Modeling Forest Growth with Management Data: A Matrix ... · A matrix approach for the Italian...

Silva Fennica 31(1) research articles

Modeling Forest Growth withManagement Data: A Matrix Approachfor the Italian Alps

Paola Virgilietti and Joseph Buongiorno

Virgilietti, P. & Buongiorno, J. 1997. Modeling forest growth with management data:A matrix approach for the Italian Alps. Silva Fennica 31(1): 27-42.

This paper reports on the possibility and difficulties in building growth models from pastForest Administration records on cut and growth in the Italian Alps. As a case study, amatrix model was calibrated for uneven-aged forests in the Valsugana valley of theTrentino province. The model gave reliable predictions over 30 years, and plausiblelong-term forest dynamics, including steady-states that are similar to virgin forests. Theresults support the view that the current forests are deeply altered as to composition,relative to what would obtain from natural growth. They also support the concept of longcyclic changes in natural stands, gradually approaching a climax state. Shortcomings ofthe data are that they do not come from an experimental design, they are not alwaysaccurate, and they must be supplemented with other information, especially concerningmortality. Still, these cheap and available data can lead to workable models adapted tolocal conditions, with many management applications.

Keywords growth models, matrix models, forest dynamics, Italian AlpsAuthors' address Virgilietti Universita degli Studi di Padova, Dipartimento Territorio eSistemi Agroforestali, Padua, Italy. Buongiorno University of Wisconsin-Madison, Depart-ment of Forestry, 120 Russell Laboratories, 1630 Linden Drive, Madison WI53706-1598,USA Fax (Buongiorno) +1 608 262 9922 [email protected] 7 January 1997

1 Introduction Although forms of uneven-aged managementwere used in some areas of the Alps for centu-ries, foresters began to be really interested in

Uneven-aged forests constitute about 25 % of them after the Second World War, following thethe Italian forests (excluding coppices), cover- spread of the nature oriented silviculture. Theing more than 554 400 ha, and irregular forests view is that there is a better chance to guaranteecover another 391 900 ha, altogether 41 % of the the sustainability of cultivated forests, with lesshigh forests (ISAFA-MAF 1988). expenditures in energy and money, if they are

27


managed to get structures and functions similarto those of natural forests (Bernetti 1977, Sus-mel 1980, Ciancio and Nocentini 1994).

In temperate oceanic and suboceanic climates,where beech, fir and spruce find optimal livingconditions, the multispecific, uneven-aged for-mations represent the final stage (climax) of along evolution. In the subcontinental climates,for the forests of spruce, fir and larch, the finalstructures are uneven or irregular, while in conti-nental climates, even-aged forests dominate (Sus-mel 1980). Therefore, in the Alps, uneven-agedforests are, in most cases, the natural formationtoward which, according to the principle of na-ture oriented silviculture, the management of for-ests should tend.

One of the first regions where these ideas wereapplied in Italy was the Trentino Province, wherefrom the fifties, the methode du contröle1 was usedwidely. The method requires precise and frequentinventories, and accurate data of harvests in theinterval between two inventories (ISEA 1986).

The results of the application of this methodfor about 40 years in Trentino are altogetherpositive, with a growth in wood biomass, anincrease in forest biodiversity, a decrease in theneed for artificial regeneration and a reductionof pathologic attacks. The problems are the highmanagement costs, due particularly to the neces-sity of having frequent complete inventories(Wolynski 1993).

In this context it seems worthwhile to researchmethods of predicting future stand states and theeffects of management, with minimum data andexpenses. Growth models could serve for thatpurpose; and matrix models in particular, be-cause of their relative simplicity and the possi-bility to build them with inventory and harvestdata that are already available.

The purpose of this paper is to report on thepossibilities and difficulties in building such amatrix growth model with the data collected in thepast by the Forest Administration for managementpurposes. In the specific forests of Valsugana, asin all the Italian Alps, there are no permanentuneven-aged plots that can serve for this.

