Modelling human endurance: power laws vs critical power

Abstract

The power–duration relationship describes the time to exhaustion for exercise at different intensities. It is believed to be a “fundamental bioenergetic property of living systems” that this relationship is hyperbolic. Indeed, the hyperbolic (a.k.a. critical-power) model which formalises this belief is the dominant tool for describing and predicting high-intensity exercise performance, e.g. in cycling, running, rowing or swimming. However, the hyperbolic model is now the focus of a heated debate in the literature because it unrealistically represents efforts that are short (< 2 min) or long (> 15 min). We contribute to this debate by demonstrating that the power–duration relationship is more adequately represented by an alternative, power-law model. In particular, we show that the often-observed good fit of the hyperbolic model between 2 and 15 min should not be taken as proof that the power–duration relationship is hyperbolic. Rather, in this range, a hyperbolic function just happens to approximate a power law fairly well. We also prove mathematical results which suggest that the power-law model is a safer tool for pace selection than the hyperbolic model and that the former more naturally models fatigue than the latter.