growth-and-yield model for uneven-aged cedrus atlantica stands in morocco

Forest Ecology and Management, 36 (1990) 253-266 253 Elsevier Science Publishers B.V., Amsterdam

Growth-and-yield model for uneven-aged Cedrus atlantica stands in Morocco

Mohamed KhatourP and Brain Dennis 2 ~ Institut Agronomique et Veterinaire Hassan II, BP 6202 Rabat-Instituts (Morocco)

2College of Forestry, Wildlife and Range Science, University of Idaho, Moscow, ID 83843 (U.S.A.)

(Accepted 8 June 1989)

ABSTRACT

Khatouri, M. and Dennis, B., 1990. Growth-and-yield model for uneven-aged Cedrus atlantica stands in Morocco. For. EcoL Manage., 36: 253-266.

A model was developed to predict growth and yield in uneven-aged cedar (Cedrus atlantica Ma- netti) stands in the Ajdir Forest in Morocco. The model predicts yield from differential equations and diameter distributions. Using data from temporary plots, a system of eleven first-order nonlinear differential equations was developed. These equations relate the rates of change of ingrowth, mortality, and survivor growth to the stand conditions. Numerical integration gives growth-and-yield projections through time. Predicted stand tables are produced by estimating Weibull distribution parameters from the results of the system of stand-level equations. The model is written in Fortran and runs on a microcomputer.

INTRODUCTION

The cedar species Cedrus atlantica Manetti is North Africa's most valuable source of timber. The cedar forests are the principal productive forests adapted to the cold of a humid Mediterranean climate (Emberger, 1955) between 1600 m and 2800 m elevation. Earlier ecological studies of these forests have been non-quantitative (Negre, 1952; Emberger, 1955; Lepoutre and Pujos, 1963; Pujos, 1964), but recently some authors have studied quantitative as- pects of the growth and yield of the species (M'hirit, 1982; Messat, 1986; Khatouri, 1988).

Cedar forests are subject to great cutting pressure. The silvicultural treat- ments and cutting practices adopted by foresters and local inhabitants through many centuries have produced a diversified and irregular structure. As the area of the forests is decreasing and the demand for forest products is increasing, the need for intensive management is becoming urgent. A component of such management is the development of quantitative methods for predicting current and future growth and yield.

0378-1127/90/$03.50 © 1990 - - Elsevier Science Publishers B.V.

254 M. KHATOURI AND B. DENNIS

The objective of this study was to develop a growth-and-yield model to aid forest management planning for uneven-aged cedar stands in the middle At- las in Morocco. The methodology used should be useful for other species of irregular structure in other geographic regions.

M O D E L I N G A P P R O A C H

In this study, the whole-stand modeling approach was followed (Munro, 1974). The principle of compatibility between the stand average and diameter-distribution models was used (Knoebel et al., 1986; Lynch and Moser, 1986). First, stand equations in the form of a system of differential equations were developed for estimating the average rates of changes in the stand attributes. Stand and stock tables were then derived by estimating the diameter distribution from the predicted-stand level attributes. This approach was chosen because if offered a good compromise between model complexity and practical needs.

Stand-level equations

In even-aged stands, attributes such as number of trees, basal area, arith- metic mean diameter, and volume are often predicted directly as functions of age, site index or dominant height, and/or stand density. This method cannot be applied in uneven-aged stands, since age is not a meaningful variable and site indexes are questionable for prediction purposes. Therefore, we adopted the concept of compatibility in growth and yield, as introduced by Clutter ( 1963 ) and extended to uneven-aged application by Moser ( 1967 ) and Moser and Hall (1969).

If Y denotes the vector containing the values of yield attributes, the instantaneous rate of change in Y can be expressed as a system of differential equations:

d Y / d t = f ( Y,S), ( 1 )

where S is a vector containing parameters of site quality. Integration of this system over the growth period with the initial condition results in the elements of Y at time t. The instantaneous rates of change in the elements of Y can be approximated by their respective average annual changes, since one year represents a relatively small increment of the time-scale.