Gurnauds methode du controle, as applied in the Trentino requiresthat the harvest be decided stand by stand, according to forestconditions which must be closely monitored (Gurnaud 1886, Ber-netti 1977).

The first part reviews briefly model taxonomy,with particular attention to matrix models, and itdescribes the specific model structure chosenhere, followed by a comparison with the culturalmodel applied currently. The second part presentsthe calibration of the matrix model with data fromforests of the Valsugana valley in Trentino prov-ince, and describes the difficulties caused bysome data, and their solution. Part three reportsthe results of model validation for 30 years fore-casts, and the long term stand dynamics predictedby the model. The steady state implied by thematrix model is also computed, and compared tothe tree distribution in virgin forests, and thatassumed by the current cultural model.

2 Choosing a Model Structure

As used here, the term "growth model" refers toa set of mathematical equations, to predict thegrowth of a tree, stand, or forest. Growth modelsrequire empirical data to be realistic. After cali-bration for a specific area, they can be of greathelp in forest management (Vanclay 1995).

There are three general types of growth mod-els, with a number of intermediate forms. "Wholestand models" group all the trees in a singleclass, with little detail on its composition. Theymay be equations that predict future stand basalarea as a function of current stand basal area. Or,consist of equations to forecast the parameters ofa tree distribution function. They are useful withsimple stand structures, with few species, andthey are most suitable for even-aged stands.

Instead, in "stage class" models, trees are di-vided in several classes of species and size, andthe future state is predicted for each class. Thereare explicit equations for the growth of treesbetween size class and for regeneration. Theyare more flexible than whole stand models andfit even, uneven, mono, or multi specific forma-tions. They have enough detail for many appli-cations, remaining relatively simple and compu-tationally efficient.

Last, "single tree models" describe a forest bya list of trees, each one defined by several pa-rameters (species, diameter, height, location andso on). These models are very detailed, but they

28

Virgilietti and Buongiorno Modeling Forest Growth with Management Data:...

are costly to develop and present problems oftractability, given the numerous equations andtheir complexity.

There is an obvious continuum among modeltypes: increasing the number of classes in a standclass model to one class per tree gets a singletree model, and vice versa. A specific stage classmodel, with a matrix structure, was selected here,because it strikes a balance between the detailrequired to identify meaningful management al-ternatives and the tractability needed to developand apply the model (Vanclay 1995, Getz andHaight 1989).

2.1 Matrix Models

A general form of a linear matrix model is:

yt+i = G (yt - ht) + c (1)

where yt is a vector of the number of trees perunit area in a stand at time t, ht is the number oftrees harvested at time t, G is a matrix of coeffi-cients and c is a vector of constant parameters.These coefficient are derived from two sets ofdata: the fraction of trees that move from onediameter class to another and an equation thatrelates ingrowth to the stand state (Buongiornoand Gilless 1987, p. 101).

The first matrix models were used by Leslieand Lewis to investigate the effect of age struc-ture on the growth of animal populations (Lewis1942, Leslie 1945 and 1948). Usher (1966 and1969) adapted Leslie's model to forest stands. Inthese first models, recruitment was directly pro-portional to the number of trees after harvest. Tofind the maximum sustainable yield, a growingstock constraint was needed to bound the prob-lem. Buongiorno and Michie (1980) avoided thisby expressing ingrowth as a linear function ofthe number of trees and total basal area, relyingon Ek's (1974) observations. This linear matrixmodel structure can readily accommodate standsof many species (e.g. Lu and Buongiorno 1993).

Like ingrowth, transition probabilities may alsodepend on stand state, and a common criticismof linear matrix models is that they assume astationary upgrowth matrix (Harrison and Michie1985). To avoid this, Solomon et al. (1986), and

Mengel and Roise (1990) made the matrix G afunction of stand state. Buongiorno et al. (1995)verified, for uneven-aged forests in the FrenchJura, the existence of an inverse relationship be-tween transition probabilities and basal area. Withthese relations, the model is no longer linear,although it is linear for a given basal area. Nonlinearities complicate considerably the study ofmanagement alternatives.

