On the mathematical description of lactation curves

SUMMARY The lactation curve may be represented mathematically by the general equationY= Aϕ1(t)ϕ2(t), whereAis a positive scalar, ϕ1,(t) is a positive monotonically increasing function with an asymptote at ϕ1= 1, and ϕ2is a monotonically decreasing function with an initial value of unity and an asymptote at ϕ2= 0. Functions considered as candidates for ϕ1were: (Mitscherlich), (Michaelis-Menten), (generalized saturation kinetic), 1/(1 +b0 (logistic), b0exp (Gompertz) and [1+ tanh(b0+ b1t)]/2 (hyperbolic tangent). Candidates for ϕ2were e–ct(exponential) and 1/(1 +ct) (inverse straight line). The 12 models thus obtained andY= Atbe–ct(Wood's model) were fitted to whole-lactation data from 23 animals. Mitscherlich x exponential, Michaelis-Menten x exponential, logistic x exponential, logistic × inverse straight line and Wood's model all fitted well. For these models, expressions for time to peak, maximum yield, total yield over a finite lactation and relative decline at the midway point of the declining phase were obtained. The Mitscherlich x exponential model generally fitted better than Wood's model and, unlike Wood's model, gives simple algebraic formulae for all these summary statistics.