In uneven-aged stands, the dynamics of stand development are determined by three basic components: ( 1 ) ingrowth (trees recruited into the smallest diameter classes ); (2) mortality (trees dying); and (3) survivor growth (size increase of the existing trees). In the model, these components are incorpo- rated into growth rates of three stand attributes: number of trees; basal area; and sum of diameters. The rates of change of these attributes are given by:

A GROWTH-AND-YIELD MODEL FOR UNEVEN-AGED CEDRUSATLANTICA 255

d(N) /dt=d(N~) /dt-d(Nm) /dt (2)

d (SD)/dt = d (SDi ) /d t - d (SDm)/dt + d (SDs)dt (3)

d (SBA)/dt = d (SBAi)/dt- d (SBAm)/dt+ d (SBAs)/dt (4)

where: N is the total number of trees; SD, sum of tree diameters; SBA, sum of tree basal areas; Ni, number of ingrowth trees, Nm, number of dead trees; SDi, sum of ingrowth tree diameters; SDm, sum of diameters of dead trees; SDs, sum diameter growth of surviving trees; SBAi, sum of ingrowth tree basal areas, SBAm, sum of dead tree basal areas; and SBAs, sum of basal-area growth of surviving trees. Prediction equations for each element on the right side of equations (2), ( 3 ), and (4) were developed from data as functions of stand density and site variables. Simultaneous integration produces predictions of the average stand attributes at any particular time. The predicted attributes are then used to estimate the parameters of the diameter distribution model in order to provide stand tables. The model development section describes the form of the functions and independent variables used in each equation of the system of differential equations.

Diameter distribution

A probability density function (PDF) provides yield estimates over specified diameter-class limits under the approach of Strub and Burkhart ( 1975 ). The general form of such an estimate is

Y=N i g(x) f(x,O) dx (5) l

where Y is the yield attribute, N is the number of trees in the area, x is tree diameter at 1.3 m, g(x) is a function of tree diameter defining the yield attribute to be estimated, f(x,O) is the PDF describing the diameter distribution, O is the parameter of the PDF, and l and u are the lower and upper diameter limits for the attribute defined by g(x). Some of the yield attributes defined by g(x) are number of trees (when g(x) = 1 ), basal area (when g (x) = ( n/4 )x 2 ), and volume (when g(x) = volume equation ). With an appropriate r'DF, integration over the whole range of diameters gives the total value of the yield attribute of the stand. Use of the above equation requires the knowledge of the PDF parameters O and the number of trees N present in the area at time t.

We used the parameter 'recovery' method introduced by Hyink and Moser (1983 ) to estimate the PDF parameters. The values of the parameters were selected to produce yield attributes (from equation ( 5 ) ) which match aver-


age attributes predicted from the stand-level model. Basically, the method is similar to the moments technique for estimating the parameters of any PDF (Johnson and Kotz, 1970). The primary advantage of this method is that the whole-stand and diameter-distribution values of specified attributes are compatible. We estimated the first and the second moments of a two-parameter Weibull distribution from the average diameter and the basal area of a stand. See Frasier ( 1981 ), Lynch ( 1982 ), Matney and Sullivan ( 1982 ), Burk and Burkhart (1984 ), and Knoebel et al. (1986 ) for additional examples of the parameter-recovery method.

M O D E L D E V E L O P M E N T

Data collection

This study focused on the Ajdir Forest in the Middle Atlas mountains in Morocco, in Khenifra Province, approximately 30 km east of Khenifra city (32°5 'N lat., 5°6 'W long.), and varies in elevation from 1500 m to 2300 m above sea level. The total area of the Ajdir Forest is 23 473 ha (Anonymous, 1978 ). This study was concerned only with the cedar series, which constitute approximately 15 000 ha. The Ajdir Forest is characteristic of the type of cedar stand encountered in the whole Middle Atlas region. A recent forest management plan (Anonymous, 1978 ) provided good maps and information on vegetation, geology and climate which aided the selection of the measurement plots.

