A two-runners model: optimization of running strategies according to the physiological parameters

TL;DR

This paper introduces a mathematical model for middle-distance running strategies, optimizing velocity and energy use based on physiological parameters, to determine optimal overtaking points and improve race outcomes.

Contribution

It presents a novel optimal control model incorporating physiological and physical factors to analyze and optimize running strategies in competitive races.

Findings

Model predicts optimal overtaking points based on athlete differences

Simulations show how staying behind can improve race performance

Comparison with real race data validates the model

Abstract

In order to describe the velocity and the anaerobic energy of two runners competing against each other for middle-distance races, we present a mathematical model relying on an optimal control problem for a system of ordinary differential equations. The model is based on energy conservation and on Newton's second law: resistive forces, propulsive forces and variations in the maximal oxygen uptake are taken into account. The interaction between the runners provides a minimum for staying one meter behind one's competitor. We perform numerical simulations and show how a runner can win a race against someone stronger by taking advantage of staying behind, or how he can improve his personal record by running behind someone else. Our simulations show when it is the best time to overtake, depending on the difference between the athletes. Finally, we compare our numerical results with real data from the men's 1500 -- m finals of different competitions.