A matrix model of uneven-aged forest management

7/29/2019 A matrix model of uneven-aged forest management

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Fores t Sc i . , Vo l. 26, N o. 4, 1980, pp. 609625Copyright 1980, by the Society of American Foresters

A Matrix Model o f Uneven-Aged

Forest ManagementJ o s e p h B u o n g i o r n o

B r u c e R . M i c h i e

A b s t r a c t . A matrix model of a selection forest was developed. Parameters of the model rep-

resent (i) stochastic transition of trees between diameter classes and (ii) ingrowth of new trees, which depends upon the condition of the stand. Parameters were estimated from NorthCentral region hardwoods data. The m odel was u sed to pre dict longterm growth of undisturbed andmanaged stands. A linear programming method was used to determine sustainedyield manage-

ment regimes which would maximize the net present value of periodic harvests. The method

allowed for the joint determination of optimum harvest, residual stock, diameter distribution, andcutting cycle. F o r e s t S c i . 26:609625.

A d d i t i o n a l k e y w o r d s . Selection forest, sustained yield, linear programming, optimization.

S e v e r a l m e t h o d s have be en dev eloped to project the evolution of uneven-agedforest stands. They can be classified in two broad groups (Fries 1974) accordingto whether the elementary unit considered by the model is a tree (single-treemodels) or the stand (whole-stand models). Single-tree models such as thosedeveloped by Botkin and others (1972), Ek and Monserud (1974), and Shugartand West (1977) have proved to be very powerful means of representing com

peti tion betw een trees , m orta li ty , varia tions in specie s com posit io n, and environmental influences on forest growth. Whole-stand models, instead, are by nature m uch more aggregated, representing forest stands with very few param eters.N evertheless, the am ount of inform ation they provid e is usually suff ic ient toansw er key qu estions of im portance to forest m anagers. O ne of the oldest whole-stand models used to predict uneven-aged forest growth is the stand table pro

jec tion (H ush and o thers 1972), bu t m ore com pact m odels have since been devisedby M oser (1967), L eary (1970), E k (1974), and L eak and G raber (1976). Com parisons of th e fo recasting perform ances of a w hole-stand m odel again st those ofa single-tree distance-d epen den t m odel (Ek a nd M onserud 1979) have shown thatthe former forecasts alm ost as well, and m uch m ore cheaply, stand characteristics

of interest to forest m anagers, if conditions a re not exce ssively different from thedata used in calibrating the models. These whole-stand models, coupled withoptimization tech niqu es, have pe rm itted the analysis o f m anagement alternatives(Adams and Ek 1974 and 1975, Adams 1976). For these reasons whole-standmodels have the potential of quickly becom ing practical forest managem ent tools.

This promising future has stimulated the research leading to this paper. Oneimpo rtant objective of this study was to d evelop a model of uneven-aged stand

The authors are res pectively A ssociate Professor and Research Assistant, Department of Forestry,Un iversity o f Wiscon sinMad ison. Research supported by Mc lntireStennis grant 2253 and by theSchool of Natural Resources, College of Agricultural and Life Sciences, University of Wisconsin,

Madison. The authors wish to thank OwensIllinois, Inc., Northern Woodlands Division, for providingbasic data for this study. T he con structive com ments of A. R. E k, J. C. Stier, and three anonym ous reviewers are also gratefully acknowledged. Manuscript received 4 September, 1979.

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management which, while accurate, would be conceptually and operationallysimple. T he resulting m odel consists e ssentially of a growth table o r matrix. Eachelement of the model has a direct interpretation. Determination of the modelcoefficients requires at most ordinary linear regression, and in some circumstances multiplications and divisions are sufficient. Future states of the forest andthe im pact of alternative treatm ents c an be obtained analytically. E conom ic levelsof growing stock and economic stand diameter distributions can be determined

simultaneously by ordinary linear programming. Finally, the seldom consideredquestion of the economic length of the cutting cycle in uneven-aged forest management can also be treated.