Eighty-seven temporary plots were selected to provide data for model construction and parameter estimation, since no permanent plots exist in the region (Table 1 ). Another 22 plots were selected independently for validation purposes. Sample plots were located in stands representative of the forest type and covering a wide range of stand conditions. The plots all had fixed areas ranging in size from 0.02 to 0.10 ha. Large discrepancies in density between stands required the use of different plot areas. Almost all the area had been subjected to some form of partial cutting. The plots from which model coefficients were estimated were, for the most part, chosen from areas where there had been minimal disturbance. All the plots in which cutting had occurred in the last 10 years were eliminated from model construction.

Most of the stands were uneven-aged, poorly stocked, and, in certain loca- tions, had mixed species composition, consisting primarily of cedar and green oak. For this reason, age was not considered to be useful for prediction purposes and was not considered in data collection and model construction. Gen- eral site descriptor variables included slope, aspect, elevation, soil parent material, and data on the vegetation community (which was based on the procedures given by Anonymous, 1978). These variables, recorded as an al- ternative for describing site potential ( Stage, 1973; Wykoff et al., 1982 ), could


TABLE 1

Number of plots used for model parameter estimation for various levels of cedar community, soil parent material, aspect, slope, and elevation in the Ajdir Forest in Morocco

Cedar community ~ VO5 VO3 VO 1 VO4 Total

46 26 10 5 87

Soil parent material Calcareous Dolomites Sandy Stones Total

40 34 13 87

Aspect None N NE E SE S SW W NW Total

5 15 7 4 7 12 i 1 14 8 87

Slope (degrees) 00-05 06-10 11-15 16-20 21-25 26-30 Total

8 10 12 15 14 17 87

Elevation (m) 1451-1550 1551-1650 1651-1750 1751-1850 1851-1950 1951-2050 Total

7 21 8 18 21 11 87

~VO5, Ajdir; VO3, JBel Saa; VO1, Kerrouchen; and VO4, Itzer.

have a masking effect on the correlation between microclimate factors (hu- midity, temperature, soil fertility, precipitation, etc... ) and growth of trees.

All trees in the plot were numbered and their diameters at 1.3 m recorded. All cedar stems with a diameter above 10 cm were cored at 1.3 m above ground level with an increment borer. Data from these core measurements allowed calculation of ingrowth and survivor growth estimates during the last I 0-year growth period. Trees that had died in the last 10-year period were identified (subjectively based on experience of local foresters) and their diameters at 1.3 m were measured.

Stand-level equations

The average annual rates of change of the stand attributes described on the right-hand side of the system of equations (2), (3) and (4) were first computed from the measurements collected on the 87 study plots, and then used to fit growth-rate equations with linear and nonlinear regression techniques. Various nonlinear equations were selected from many candidate expressions


having different basic variables and transformations, mainly by analyzing re- sidual plots. Number of trees, sum of diameters, and basal area were found to be the main predictors of stand growth. Table 2 gives the equations ( (6 ) - ( 13 ) ) which apply.

Certain simple and inexpensive site measurements, such as the cedar community type, soil parent material, slope, aspect, and elevation, are reported as being related to the growth and yield of cedar stands (Anonymous, 1978; M'Hirit, 1982). However, these variables did not contribute significantly to explaining the large variability observed in the data. One possible reason for this was that the cedar stands in the Ajdir Forest are not fully stocked. In- stead, the main predictor variables were the variables characterizing the stand density.

There is poor natural regeneration in many parts of the cedar stands in the Ajdir Forest. Forty-seven percent of the measured plots had no regeneration (or ingrowth) during the last 10-year growth period, and plots without ingrowth occurred in almost all stand types. The presence or absence of regeneration in plots was accounted for by a mixed-model approach. The data were separated into two sets based on the presence or absence of trees with diameters less than 10 cm. The data set having such trees was used for fitting the

TABLE 2

Stand growth-rates equations used for the projection of growth and yield in the cedar stands

Equations I R 2 Sy.x Eqn.

