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591 - 10.9 Forest Management 59 Let xi(i = l 2 n be the...

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Unformatted text preview: 10.9 Forest Management 59 Let xi (i = l, 2, . . . . n) be the number of trees within the ith class that remain aftt each harvestr We form a column vector with the numbers and call it the nonharve: vector: Xi x2 x: xn For a sustainable harvesting policy, the forest is to be returned after each harvest to th fixed configuration given by the nonharvest vector x. Part of our problem is to find thos nonharvest vectors x for which sustainable harvesting is possible. Because the total number of trees in the forest is fixed, we can set x1+xz+---+x,1:s (l where s is predetermined by the amount of land available and the amount of space eacl tree requires. Referring to Figure 10.9.2, we have the following situation. The fores configuration is given by the vector x after each harvest. Between harvests the tree: grow and produce a new forest configuration before each harvest. A certain number 0 trees are removed from each class at the harvest. Finally, a seedling is planted in place of each tree removed, to return the forest again to the configuration x. Trees removed Forest afier growth Trees not removed Gromh . .3 Same forest configuration Plant seedlings .§\A§Ailii A Forest before growth Forest afier harvest (nonharvest vector x) (nonharvest vector X) A. Figure 10.9.2 l» DE> DE» —L>> (if. M a. ‘/ _‘EE> DD> DE> DD> 9-» fia> DD» fix» DE> -l> -l> -i> -l> C» +22» ~\. f“; Consider first the growth of the forest between harvests. During this period a tree in the ith class may grow and move up to a higher height class. Or its growth may be retarded for some reason, and it will remain in the same class. We consequently define the following growth parameters gi fori : I, 2, , n — 1: gr 2 the fraction of trees in the ith class that grow into the (i + I)—st class during a growth period For simplicity we assume that a tree can move at most one height class upward in one growth period. With this assumption, we have 1 — g = the fraction of trees in the ith class that remain in the ith class during a growth period ...
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