T h e M o d e l

Th e growth m odel pre sente d here has its roots in L eslie s and L ew is growthmodels (Lewis 1942, Leslie 1945 and 1948) which were originally designed toinvestigate the effect of age structure on the growth of animal populations. Bosch(1971) applied Leslies model to analyze the growth of California redwoods, andWadsworth (1977) used it to predict the growth of tropical forests. Usher (1966,1969a and b, 1976) modified Leslies model to analyze managed stands, while

Lefkovitch (1965, 1966) considered the general problem of class-specific harvesting in population models of the Leslie type. Questions of optimum class-specific harvesting policies have been investigated by Beddington and Taylor(1973), Beddington (1974), Rorres and Fair (1975), Doubleday (1975), and Rorres(1976, 1978). All of these studies rely on variants of either Leslies or Ushersmodels of animal or tree popu lations. Both models are attractive b ecause of theirsimplicity of interpretation and use. In these models population states are described by vectors while transitions from state to state are described by matrices.Consequently, management problems can often be solved analytically by directapplication of linear algebra. However, a basic problem inherent in these modelsin terms of representing the behavior of a stand of trees is that they lead toexponential growth of the numb er of trees in each size class. Expo nential growth

might nevertheless be acceptable for short-term projections, but when optimalharvesting strategies are sought, global optimization is not possible. For example,a diameter distribution may be found which maximizes volume of production perunit of time, given a certain basal area of the stand, but the optimal basal areaitself is undefined and must be set arbitrarily (Rorres 1976). This situation, ofcourse, leaves the problem of optimum growing stock unsolved. Therefore, amodel of uneven-aged forest growth was sought which had the inherent simplicityof Leslies and Ushers models but which would describe forest growth moreaccurately. This was achieved by modifying Ushers model to make ingrowthonly partially dependent upon harvest, and to allow for ingrowth to respond tochanges in stand density and diameter distribution. As a result, and depending

upon its condition, the stand could grow at an increasing, constant, or decreasingrate.Following the convention of previous authors, the trees in a stand are divided

into a finite number, n, of size classes specified by the diam eter of the trees. Theexpected number of living trees within each size class at a specific point in time,t, is de note d by y 1(, y .u , . . . , y nl. T herefo re the_ entire stand of living trees isrepresented at time t by the column vector

yt = ty] i = l

During a specific growth period 6the trees in a given diam eter class i may remainin the same class or advance to a larger size class. They may also die during theinterval 6, or they may be harvested. We will denote by hit the number of trees

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harvested from diameter class i during the interval 6. Therefore the entire harvestis represented by the column vector

ht = [hit] i = 1, . . . , w.

Furthermore, let us denote by at the probability that a live tree in size class / at

time t which is not harvested during the interval 8 will be still alive and in thesame size class at time t + 6. Also, we will denote by bt the probability that alive tree in size class i 1 at time t which is not harvested during the interval6will be alive and in size class i at time t + 6.1Finally, I tdesignates the expectedingrowth, i.e ., the exp ected num ber of trees en tering the smallest size class duringthe interval $. The situation of the stand at time t + 6 may then be entirelydetermined from the situation at time t, the harvest during 6, and the ingrowthduring 6by the n equations:

yu+e = i t + oiiyu ~ h u)y2t+e = h2{yjt ~ ^it) + &2,(y2t ~ h2t)

y-nt+6 bn( yn^ it hn_lt) + o.n(ynt hnt).To complete the model a specific form must be given to the ingrowth function I t .The simplest alternative would be to set i.t to a constant. This may be appropria te fo r so me m anaged fo rests . H ow ever, a more flexible form would re cognize that ingrowth is affected by the condition of the stand. Eks (1974) observations suggest that ingrowth is inversely related to the basal area of the standand that, for a given basal area, ingrowth is directly related to the number oftrees, that is to say, other things being equal, ingrowth appears to be favoredby sta nds of sm all tr ees. Adoption of this concept led to an expecte d ingrowthfunction of the form

I t = p Q+ /3i jr Bi( yit - hit) + pz 2 (y - hit) (2)-i = l i=l

with I t ss 0. Where B t is the basal area of the tree of average diameter in sizeclass i, while /30, /3l5 and fi2 are constants which one would expect to be, respectively, positive, negative, and positive.

Using (2) as the expression for l t leads to a new expression for the numberof trees in the sm allest size class as a function of the num ber of trees in allsize classes and of the harvest : 2

! This assumes that the interval 0 is chosen in such a way that no tree grows more than one size

class during the period 6. Alternatively, other coefficients can be added to represent the probabilityof a tree growing into higher size class es. Letting be the probability that a live tree in size class

i at time t which is not harvested during the interval 6 be dead at time t + 6, we have mx = 1

a t bi-i-i for / = 1, . . . , n 1 and m n - 1 a n. This unharvested part of mortality is lost.

