Essentials of the law of geometrical progression studies on the fundamental structure of biological populations.III.

Abstract

1) The authors once reported the relation between Motomura's law of geometrical progression and Preston's lognormal curve, This is the further report about such a problem. As essentials of Motomura's law, the sample size (Numata 1950,Nobuhara 1953,) and the method of testing fitness (Ito and Numata 1954) and so on were pointed out till now. Motomura started perhaps not paying sufficient regard to the theoretical model of biological universe, and his law is likely to be adequate approximately for the data only when the sample size is considerably small sample of an universe of lognormal and other type. 2) Recently, Shinozaki and Urata (1953) showed the close relation of Motomura's law to Corbet's harmonic series, Fisher's logarithmic series, and Preston's lognormal curve based on their heterogeneity concept. There, Motomura's law : log n+a(x_n-b)=0,where x_n is the order of a species whose number of individuals is n. The order of a species : x_n=b-1/a log n. The number of species whose number of individuals is n : S_n=x_n-x_n^<-1>, then Sn=1/a log (n+1)/n. Now, if the commonness of species followed the law of geometrical progression, the number of species in each octave would be indicated as Table 1 and Fig. 1. This is much different from Preston's model of population. 3) The data of beech forest at Mr. Dsisen, Tottori Prefecture explain that there are three types of population, that is, GP type which fit for the geometrical series, non-GP type or S type which is derived from Shinozaki and Urata's homogeneity concept, and intermediate type (Fig. 2) And GP type populations were found at ridges where the habitat conditions were bad, and non-GP types, at valleys where those conditions were better. The summation of 20 GP type populations indicated S type as shown in Fig. 3 Now, a random sample from S type population is considered to be adequate for the geometrical series (Shinozaki and Urata 1953). This state of affairs is similar to the author's previous discussion about the relation between Motomura's law and Preston's lognormal curve. 4) The data of commonness of species in that beech forest were adequate for the geometrical series only when the size of samples was small or some proper stratification according to life-forms was done (Fig. 4 and 5), and the cases not adequate for the geometrical series resemble S type population. Namely, Motomura's law is fit for small and homogeneous groups of life-forms based on individual niches. In such a meaning, the geometrical series is a "law of niche", and the universe structure theory as Preston's lognormal curve is a "law of habitat" which consists of several niches. It is common to the state of affairs mentioned above that the geometrical series is adequate for the data when the size of sample is small or habitat conditions are severe and homogeneous.