2次元林分遷移の方程式とその応用に関する研究

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An actual forest stand, even in the case of an even-aged pure stand, consists of trees ofdifferent sizes, and each stand has its own pattern of distribution of diameters and heights. Forreasons of diversity of the distribution, we can give the following external factors : restrictions oftopography and site conditions, influences of environmental factors such as weather, diseases,insects, and disasters, and changes made artificially such as by thinnings. In addition, that there isan internal factor which originates in the characteristics of growth. This paper considers thedistribution of tree sizes caused by the latter factor when the effects of external factors are verysmall and need not be considered. It is fundamentally important for growth prospects, to understandthe changes of stand structure originating in the characteristics of growth. A forest stand whosechanges of structure with age are subject only to the characteristics of growth does not exist inreality. It seerns to be grorving under ideal external conditions, so we call it "a stand growingunder ideal conditions".To infer the stand structure and the changes with age of "a stand growing under ideal conditions" from the growth progression of actual forest stands, there is a need of abstract ideas on the growth phenomena and a model explaining them. SUZUKI explains the changes of diameter distribution with forest age as a diffusion phenomena and presents the forest transition equation and a growth model based on this theory (1966, 1967, 1971, 1974, 1983).In this study, following SUZUKI's guidance, the height of trees is added to the transition equation as a new factor and the two-dimensional forest transition whose factors are diameter and height, is discussed.Althoueh the two-dimensional forest transition equation, Equation (8), was derived by SUZUKI (1966), this paper shows that its fundamental solution is a two-dimensional normal distribution, Equation (14), when all of the coefficients of the equation are tunctions of forest age only. The development of the theory is shown in Chapter 2.To show the two-dimensional forest transition equation as a concrete model,we need to determine the function of each coefficient of the equation and to know the range of the parameters of each function. In Chapter 3, a diameter growth model, a heieht growth model, and a model concerning the coefficient of correlation between diameter and height are presented. When we construct a stochasric growth model, we must know the difference between the differential equation which expresses the phenomena in the entire growth period and one which express the growth phenomenon at each forest age. For example, we can show the diameter growth and the height growth in an even-aged pure stand. When we do not consider the randorn fluctuations, the phenomena in the entire growth period of both diameter and height can be described by the differential equation of MITSCHERLICH's growth law. However, at each forest age, although diameter growth in a very short time interval is expressed as a linear function of diarneter, Equation (110), height growth in a very short time interval is constant regardless of the amount of heieht, Equation (146). A.s explained in Paragrarrh 3.2, this difference has a decisive influence on the increase of the variance of distribution.The diarneter growth model whose construction is described in Paragraph 3.3 is an expansion of that of SLOBODA(1976,1977,1984). He proposed that diameter growth in a very short time period is a linear function of the diameter at each forest age in an even-aged pure stand when random fluctuations are not considered, but he adopted a line through the origin in his concrete model. However, the results of a permanent plot of Cryptomeria jappnica D.Don in the Tokyo University Forest in Chiba Prefecture showed that the line does not always pass through the origin and generally has a intersection with the diameter axis. It also was found that there is a linear relationship between the location of an intersection on the diameter axis and the mean diameter, In this case, the increase of variance of the diameter distribution can be expressed as the product of the RICHARDS growth function and a term related to the random fluctuations, Equations(136, 137, and 138). However, after the middle stage of the growth period, it can be expressed approximately by the RICHARDS growth function.As mentioned berore, a height growth model, whose construction is described in Paragraph 3.4, is quite different from a diameter growth model in expression at each forest age。 This difference causes different trends in the increase of variance and explains the commonly accepted view that thecoefficient of variation of height distribution is about half of that of diameter distribution. This difference in the increase of variance is related closely with the changes of inclination of the diameter-height curve with forest age.Changes of the coefficient of correlation between diameter and height with forest age is discussed in Paragraph 3.5 where diameter growth follows the model proposed in Paragraph 3.3, and height growth follows the model proposed in Paragraph 3.4. The coefficient of correlation between the diameter increment and the height increment can be considered to be constant regardless of the sizes of diameters and heights. If we suppose that this coefficient is constant regardless of forest age, the coefficient of correlation between diameter and height decreasesmonotonically as forest age increases.In Chapter 4, the trend of the fundamental solution of the two-dimensional forest transitlon equation with forest age is inferred from the results of permanent plots of C.japonica in the Akita Region. In that analysis, only trees in the upper story were the object of study because, in some actual forest stands, the distribution of the nurnber of trees in the upper story by diameter and height classes was similar to the two-dilnensionai nonnal distribution.In actual forest stands, generally it is observed that the diameter distribution is skewed positively, whereas height