I n g r o w t h

dNJdt= 15.541 exp (--0.038SBA) ifNTLlo> 0 dN~/dt = 0, otherwise d(sD,)/dt=O.125 d(Ni)/dt, d(SBgi)/dt=O.O13 d(Ni) /dt , Mortality d(Nm)/dt= (P)" (F)

where P= 1/{ ( 1 +EXP( 1.444-- 0.001N) } where F = 0.388SBA{ 1 -- EXP ( -- 0.001 N) }

d(sDm)/dt= 1.573 (MBA)d(Nm)/dt, d(SBAm)/dt=0.448 (MBA)'d(Nm)dt, Survivor growth d (SD,)/dt = 0.694MD-o.s57{ 1 -- E X P ( - - 0.002N ) } d(SBA~)/dt=0.353MBA-°'°TSExP{--0.001N}

0.41 2.730 (6)

0.99 0.069 (7) 0.98 0.015 (8)

- - - - ( 9 )

0.91 0.353 (10) 0.65 0.228 (11)

0.82 0.313 (12) 0.85 0.237 (13)

i Where: N is the number of trees at the beginning of the measurement period (trees/ha); MD, mean of tree diameters at the beginning of measuring period (m); SBA, sum of tree basal areas at the beginning of measurement period (m2/ha); MSA, mean of tree basal areas at the beginning of measuring period (m 2/ha ); Ni, number of ingrowth trees ( trees/ha ); SDi, sum of ingrowth tree diameters ( m / ha); SBAi, sum of ingrowth tree basal areas (m2/ha); Nm, accumulated number of dead trees (trees/ ha); SDm, sum of dead tree diameters (m/ha) ; SBAm, sum of dead tree basal areas (m2/ha); SBA~, sum of surviving trees basal areas (m: /ha) , SD, is the sum of surviving trees diameters (m/ha) ; and NTLI0 is the number of trees less than 10 cm.

A GROWTH-AND-YIELD MODEL FOR UNEVEN-AGED CEDRUS ATLANTICA 2 5 9

ingrowth equation (equation (6) ). The signs of the coefficients in equation (6) indicate that ingrowth is inversely related to stand basal area. Once the seedlings are established, the only stand variable that affects their development is stand density. The changes in sum of diameters and sum of basal areas for ingrowth trees were found to be related to the change in the number of ingrowth trees (equations (7) and (8), respectively).

Different functional forms have been used for modeling tree mortality in uneven-aged stands (Ek, 1974; Hamilton and Edwards, 1976; Lynch, 1982; Wykoff et al., 1982 ). Only 24% of the measured plots had experienced some tree mortality during the last 10-year growth period~ We combined two nonlinear functions in this study. First, for all measured plots, a response variable was set equal to 1 if mortality occurred in the plot, and equal to 0 otherwise. A logistic regression was then performed to estimate the probability of the occurrence of mortality as a function of tree density (part P of equation (9)) . Second, a nonlinear model was fitted to the data set containing only the plots that had mortality during the last 10-year period to predict the change in the number of dead trees (part F of equation (9) ). The combined mortality rate estimate is computed as the product of the two functions (equation (9) ).

The variables most useful for predicting mortality rate were the number of trees and the stand basal area. The changes in the accumulated diameters and basal areas of dead trees were found to be linearly related with the product of the mean tree basal area and the mortality rate (equations (10) and ( 11 ), respectively).

A form of a negative exponential model was selected to predict the rate of survivor growth (equations (12) and ( 13 ) ). Note that a large growth rate of the diameters (sos) of surviving trees is associated with stands of small mean tree diameters, and a large number of trees per ha (equation ( 12 ) ). The survivor stand basal area (SBAs) growth rate decreases as the stand basal area increases (equation (13) ). Biologically, such a relationship would be expected, since growth rate tends to decrease with increasing mean tree basal area or mean tree diameter.

• . • ~ . .

Dtameter-dtstnbutzon generation

The Weibull distribution was used to generate the stand attributes by diameter classes (stand table ). The Weibull density function is

f(x',a,b,c) = (c/b){ ( x - a ) / b } c-~ e x p [ - { ( x - a ) / b } c]

O<__a<_x<_oo; b,c>O (14)

=0, elsewhere

where x is the tree diameter, a is the location parameter, b is the scale parameter, and c is the shape parameter. The Weibull PDF can assume a variety of


shapes. Substituting sample moments representing average stand attributes (mean diameter and mean squared diameter) produces the following equations for the moment-based parameter estimates:

l )=a+b F( l + l /c) (15)

-D -7 =a2 + 2ab F( 1 + l /c) + b 2/ '( 1 + 2 / c ) (16)

Here/9 is the mean diameter, ffz is the mean squared diameter, and F(. ) is the gamma function. Solving for b in equation (15 ) and substituting into equation (16), the system reduces to one nonlinear equation with one unknown (parameter c), given that parameter a is fixed. The solutions were obtained from a Fortran subroutine developed by Knoebel et al. (1986), which uses the secant method for solving a nonlinear equation with one unknown.