Growth models have many uses. A direct appli-cation, and a check of the realism of the model, isto predict stand dynamics and find the steadystate in natural conditions, that is the climax, if itexists. For linear matrix models, the steady statecan be computed directly from the equations ofthe model, and it is independent of initial standconditions (Buongiorno and Michie 1980).

Moreover, one can study the consequences ofdifferent cutting regimes, and thus find thosethat best meet the management objectives. Afundamental goal in forest management is toproduce a constant periodic harvest without de-pleting the growing stock (sustained yield condi-tions). But other objectives are generally present.Though early applications dealt mostly with eco-nomic or productivity objectives, more recentlythe models have also been applied to optimizeecological objectives such as diversity (Gove etal. 1994, Buongiorno et al. 1994, 1995). Thistrend reflects an increasing environmental con-cern in forest management, which is bound tocontinue in the future.

2.2 Matrix Model Structure

The growth model adopted for the Valsuganaforests is analog to that of Buongiorno et al.(1995) for the similar forest in the French Jura.But, the final model is linear. As explained be-low, though non linearities were tested, theireffects were found to be too small to matter inpractical applications. The model structure is asin Equation (1), where the stand state at time t isrepresented by the vector:

where yijt is the number of trees per hectare thatare alive of species / (=1, 2....m) and sizej (=1,

29


2...n) at time t. During a growth interval, t to M-l(here a decade), the trees in a diameter class imay stay there, advance to the next class, die orbe harvested. The harvest in a decade is repre-sented by the vector:

ht = [/*//,]

where hyt is the number of live trees cut perhectare.

The matrix G in (1) is the sum of an upgrowthmatrix A and ingrowth matrix R. The upgrowthmatrix consists of a diagonal of matrices, Aj, onefor each species:

cman

a/3

bin a

where ay is the probability that a live tree ofspecies / and in size class j at time t which isnot cut from t to t+l will be alive and in thesame size class at time t+l; b(j is the probabilitythat a live tree in size class j-1 at t which is notcut will be alive and in size j at t+l. So theprobability of mortality in species / and sizeclass j , is my•= 1— fly— bij+j.

The ingrowth matrix, R, consists of submatri-ces Rjk describing the effects of trees of species kon the ingrowth of trees of species i. Here, thehypothesis is that ingrowth of one species isrelated inversely to stand basal area, and directlyto the number of trees of the same species. Spe-cifically:

Rik =

dikB\+ek dikBi+ek ... dikBn+ek

0 0 ... 0if i = k

if i * k

where Bj is the basal area of a tree of size j andthus dikBj shows by how much ingrowth of spe-cies k decreases for each additional tree of spe-cies / and size j . And, ek shows by how much thepresence of an additional tree of species k in-creases the ingrowth of the same species.

2.3 Susmels Cultural Model

In the Trentino, harvest policies are establishedcurrently by referring to the cultural model de-veloped by Susmel (1980). This model is meantto describe diameter distributions and other for-est parameters that are deemed desirable andsustainable, (but not the path followed by a standto reach that state). The model assumes that theheight of the dominant trees is a good index ofsite fertility. Then, all stand parameters are ex-pressed as a function of the dominant height.

For the mixed formations of fir, spruce andbeech (with beech less than 25 %), the equationsare:

qNBVLJmnY

= 4.3/H1/3

= 300= 0.97 H= 0.33 H2

= 2.64 H

where q is the ratio between the number of treesin a diameter class and the number of trees in thenext, assumed to be constant. N is the number oftrees per hectare above 17.5 cm of diameter,independent of fertility and constant. B is thebasal area, V the volume of the whole stand, Dmax

the maximum diameter and H is the potentialheight. The model gives no information on theproportion of the different tree species.