2 A similar procedure can be use d to make ail eleme nts ofy t+e functions not only of the number oftrees in adjacent size classes but also of the stand density as measured by the number and size oftrees in other size classes. The method consists in writing each equation of system (1), except for the first one, as

y u+g = dn(y hit) + . . . + dUl(yrlt hnt) i - 2 , n

where the d coefficients can be estimated in a similar manner as the coefficients of the ingrowth equation. However, in the application reported below, stand density turned out to have very little

effect on the transition probabilities a and b which were therefore treated as constant, within a diameter class, due to the resulting simplicity of estimation and interpretation. Stand density appearedinstead to play an important role in inhibiting or favoring ingrowth. That is to say, stand density

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yn+e /30 + d 1(y lt h u ) + . . . 4- dn(ynt h nt) (3)

where dx a 1 + f^tB1 + /32dt fi\Bi + /32 for i > 1.

The final model takes then the form

y_t+8= G ( y t - hi) + c (4)

where G and c are respectively a matrix and a column vector of constantcoefficients:

/ / \\d1 d9 . . . dn

b 2


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trees .were grouped by 5.1 cm size c lasses. T here w ere seven c lasses ranging fromtrees in the 15.2 cm diameter class composed of trees 12.6 cm to 17.7 cm, to treesin the 45.7+ cm size class composed of trees of 43.1 cm in diameter, or larger.

Estimation of the probabilities a and b in (1) could be done by simple proportions because the data set used indicates for each plot the number of trees in each

diameter class which between two successive inventories either remained in thesame diameter class, moved up one class, were harvested, or were lost to mortality . 4 The resulting matrix ofa and b coefficients, as defined in (1), was

0.72(0 .0 1 )0.23 0.70

(0 .0 1 ) (0 .0 1 )0.26 0.67

(0 .0 1 ) (0 .0 1 )

0.30(0 .0 1 )

\

0.65(0.03)0.30

(0.03)0.66

(0.04)0.30

(0.03)0.81

(0.05)

0.19(0.05)

0.86(0.06)

\

(6)

/

where the numbers in parentheses are standard errors of mean proportions. Interms of relative error, the coefficients are very accurate for the smaller sizeclasses, but the accuracy declines systematically with the size of trees. This isso because there were few large trees in the data set used. The probability (a) ofa tree staying alive and in a specific size class declines from 0.72 for the 15.2 cmsize to 0.65 for the 30.5 cm size, and increases again to 0.86 for the 45.7 cm sizeclass. On the other hand, the probability ( b) of a tree staying alive and movingup to a higher size class inc rease s from 0.23 for the 15.2 cm size class to a plateauof 0.3 for the 30.5 cm, 35.6 cm, and 40.6 cm size classes and declines to 0.19 forthe 40.6 cm size class. This pattern corresponds naturally to the classical S shapegrowth curve of tree diameter as a function of age.

The ingrowth equation (2) was estimated by linear regression from data oningrowth, number of trees in each size class and harvested trees. Ingrowth wasdefined as the number of trees reaching a diameter of 12.6 cm during a 5-yearinterval because no d ata were available for smaller trees. T he results of ordinaryleast squares estimation were

4 The data for the 133 plots which were measured in 1961 and 1964 were converted to a 5year timeinterval by linear extrapolation. The resulting data were combined with data for the 161 plots measuredin 1964 and 1969. Three trees only were recorded as having grown more than one diameter class and were counted as growing one single class. Three trees were recorded as having declined in diameter and were counted as remaining in the same diameter class. Four ingrowth trees which had been

recorded as entering classes larger than the 15.2 cm class were counted as entering the 15.2 cm class.These anomalies may reflect recording errors and there are so few of them that it was deemed unnecessary to complicate the model to account for them.

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I t = 109.0 - 9.65 V **(? - M + 0.27 y (> - A) (7)(9.9) (1.4) i=i (0.05)i=i

R 1 = 0.15 (corrected for degrees of freedom),Standard error of estimates = 72.7 trees/ha,

Num ber of observ ations = 294.