distribution is skewed negatively, and the diameter-height curve is convex along the diameter axis. However, these characteristics can be considered to originate from the existence of trees in the lower story which usually are behind the others in growth, and therefore the trend of growth is quite different froin trees in the upper story. Consequently, this paper analyzes the growth of trees in the upper story, and because they compose about eighty percent of the number of trees and about ninety-five percent of the volume of a stand, it is thought that an analysis of the trees in the upper story is sufficient for determining the growthtrend of a forest stand.Distributions of the number of trees in actual forest stands by diametor and height classes were classified into the following five patterns : normal in the upper story, quasi-normal in the upper story, normal, quasi-normal and non-normal, according to test results of the two-dimensionnormality. About twenty percent of the stands followed the non-normal pattern. Thererore, in most of the actual forest stands, the distribution of the number of trees by diameter and height classes was expressed adequately by the two-dimensional nomal distribution. It also was observed that a1l of the distribution patterns generally change to the normal in the upper story as forest age increases.For the increase of the mean diameter of trees in the upper story, the MITSCHERLICH growth curve, Equation(179), fitted well. The parameter of its upper limit was about 50 cm, and <special>parameter k</special> was about 0.02. Fitting the GOMPERTZ growth curve, Equation(180), to the increase of　variance of the diameter distribution in the upper story resulted in a good fit with an upper limit of　about <special>100 cm2</special>. Here, <special>parameter k</special> was fixed to the value of <special>parameter k</special> of the mean diameter increase. The MITSCHERLICH growth curve also fitted well to the increase of the mean height of trees in the upper story. The parameter of its upper limit was about 35 m, and <special>parameter k</special> was about 0.025, which is about 1.3 times the <special>parameter k</special> of the increase in mean diameter. For the increase of the variance of the heieht distribution in the upper story, the MIRSSCHERLICH growth curve which passes through the origin, that is, <special>parameter l</special> is unity, fitted well. Here, <special>parameter k</special> was fixed to a value of two times that of the <special>parameter k</special>of the increase of mean heieht. The upper limit was about <special>6m2</special>. The coefficient of correlation between diameter and height in the trees in the upper story decrease slightly as forest age increased, but it was very stable in stands, and it seemed to be constant in practical use. Its value varied with stands from 0.7 to 0.85. For the decrease in the number of trees in the upper story between thinnings, the exponentially decreasing curve, Equation (190), fitted well with the rate of 1.3 percent per year.The characteristics of the fundamental solution of the two-dimensional forest transition equation can be inferred from synthetic considerations of the above-mentioned trends of each growth factor. In other words, the state of transition of a two-dimensional normal distributionwhich shows the density distribution by diameter and height classes was expressed concretely. This two-dimensional normal distribution starts near the origin of the basic plane where the horizontal axis is diameter and the vertical axis is height. It moves to the upper right-hand corner on the basic plane accompanying the clockwise rotation and diffusion (Figure 29). The trajectory of the center of the distribution was proven theoretically to be a convex curve along the diameter axis, Equation (194), but actually it moved on a straight line (Figures 25, 26, and 28). The clockwise rotation of the distribution indicates that the inclination of the diameter- height curve decreases as the forest age increases (Figures 27 and 28). Because we are considering the two-dirnensional normal distribution, note that the diameter-height curve is expressed as a straight line, Equation(196).In application of the two-dimensional forest transition theory, the development of volume distribution with forest age (Figure 30) and yield prospects by loes (Table 4l) are given in Chapter5. These examples are applications of the fundamental solution. Therefore, we can confirm that the generally known characteristics of volurne distribution having a positive skewness originates from the normality of both diameter distribution and height distribution. Because the method used for the yield prospects by logs is available for a case of having initial conditions, this paper presents the basic idea of the application of this theory using the fundamental solution.This paper discusses a two-dimensional forest transition in the case of a fundamental solution, and the object of this model is a low-density stand where all of the trees have enough growing space. In the natural world, forest stands are usually subjected to various modifications or alterations. Therefore no stand may grow as the fundarnental solution does, but by understanding an actual forest stand as such a stand in which the external factors are operating in the stand growing under ideal conditions, we can obtain a good view of growth prospects.As previously stated, this paper develops the two-dimensional forest transition equation and proves that the fundamental solution of this equation is a two-dimensional normal distribution. The growth models which express the coefficient functions of this equation also were constructed, and the changes with forest age of a two-dimensional density distribution were discussed. The results were expressed easily as a matter of common knowledge. Through this study, the characteristics of the fundamental solution of the two-dimensional forest transition equation were clarified and its image was presented.
三重大学生物資源学部演習林の論文

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田中和博
三重大学生物資源学部

2次元林分遷移の方程式とその応用に関する研究

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