Farm Animal Metabolism and Nutrition

curves of this kind do not differ enough in
shape from a single curve to be able to
reproduce the kinds of shapes shown in
Fig. 10.4. They are not able to produce a
sigmoidal shape.

Logistic curves

The exponential model assumes a single
rate-limiting factor, the amount of digestible
fibre. The logistic growth function, as
applied in microbiology (Zwietering et al.,
1990), assumes that gas production is pro-
portional to both the microbial population
size and the digestible substrate (Schofield
et al., 1994). At the beginning of fermenta-
tion, the microbial population is the limit-
ing factor, at the end the substrate plays this
role. In consequence, the logistic curve is
inherently sigmoidal and the maximum rate
occurs when half the substrate has been
digested. The single pool equation takes the
form:

Vt= Vf(1 + exp(2 
 4

S(t
L)))
^1 (10.5)

The symbols have the same meaning as in

Equation 10.4 and S is a specific rate,

similar to the fractional rate constant k.

Equation 10.5 produces curves of the form

shown in Fig. 10.5.

A multiple pool version of this

equation would be:

Vt= Vfn(1 + exp(2
4

Sn(t
Ln)))
^1 (10.6)

This equation turns out to be very versatile

for fitting gas curves and, in most cases,

two pools suffice to describe a feed gas pro-

file (Schofield et al., 1994). One important

difference between Equations 10.4 and 10.6

lies in the lag term L. Equation 10.4 is a

discontinuous function and L determines

the point of discontinuity. In contrast,

Equation 10.6 is valid over all positive

values of t.

Gas Production Methods 219

Fig. 10.4.Exponential curve with discrete lag.