The coefficient of determination (R2) is small. As computed here, R - measuresthe proportion of the variance around the mean ingrowth which is not ex

pla in ed by th e variables on the right of equation (7). One might conclu de fromthis that little would be lost by modeling ingrowth as a constant . 5 However ,the standard errors in parentheses are very small relative to the coefficientsof the basal area and num ber of trees. One can the refore re ject, with a highlevel of confidence, the hypothesis that ingrowth is independent of stand condition. Instead, equation (7) suggests that ingrowth, which would be of some 109trees every 5 years were there no trees in and above the 15.2 cm diameterclass, would tend to decline as the residual basal area of the stand increases.But this decline would be smaller the larger the number of trees in the stand,i.e., the smaller the average diameter of trees. In summary, although ingrowthappears to be a highly random process there seems to be a systematic and

predic ta ble fe edback of sta nd condit io ns on it , which may be alt ered by harvest.This feedback process is represented by the first row in the matrix G and thevector which can be computed from (6 ) and (7) using e qu ation (3). Th e re sult -ing estiiuated matrices

/are

/ \0.81 -0 .043 - 0 . 2 2 -0 .4 3 -0 .6 9 -0 .98 -1 . 3 109.00.23 0.70 0 0 0 0 0 0

G =0 0.26 0.67 0 0 0 0 00 0 0.30 0.65 0 0 0 - 00 0 0 0.30 0.66 0 0 0Au f\u 0 o 0.30 0.81 0 00

\0 0 0 0 0.19 0.86

/A

\ /

(8)

In concluding this section, it is worth noting that a detailed data set such as theone used here is not mandatory to estimate the matrix coefficients. The elementsof G and can be estimated by linear regression analysis based upon equation(4). Periodic observations on permanent sample plots reporting the number ofliving trees in each diameter class at the time of observation, and the number oftrees harvested, if any, between observations, would be a sufficient data base toestimate the matrix model . 6

A p p l i c a t i o n t o N a t u r a l S t a n d G r o w t h P r o j e c t i o n s

The matrices G and the vector c presented above have been computed usingdata from a managed forest property observed over a period of 8 years. Conse-

3 In this particular case site index did not appear to significantly influence either ingrowth or the

transition probabilities. This may be due to the small variation in site quality within the data used.

Under other circumstances, and depending upon the purpose of the model, different matrices couldbe developed reflecting various site qualities.

{i Sin ce the resulting data set would most likely be of the poole'd cros ssec tion and timeseries type,and because the model is composed of a set of simultaneous difference equations, special attentionwould have to be paid to two possible estimation problems, (i) the likelihood that the coefficients ain matrix C be biased toward 1 (Ner love 1967 and 1971). and (ii) the possible cros s correlationbetween residuals relating to each equation. The first problem might be minimized by using variance comp onent m odels, as suggested by Nerlo ve. The second may require the use of a simultaneousequation estimating procedure of the Zeilner (1962) type.

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quentiy the most appropriate application of the empirical model should be toanalyze only slightly different management regimes on the same property andover a relatively short time span. To try to project with this model the evolutionof a stand assuming no harvest and over several decades is indeed a very bigextrapolation. Nevertheless, such an exercise will be presented. It should not be

viewed as a strict validation of the model, which should be based on extraneousdata and was not done here. The only quantitative indicators of model adequacyare the statistical results reported above. The purpose of this section is insteadto show the implications of the m odel un der the extreme conditions of no ha rvestand very long projection periods. One would like these long-term predictions tocorrespond at least in a general way to what is known about the actual growthof uneven-aged stands.

Assuming no harvest consists of setting ht 0 in equation (4) which thereforebecom es

y t+e = G yt + c. (9)

Given an initial condition of the stand described by y 0 one may compute the

situation of the stand after k growth periods of length 6from the solution of (9):

yke = G ky 0 + 2 GiQ (10)i =0

alternatively one may use (9) in k i terations. This second procedure has theadvantage of providing not only the stand situation at the end of k periods butalso the dynamic evolution during the intermediate years . 7 The computat ionshave been done using the average acre of the property which provided the dataas an initial condition and a time unit 6 of 5 years, consistent with the way thecoefficients (8 ) we re estima ted. Th e results are summ arized by Figures 1 and 2.As Figure 1 shows the initial diameter distribution had an inverse J shape reflect

ing many small and few large trees. The upgrowth of smaller trees resulted in aninverted U distribution after 60 years. During that time interval ingrowth sur

passed m ortali ty , re sultin g in an in crease of to ta l num ber of trees, while basalarea also increased, but at a somewhat slower rate (Fig. 2). The high basal areaforced ingrowth to decline below the mortality rate, resulting in a decline in totalnum ber of trees. De creased ingrow th coupled with continuing upgrow th led aftersome 165 years to a J shaped distribution. At that time the stand had very fewlarge trees, and basal area was at a minimum. This situation favored ingrowthand led to an increase in the number of trees in the smaller size classes. As aresult after 200 years the diameter distribution had a U shape. As shown in Figure2 , the pattern was one of very long oscillations in stand characteristics with a

period of some 250 years . The oscilla tions declined in ampli tude and te ndedtoward an equilibrium distribution which could be readily computed from equation (9). At equilibrium, regardless of the value of t, one must have

y t+e = y t = y *

where y* is the equilibrium distribution. This condition and equation (9) lead to

