生長曲線の検討(第2報) : テイラー級数展開とCrozier式

The system of growth can be defined mathematically in various ways. Assuming that the system consists of only one measure y, the system is reduced to the single differential equation :
(dy)/(dt) = f(y)
where f(y) is the function of y. Let us assume that f(y) can be developed into Taylor series :
f(y) = a_0 + a_1y + a_2y_2 + a_3y_3 + . . . ..
Retaining three trems, we have :
(dy)/(dt) = a_0 + a_1y + a_2y_2 .
This equation is almost the same as the Crozier (Crozier, 1926). Moreover, this equation comes to the Mitscherlich (when a_2 = 0) and the Logistic (when a_0 = 0).
Denoting the Crozier by
(dy)/(dt) = (K_1 + K_2y)(A - y),
the solution of the Crozier is :
y =( AB exp[(K_1 + K_2A)t] - K_1)/(B exp[(K_1 + K_2A)t] + K_2)
where A. B. K_1 and K_2 are constants. To examine the applicability of the Crozier equation, it was applied to the observed height growth. The Crozier showed a good fit to the growth not only with a clear inflection but also without one, as compared with the Mitscherlich, the Logistic and the Gompertz. The applicability of the Crozier equation in the growthcurve- fitting was recognized.