Once the Weibull parameters have been recovered, special cases of equation (5) provide a stand table. The appropriate cases giving the number of trees (Ni) and the basal area (BAi) in each diameter class, xi, are respectively:

Ni=N [ e x p ( - w ~ ) - e x p ( - W u ) ] and (17)

BAi = N (n /4 ) [a2{ (exp( - w t ) - e x p ( -wu )} +2ab {F(Wu,Vl )

-F(Wl,Vl )}+bZ{F(wt,vz)-F(Wu,V2)}] (18)

Here, w , = { ( x , - a ) / b } c, w , = { ( x , - a ) / b } c, v~=( l+l /c ) , v2=(1+2/c), F(w,v) is the incomplete gamma function, and Nis the total number of trees/ ha. A local volume table was then applied to the stand tables to obtain stock tables.

A Cedar Growth and Yield (CEDGY) program to implement the model was written in Fortran and is operational in an interactive mode on a microcomputer. The model is composed of a main program which calls many subroutines. It does not require any external subroutines. Users can easily alter subroutines to change coefficients in the system of differential equations, use other methods to estimate the Weibull parameters, or produce different outputs.

M O D E L T E S T I N G

Two procedures were applied in this study to determine the reliability of the CEDGY model predictions. First, the predictions of the model were compared with an independent data set which was not used to build the model (validation). Second, long-term projections were calculated to evaluate the overall model performance.

A total of 22 plots was selected across the range of conditions covered by the plots used in model fitting. Two data sets were obtained. The first represents the average stand conditions 10 years prior to sampling. The second set

A GROWTH-AND-YIELD MODEL FOR UNEVEN-AGED CEDRUS ATLANTICA 261

represents the observed stand average conditions at the sampling time. The stand attributes of the first data set were then used to predict the stand summary of the second data set.

The test of the model focused on the two methodological components used in this study: (i) comparisons of the average whole-stand attributes for each component of growth derived from the system of differential equations (stand- level-equations system); and (ii) comparisons of observed and fitted diameter distributions derived from the fitting of the Weibull distribution (diameter-distribution recovery).

The CEDGY model produced acceptable overall results when compared to the average observed growth in the 22 plots (Table 3). There was relatively close agreement of model predictions and observations for ingrowth and survivor growth, but some discrepancies in mortality estimates. The negative bias observed in the prediction of the number of ingrowth trees affects the overall prediction only slightly, since its contribution to the sum of diameters and basal areas is small. The mortality equation under-predicted the observed mortality rate. This was probably linked to the difficulties encountered in estimating mortality from temporary plots.

To assess the reliability of the model predictions of stand attributes by diameter classes, values of mean diameter and mean squared diameter from the first data set (22 plots, 10 years prior) were used to recover the parameters of the Weibull distribution. These parameters were then used with the initial number of trees/ha to produce the fitted initial stand and stock table showing the number of trees, basal area and volume/ha in each 10-cm diameter class

TABLE 3

Comparisons between predicted (from CEDGY model) and observed average stand components of growth of 22 cedar validation plots over a I 0-year projection period

Parameters Predicted Observed Difference

Average stand attributes Number of trees ( N / h a ) 382.04 380.00 +2.04 Sufn of diameters ( m / h a ) 127.63 126.50 + 1.13 Stand basal area (mE/ha) 48.62 47.96 + 0.66 Ingrowth Number of trees ( N / h a ) 27.48 34.14 - 6.66 Sum of diameters ( m / h a ) 3.43 3.89 - 0 . 4 6 Stand basal area (mE/ha) 0.36 0.38 -- 0.02 Mortality Number of trees ( N / h a ) 20.67 28.68 + 8.0 l Sum of diameters ( m / h a ) 3.92 6.31 - 2.39 Stand basal area ( m E/ha) 1.12 1.97 - 0.85 Survivors Sum of diameters ( m / h a ) 10.93 11.74 -0 .81 Stand basal area (m2/ha) 6.78 6.96 - 0 . 1 8


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DIAMETER CLASS (M)

Fig. 1. Observed and fitted diameter distribution of the initial stand summary from the 22 cedar validation plots.

