points indicative of thermal breadth, Td(lower) and Td(upper). There was a positive relationship between the species' preferred body temperatures (Tpref) reported in the scientific literature and both Topt and Td(lower) extracted from the species-specific beta functions. While Td(lower) estimates were not different to published Tpref values, Topt estimates were statistically higher than Tpref. This is consistent with previous observations that the point of peak performance does not match Tpref. The predicted CTmax also correlated well with published values. The model in its current form was not able to estimate CTmin, and this parameter was not explored here, but should be in future research. By providing a quantitative description of the thermal performance, the beta-distribution function offers a new theoretical basis for thermal optimality. I contend that Tpref aligns with the mathematical threshold Td(lower), where metabolic rate is at its maximum prior to thermal inhibition.