Growth model of the reared sea urchin Paracentrotus ... - SciViews

General introduction
Y∞−Y Yt () = Y+ 1+ e
0
0 −k⋅( t−t0) We will refer to it as the 4-parameter logistic model.
c. The Gompertz model, an asymmetrical sigmoidal curve
Y
1
Y•
0.8
0.6
0.4
Y•êe
0.2
Gompertz (1825) empirically observed that survival rate often
decreases proportionally to the logarithm of survival. Although this model
is still used with survival data (Ebert, 1999), it has many applications for
growth data as well (Winsor, 1932, Ebert, 1999). The differential equation
of Gompertz model is:
which simplifies to:
dY () t
= k⋅[ lnY∞−ln Y( t) ] ⋅ Y( t)
(7)
dt
−k⋅( t−t0) e
−kt ⋅
e
t
b
∞ ∞ ∞
Yt () = Y.e = Y. a = Y. a
(8)
The last parameterization is simpler and used more often. The first one is
derived from the differential equation (eq. 7) and gives a better comparison
with the logistic model, since t0 also corresponds to the abscissa of the
inflexion point, which is not in a symmetrical position here (Fig. 9).
t 0
i
1 2 3 4 5 6
Figure 9. Example of a Gompertz curve with k = 1, Y∞ = 0.95, t0 = 1.5. Inflexion point i, at
{t0, Y∞ /e}, is lower than in the logistic curve.
t
(6)
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