生長曲線の検討(第2報) : テイラー級数展開とCrozier式

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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.