Application of Susmel's model begins byjudging the potential height of the forest, themaximum dominant height compatible with theecological characteristics of the site. If it is equalto, or 1 to 2 meters higher than the current ob-served dominant height, the cultural model isused as is for management. If it is more than2 meters higher than the observed height, inter-mediate models, based on the average of ob-served and potential heights, are adopted. OnceH is determined, the normal distribution is cal-

30


Table 1. Summary of stand statistics.

Trees/ha MeanSD

Basal area (m2/ha) MeanSD

Volume (m-Vha) MeanSD

Growth(m3/ha/yr) MeanSDN

Spruce

147.458.120.47.1

209.092.08

4.73.984

Fir

78.369.68.57.5

105.493.31.52.177

Larch

24.835.42.63.2

32.138.50.60.747

Beech

19.919.71.01.07.98.5

0.160.4349

Note: SD = standard deviationN = number of observations in 30 compartments measuredat various times from 1960 to 1990.

culated2. Then, comparing the current distribu-tion with the normal, the harvest for each diame-ter class is decided; in any case this amountshould not be more than 25 % of the volume ofthe stand. The fundamental difference betweenSusmel's cultural model and the model soughthere is that the former is normative. It defines thestructure that the manager should work for: ittells what the forest ought to be. The latter in-stead is positive. It is meant to predict how astand will grow, given its current structure, andwhat will result from different managements: ittells what the forest will be, not what it should be.

3 Estimation of the MatrixGrowth Model

3.1 The Data

The matrix model was calibrated with data from30 compartments located in three forests of thecentral Alps, in the Valsugana valley of the Tren-tino Province, Italy. This region has a very long

2 The normal distribution is determined by first choosing q andDmax. Then, a unit distribution is computed, with one tree in thelargest diameter class and the number of trees in each successivesmaller class increasing at the geometric rate q. The normal distri-bution is obtained by multiplying the number of trees in eachdiameter class of the unit distribution by BIBU, where B is thenormal basal area (obtained from the dominant height H), and Bu isthe basal area of the unit distribution.

forestry tradition, and harvest and stand datahave been collected regularly for managementpurposes, though not for research, for more thanthirty years. The specific compartments were infour communes: Telve (5 compartments), Telvedi Sopra (4 compartments), Castel Tesino (9 com-partments), and Cinte Tesino (12 compartments),all near the town of Borgo Valsugana. The com-partments were selected with the help of ForestAdministration technicians and were meant tobe representative of the forests in the four com-munes. In total, they covered 541.95 ha, or 8.5 %of the total forest area in the four communes.They averaged 18 hectares, were located at alti-tudes from 950 to 1700 m (average 1320 m), andmost had steep slopes. They all had an uneven-aged or irregular and multi stratified structure3

on more than half of their area. During the past30 years the compartments were managed withselective cuttings.

The stands consisted mostly of spruce (Piceaabies L.), fir (Abies alba Miller), larch (Larixeuropea L.) and beech (Fagus sylvatica L.). Thedata, summarized in Table 1, came from period-ic inventories and records of harvested and sal-vaged trees. In all the compartments the firstinventory was done between 1960 and 1962 andthe last between 1990 and 1992. Twelve com-partments had four inventories and eighteen hadthree, leading to 72 pairs of successive observa-tions, or sequences. Three sequences weredropped due to changes in the compartmentboundary, leaving 69 sequences.

According to the management rules, dead treesshould be recorded and salvaged. However, therecords for the salvaged trees clearly did notreflect all the mortality. For example, in the old-est records there was often no salvage for a dec-ade. One reason could be that the steep slope ofmany compartments made salvaging operationsdifficult and costly. For beech, salvaged treeswere very seldom recorded and inventories andharvest records were also less accurate, probablybecause beech had the least commercial value.