7 Another advantage of the iteration procedure over the use of equation (10) is that the latter may, for some values ofy0 and k yield negative elements for.y. This occurs if stand density gets very high, leading the ingrowth function (7) to take negative values. As specified by equation (2), ingrowth should take only positive values, which implies from (1) that a lower bound for y u+ is (j u

hu). This constraint can most readily be taken into account in the iteration procedure. In our expe-rience with the model this situation occurred only in projecting undisturbed stand growth, but not under management conditions.

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DIAMETER CLASS ( cm)F i g u r e 1. Predicted longterm growth o f an undisturbed northern hardwoods forest stand. Figures

in parentheses indicate years from the beginning of projection, dashed line refers to limiting equi-

librium distribution of trees.

y* = (J - G)-1c (11)

where I is the identity matrix of order n. As seen from (11) the equilibriumdistribution depends only on the growth potential of the stand as defined by Gand c, and it is independent of the initial stand conditions. The equilibrium standdistribution for ihe average hectare on the property analyzed here is reported inTable 1 . It has a U shape with approximately the same number of trees in the15.2 and 20.3 cm size classes as in the 40.6 and 45.7+ cm class.

In concluding this section, a few remarks are in order. First, all trees of 43.1cm diameter or larger were pooled into a single class and assigned a diameter of45.7 cm to compute basal area. As a result the basal area of stands with few largetrees tended to be underestimated. This is likely to exaggerate the fluctuations inbasal are a appearing in Fig ure 2 . Second, only trees of 12.6 cm or larger areexplicitly accounted for in the model. Consequently, the equilibrium distributionpredic te d by the model , which has a very fla t U shape is most likely to be atruncated inverse J distribution, with many more trees in the less than 12.6 cmdiameter class than in other classes. This is consistent with the general observation that the diameter distribution for mature uneven-aged stands has an inverseJ shape. It seems clear, however, that the existing diameter distribution on theproperty which provid ed the data (F ig . 2, dis tr ib ution (0)) is not th at of a matu reundisturbed stand, but rather the result of periodic harvests. Finally in conditionsof managed stands, stand basal area ranges between 10 and 30 m 2 /ha, and cuttingcycles do not exceed 30 to 40 years. Consequently, only a short portion of thecurves in Figure 2, mostly the rising portion at low basal areas, is relevant to the

study of management regimes to which we now turn.

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TA BL E 1. Lo ng -ter m equilibrium distribution on average hectare, without harvest.

Diameter class(cm) Trees Basal area

number m 2

15.2 35.60.7

20.3 27.9 0.9

25.4 22.5 1.1

30.5 19.5 1.4

35.6 17.1 1.7

40.6 26.7 3.5

45.7 + 37.8 6.2

Total 187.1 15.5

in which case all trees in diameter class m and above would be harvested,

but none o f th e oth ers . A noth er specia l case consis ts of having all cq = 0which would correspond to the undisturbed situation analyzed in the previoussection.

The harvest vector has then the expression

ht =Hyt . ' (12)

And the number of trees resulting from growth and harvest may be computedby substi tu ting htby expression (12) in the growth model (4):

y t+e --- G (I - H)yt + c. (13)

In general, however, the harvest will be applied only every y periods, corre

sponding to a cutting cycle ofyd years. The situation of a stand at the end of acutting cycle, given an initial condition y 0 and the propo rtional harvest Hy0applied during the time interval (0. 6) is then, from (10):

= G {I - H ) v q + V G'c. (14)i= 0

Alternatively, if it is desired to know the evolution of the stand between successive harvests, equations (13) and (4) can be applied iteratively as

yt+e = G {I - H ) y 0 + cyt+2e : Gyt+Q + c' . . . (15)

y't+ye = G yt + iy iis + c yt-Hy+l)6 = G ( I H ) v t+ye + C.

