混合マルコフ過程を応用した本数曲線の誘導

Applying the Markov Chain theories，the auther derived the formual for treenumber curve in even-aged forest stands. The assumptions for this derivation are that a forest tree will be cutted when the cumulative cutting condition for the tree counts k times，and that each tree of the stand will increases its condition M times in average for a unit time interval. Under these assumptions, the auther derived the probability functions
f_k(t)=(<(Mt)>^k)/(k!)･exp[-Mt]
F_k(t)=(M<<(Mt)>_k>^-1)/((k-1)!)･exp[-Mt],
where f_k(t) is the probability function which the cutting condition for a tree counts k times across during t unit time interval. f_k(t) is the probability function which the cutting condition for a tree counts k times at exactly t time unit. This F_k(t) gives the life span distribution of forest trees. According to this life span distribution, the auther derived the probability function
r(j)=<∫_j>^∞ (M<(Mt)>^<k-1>)/((k-1)!)･exp[-Mt] dt,
where r(j) is the probability which a tree will remain over j years old. This function is reduced to the well-known X^2 distribution. Multipling r(j) and the initial number of trees N_0, we can estimate the tree-number curve in an eyen-aged forest stand.