3 In the management plans a forest is classified as uneven-aged onlywhen trees are distributed according to an inverse J curve. Irregularand multistratified forests present some irregularities in the dia-meter distribution, but are still considered uneven-aged forests,since there are trees of all ages in a small area.

31


The parameters of the model were calibratedfor a time interval (t to M-l) of 10 years, the usualcutting cycle in these forests. Ingrowth was de-fined as the number of trees that passed the 17.5cm diameter threshold and survived in 10 years,because smaller trees were not recorded. To min-imize the probability that a tree would growmore than one class in one interval, diameterclasses of 10 cm were used. In fact, the oldestrecords recognized three broad classes only, andtrees had to be distributed by 10 cm classes inproportion with earlier observations. Then, foreach species and sequence ingrowth and transi-tion probabilities were calculated, and standard-ized for 10 years by linear interpolation. Thesequences for which either the ingrowth or thetransition probabilities were negative weredropped. This occurred the most for larch andbeech, the species with fewer trees. In all, 81 %of the sequences gave usable data for spruce, 74% for fir, 39 % for larch, and 44 % for beech.

3.2 Ingrowth Equations

The ingrowth equations predict the number oftrees of each species that enter the smallest sizeclass and that are still alive after 10 years. Theparameters, estimated by ordinary least squares,are in Table 2. The underlying hypothesis wasthat ingrowth of a species would be negativelyaffected by stand basal area and positively by thenumber of trees of that species (Buongiorno etal. 1995). The estimated coefficients had the ex-pected signs. For all the species, the positiveeffect of the number of trees of the same specieswas significant at least at the 5 % significancelevel. Instead, the negative effect of basal areawas not significant for fir and beech, and signifi-cant at 10 % level only for larch. One possibleexplanation is that beech and fir are shade toler-ant species, so that their ingrowth was not influ-enced by stand density, at least in the range ofbasal areas within the data.

The coefficient of determination, R2, was be-tween 0.31 and 0.44 for spruce, fir and larch,while it was negligible for beech (0.09). There-fore, for beech, little would be lost if the in-growth were assumed constant, independent ofstand density and number of trees. Experiments

Table 2. Equations of ingrowth, by species.

SpruceCoefficientSEtR2

N

FirCoefficientSEtR2

N

LarchCoefficientSEtR2

N

BeechCoefficientSEtR2

N

Independent variableSpecies(tree/ha)

0.150.0473.183**0.305

54

0.12260.02

6.033**0.414

50

0.0760.0184.145**0.439

26

0.31590.1483

2.13*0.094

28

Basal area(m2/ha)

-1.470.395

-3.727**

-0.3970.271

-1.463

-0.220.122

-1.794

-0.3180.476

-0.668

Constant

35.2711.2733.128**

10.7476.1721.741

6.0692.8612.121*

16.13311.431.411

Note: SE = standard error.t = Students statistic, * and ** = significant at the 5 % and1 % level.R2 = coefficient of determination.N = number of observations.

with different data sets showed that the regres-sion for this species was not robust. Therefore,the final model did assume that ingrowth of beechwas constant and equal to the average recordedingrowth per decade. There are reasons for beechto differ from the other species, since it is presentmostly as coppice.

3.3 Upgrowth Equations

The upgrowth hypothesis was that the probabili-ty of transition (trees moving from one size tothe next) was affected by stand density and treesize. The regression results in Table 3 did find anegative effect of basal area on the upgrowthrate of spruce and fir, and a significant negative

32

Virgilietti and Buongiomo Modeling Forest Growth with Management Data:...

Table 3. Equations of the probability of transition in 10year.