The computations described by (15) have been applied to predict the long-termevolution of harvested stands on the property from which the data were taken.From the sample plots it appeared that stands were harvested every 35 years( 7 = 7, 6 = 5 years), and that the stand conditions, prior to harvest, and theintensity of harvest in each diameter class were on average as described byTable 2. Continuation of the same fixed-proportion harvesting regime wouldlead the stand to grow in the manner described by Figures 3 and 4. As thesefigures show, the harvesting regime results in diameter distributions whichoscillate m uch less than those predicted for natural stand growth (Figs. 1 and

2). The harvested stand appears also to tend more rapidly towards an equi-

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TA BL E 2. Current and long-term equilibrium condition o f the average harvestedhectare on a property m ana ged u nder a fixed-proportional harvest and a 35-yearcutting cycle.

Current condition Longterm equilibrium

Diameter class(cm)

Growingstock1(trees) Ha rvest (trees)

Growingstock(trees) Harvest (trees)

number per cen t'1 num ber num ber percen t~ number

15.2 278.7 43 119.4 133.2 43 56.8

20.3 141.3 45 63.5 129.0 45 58.1

25.4 73.1 52 38.3 86.7 52 45.2

30.5 32.1 69 22.0 56.3 69 38.8

35.6 12.4 75 9.1 34.6 75 26.0

40.6 3.5 53 1.7 29.7 53 15.8

45.7+ 3.0 95 .2 17.5 95 16.8

Basal area (in2) 17.9 8.9 25.3 15.5

1 Growing stock is measured prior to harvest.

3 Proportion of trees harvested from the growing stock in each size class.

librium state. This equilibrium state is such that stand growth restores thestand to the prior-to-harvest condition y* in one single cutting cycle, y* can bedetermined directly from (14) by setting y* = y t+ye = y t , which leads to

y* = ( / - G y + GyH)~1 Glc (16)i=o

and h* = H y *, the equilibrium harvest. It can be observed again that the equilibrium situation of the stand is independent of the initial condition but dependsonly on the growth matrix, the length of the cutting cycle, and the intensity ofthe fixed-proportion harv est. The values ofy * and h* for the prope rty consideredhere are reported in Table 2.. It shows that pursuing the current fixed-proportionharvest with a 35-year cutting cycle would maintain the current inverse J shapeof the diam eter distribution, but the re would be m ore large and few er small trees.As Figure 4 indicates, the harvest would remain fairly constant, at some 10 to 15m2/ha of basal a rea, exc ept for the first harve st.

E c o n o m i c H a r v e s t i n g R e g i m e s

The fixed-proportion harvest regimes analyzed above correspon d m athematicallyto modifying the growth matrix G when a harvest is applied. The method hassome advantages in that no inequality has to be introduced in the model. In

parti cula r, since the harvest is alw ay s a fra ction of the grow ing stock, th ere is norisk of obtaining solutions suggesting negative harvests. As a result all computations may be done by simple application of the rules of linear algebra. H ow ever,specifying the harvest through a H matrix leads to difficulties when the harvestis treated as a variable.

In timber management a harvest regime is often sought which will satisfy twoobjectives. First, it should produce a constant periodic harvest without depletingthe growing stock. Second, the harvest should be such as to maximize a production objective, either in purely physical terms such as maximizing the volumeproduced per unit of time, or in econom ic te rm s in which case max im izing the

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DIAMETER CLASS (cm )

F i g u r e 3. Predicted longterm growth of a northern hardwo ods stand subject to a fixedproportionharvest and a 35year cutting cycle. Figures in parentheses indicate years from the beginning ofprojection, dashed line refers to limiting equilibrium distribution of trees.

p resent value of pro duction would be an appro pria te objective, assuming adequate prices and interest rate.

In the rem ainder o f this section the grow th o f a stand ove r time will be describedby (4), genera lized to account fo r cutt in g cycle s of various length:

yt+re = G y(y t ~ ht) + Y G\c (17)i = 0

where the harvest ht is applied every yO years.

The sustained yield condition requires that y t = y t+ye = y* and that ht =ht+ye h*> which leads to

y* = Gy(y* - h*) + 21=0

which may be expressed as

h* = (G r)_1(G 7 - l)y* + ( G r f G ' f . ( I9)i = 0

Equation (19) gives directly the constant periodic harvest h* which must beapplied every y9 years to maintain any specified stand structure y*. However,a solution of (19) is meaningful only if

h* y*, and h* & 0. (20)

If these conditions do not hold, then the stand structure y* is not sustainableunder the cutting cycle yO.

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A matrix model of uneven-aged forest management