(Figs. 1 and 2). Comparison of observed and predicted diameter distributions at the end of the 10-year projection period indicates that the model produces acceptable forecasts of stand and stock tables. With the exception of small discrepancies in the small-diameter classes, there is a generally close agreement between observed and predicted diameter distributions.

The long-term projections of the initial stand diameter distribution of the 22 cedar validation plots are shown in Figs. 3 and 4. The initial distribution is typical for uneven-aged stands. The initial values of the prediction ar 41 m2/ha for basal area and 375 trees/ha. The trend of average stand attributes through time is reasonable. Increased basal area and volume growth is associated with a decreased number of trees/ha. The initial diameter distribution appears to tend towards a positively skewed distribution over time, with fewer trees in small-diameter classes (Fig. 4). This behavior is expected in the Ajdir Forest because of the natural regeneration difficulties for cedar.

A GROWTH-AND-YIELD MODEL FOR UNEVEN-AGED CEDRUS A TLANTICA 263

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DIAMETER CLASS (M)

Fig. 2. Predicted (from CEDGY model) and actual diameter distribution at the end of a 10- year projection period for the 22 cedar validation plots.

C O N C L U S I O N

The CEDGY model provides reasonable projections of growth and yield throughout the range of conditions likely to be encountered in the Ajdir For- est in Morocco. Ideally, data for developing a growth-and-yield model should come from numerous remeasured permanent plots covering the range of ecological conditions and silvicultural techniques in the region. Since such plots do not exist in this forest, temporary plots were used. Past diameters of stand- ing trees were estimated with good precision from increment cores. Mortality, however, was estimated by subjective judgement based on the experience of local foresters. These estimates are not reliable for precise long-term projections.

The fact that cedar trees are often present in mixture with oak trees may have an influence on the growth and yield of cedar stands. Past conditions of


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green-oak stocking could not be obtained in the plots used to construct the model. The effect of green oak on cedar growth was therefore ignored in this study. Remeasured plots are the only way to assess this effect.

The major methodological difficulties encountered in the development of the growth-and-yield model were linked to the lack of successive measurements from permanent plots. Such plots should be installed in all productive stands in order to have more accurate information on the growth dynamics in cedar stands.

The model developed in this study can be used for many purposes: for ex- ample ( 1 ) to produce stand and stock tables from raw data obtained during an inventory cruise; (2) to fit a theoretical distribution to the observed stand


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DIAMETER CLASS (M)

Fig. 4. Predicted long-term diameter distribution (by the CEDGY model) of the initial stand summary from the 22 cedar validation plots. Figures in parenthesis indicate years from the beginning of the projection.

summary; (3) to predict average plot and stand attributes at a future time; (4) to predict stand and stock tables of a plot or stand at a future time; and ( 5 ) to evaluate management planning alternatives under simulated time.

ACKNOWLEDGEMENTS

The development of this model was supported by the United States Agency for International Development, through the University of Minnesota-Institut Agronomique et Veterinaire Hassan II, Morocco, Project.

REFERENCES

Anonymous, 1978. l~tudes des travaux d'amenagement dans les cedraies de: Bekrit, Senoual, Ajdir, Kerrouchen, et Itzer. Direction des Eaux et Forets et de la Conservation des Sols, Rabat, Morocco.


Burk, T.E. and Burkhart, H.E., 1984. Diameter distributions and yields of natural stands of loblolly pine. Va. Polytech. Inst. State Univ. Publ. No. FWS- 1-84, 46 pp.

Clutter, J.L., 1963. Compatible growth and yield models for loblolly pine. For. Sci., 9" 354-37 I. Ek, A.R., 1974. Nonlinear models for stand table projection in Northern hardwood stands. Can.

