On the hydrodynamics-based power-law function and its application in fish swimming energetics

The standard metabolic rate and metabolic transport differential represent parameters of energetic costs and are derived from either the conventional exponential function or the hydrodynamics-based power-law function (HPF). These two models are used independently to describe the relationship between total metabolic rate and sustained swimming speed for fish. Inappropriate estimates of the standard metabolic rate and inaccurate estimates of the metabolic transport differential have been reported in the literature owing to a lack of practical guidance in methods, especially in regard to the estimation, interpretation, and use of the parameters of the HPF. Hence, the field of fish swimming energetics could benefit from clarification and standardization of parameter estimation and use. The distinction between the parameters in the two models is addressed to ensure appropriate use of the HPF in evaluating the standard metabolic rate and metabolic transport differential. The analytical derivation of the parameters of the HPF is also addressed to ensure that they are evaluated correctly and robustly. Finally, a two-step approach to standardizing the estimation of the standard metabolic rate and metabolic transport differential is discussed, where step 1 is to choose the estimate of the standard metabolic rate derived from the model that best predicts the first derivative of the total metabolic rate with respect to sustained swimming speed and step 2 involves evaluating the metabolic transport differential without substituting the standard metabolic rate from one model into the equation for the other model.