S1 Appendix. Solution of the integral equation for
Pmax(T).
The maximal power Pmax(T) is determined by the integral equation
Pmax(T) + Psup(T) = 1
T
Z T
0
Pmax(T − t)dt = 1
T
Z T
0
Pmax(t)dt
(1)
with Psup(T) given by Eq. (3). This equation can be easily transformed into a
differential equation by defining the indefinite integral E(T) of Pmax(T) so that the
derivative E′(T) = Pmax(T). Without loss of generality, we can chose the initial
condition E(0) = 0. The differential equation for E(T) is then
E′(T) + Psup(T) = E(T)
T
(2)
which has the general solution
E(T) = TPm + TPsup(tc) − T
Z T
tc
Psup(t)
t
dt
(3)
where we imposed the initial condition E′(T = tc) = Pm so that Pmax(T = tc) = Pm as
required by definition of Pm. Performing the integral with the constant function
Psup(t) = Ps for T ≤ tc yields
E(T) = T [Pm + Ps − Ps log(T/tc)]
(4)
and using Psup(t) = Pl(t − tc)/t + Pstc/t for T ≥ tc yields
E(T) = T

Pm + Ps + (Pl − Ps)T − tc
T
− Pl log(T/tc)

.
(5)
Taking the derivative of this solution, we finally obtain the solution
Pmax(T) = Pm − Ps log(T/tc)
(6)
for T ≤ tc and
Pmax(T) = Pm − Pl log(T/tc)
(7)
for T ≥ tc. This is the result given in Eq. (6).
PLOS
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A minimal power model for human running performance.
Mulligan, Matthew,Adam, Guillaume,Emig, Thorsten