J. For. Res., 4: 23-27. Emberger, L., 1955. Une classification biogeographique des climats. Rec. Trav. Lab. Fac. Sci.

Montpellier, Ser. Bot., Fasc., 7: 3-45. Frasier, J.R., 1981. Compatible whole-stand and diameter distribution models for loblolly pine

plantations. Ph.D. Dissertation, Virginia Polytechnic Institute and State University, Blacks- burg, Virginia (unpubl.).

Hamilton, D.A. Jr. and Edwards, B.M., 1976. Modeling the probability of individual tree mortality. USDA For. Serv. Res. Pap. INT-185, 22 pp.

Hyink, D.M. and Moser, J.W. Jr., 1983. A generalized framework for projecting forest yield and stand structure using diameter distributions. For. Sci., 29: 85-95.

Johnson, N.L. and Kotz, S., 1970. Continuous Univariate Distributions. I. Distributions in Sta- tistics. Wiley, New York, 300 pp.

Khatouri, M., 1988. Whole stand and diameter distribution growth and yield models for uneven-aged Cedrus atlantica stands in Morocco. Ph.D. Diss., University of Idaho, Moscow, 152 pp.

Knoebel, R.B., Burkhart, H.E. and Beck, D.E., 1986. A growth and yield model for thinned stands ofyeUow-poplar. For. Sci. Monogr., 27:62 pp.

Lepoutre, B. and Pujos, A., 1963. Facteurs climatiques d6terminant les conditions de g6rmina- tion et d'instalation des plantules de c6dre. Ann. Rech. For. Maroc, 7:21-54.

Lynch, T.B., 1982. Diameter distribution growth and yield models for mixed species forest stands. Ph.D. Diss., Purdue University, W. Lafayette, Ind., 216 pp.

Lynch, T.B. and Moser, J.W., Jr., 1986. A growth model for mixed species stands. For. Sci., 32" 697-706.

Matney, T.G. aad Sullivan, A.D., 1982. Compatible stand and stock tables for thinned and unthinned loblolly pine stands. For. Sci., 28:16 l - 171.

Messat, S., 1986. Matrix growth model for uneven-aged cedar stands of Sheb Forest in the Moyen Atlas of Morocco. Doctorat-es-Sciences Agronomiques, I.A.V. Hassan II, Rabat, Morocco, 132 pp.

M'hirit, O., 1982. Etude 6cologique et forestibre des c6draies du Rif marocain. Th~se de Doc- torat es Sciences Naturelles, Universit6 de Droit, d'Economie et des Sciences, d'Aix-Mar- seille, France, 436 pp.

Moser, J.W. Jr., 1967. Growth and yield models for uneven-aged stands. Unpublished Ph.D. Diss., Purdue University, W. Lafayette, Ind., 149 pp.

Moser, J.W. Jr. and Hall, O.F., 1969. Deriving growth of loblolly pine. For. Sci., 10:105 - 115. Munro, D.D., 1974. Forest growth models. A prognosis. In: J. Fries (Editor), Growth models

for tree and stand simulation. R. Coll. For., Stockholm, Res. Note 30. Negre, R., 1952. Les associations v6g~tales du J'bel Sa~ (Moyen Atlas d'Itzer). Bull. Soc. Sci.

Nat. Maroc, 32: 139-165; 33: 27-38; 41: 19-62. Pujos, A., 1964. Les milieux de la c6draie Marocaine. Ann. Rech. For. Maroc, 8:283 pp. Stage, A.R., 1973. Prognosis model for stand development. USDA For. Serv., Intermount. For.

Range Exp. Stn., Ogden, Utah, Res. Pap. INT-137, 32 pp. Strub, M.R. and Burkhart, H.E., 1975. A class interval free method for obtaining expected yields

from diameter distributions. For. Sci., 21: 67-69. Wykoff, W.R., Crookston, N.L. and Stage, A.R., 1982. User's guide to the stand Prognosis Model.

USDA For. Serv., Intermount. For. Range Exp. Stn., Ogden, Utah. Res. Gen. Tech. Rep. INT-133, 112 pp.

growth-and-yield model for uneven-aged cedrus atlantica stands in morocco

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