SpruceCoefficientSEtR2

N

FirCoefficientSEtR2

N

LarchCoefficientSEtR2

N

BeechCoefficientSEtR2

N 66

Independent variableBasal area

(m2/ha)

-0.00330.0014-2.342*

0.018269

-0.00750.0023-3.252**

0.036229

0.00130.0035

0.370.134

126

-0.0050.007-0.690.044

Diameter(cm)

0.00070.0005

1.324

0.000040.00076

-0.05

-0.00540.00117-4.615*

0.00560.0034

1.626

Constant

0.4110.03910.48**

0.5050.063

8.01**

0.5140.0935.489*

0.2460.1951.261

Note: SE = standard error.t = Students statistic, * and ** = significant at the 5 % and1 % level.R2 = coefficient of determination.N = number of observations.

relation between transition probabilities and thesize of larches, suggesting a slower growth ratefor bigger trees. However, the very low R2 dem-onstrate that stand basal area and tree size ex-plain very little of the variation in the rate ofupgrowth. So the transition probabilities couldbe assumed constant without much loss in pre-dictive accuracy. In the final model, the transi-tion probabilities were set at their mean forspruce, fir and beech, and at their mean for agiven diameter for larch.

3.4 Mortality

The average recorded salvage rate was 0.4 % peryear for spruce and fir, and 0.2 % per year forlarch. The data for beech did not allow to calcu-late mortality, because salvaged trees were re-corded only twice. The recorded salvage rateswere considerably less than the true mortalityfor this type of forest, estimated to be between0.8 % and 1.2 % per year (Volin and Buongiomo1996). Therefore, an overall mortality rate of1 % per year was assumed here, for spruce, firand larch. In the Alps windfalls are frequent.Even though the recorded data of mortality couldnot serve to establish an accurate equation, theavailable records showed that mortality was morefrequent in the largest diameter classes. More-over, for similar forests in the French Jura, wheresnow breaks and windfalls are also an importantcause of mortality, there is evidence of a positivecorrelation between mortality and tree size (Buon-giomo et al. 1995). Thus, in the final model,mortality was assumed to increase with tree di-ameter at the same rate as in the forests of theFrench Jura. The Jura study did not find anyeffect of basal area on mortality, and none wasassumed here. For larch, absent in the Jura study,the equation for spruce was used, as it wasdeemed to be the closest species, ecologically.

Finally in a coppice mortality is higher, due tothe intense competition in sprout clumps (Hawleyand Smith 1954, p. 194). Thus, an average mortal-ity rate of 2.8 % per year was assumed for beech,varying with size at the rate shown in the equa-tions of Buongiomo et al. (1995). The completefinal model parameters are shown in Table 4.

4 Model Validation and Appli-cation

4.1 Short-term Validation

A formal validation of the model would requireprediction of the growth of an entirely differentset of compartments. However, this was not pos-sible here. All the available data were needed todevelop the model. So, as in much practical mod-eling, there is not a sharp distinction between

33


model construction and testing.Still, it is worth examining how the final mod-

el, a composite of several local regressions (foringrowth and upgrowth) and extraneous estimates(for mortality) predicts the growth of stands inthe forests where the model is meant to be ap-plied. To that end, the parameters in Table 4were used to predict the evolution of the com-partments with a full series of observations overthirty years. Thus, the simulated compartmentswere a subset of the compartments used to cali-brate the model, but still representative of theforests of interest. Equation (1) was applied re-cursively three times to predict the stand state atthe last inventory (1990/92), given the state atthe first inventory (1960/62), and given the in-termediate harvests.

Fig. 1 shows the average stand state predictedby the model after thirty years, and the distribu-tion of the observed stands. In general the pre-dicted average number of trees of each speciesand diameter was within the 95 % confidenceinterval of the average observed number of trees.Only for spruce was the predicted number oftrees significantly smaller than the observed, inthe smallest size class. Thus, the model seemsable to make acceptable thirty-year forecasts offorest growth, but it is more accurate in totalthan for each separate species.

4.2 Long-term Stand Dynamics

The model was also used to predict how a mini-mally disturbed forest would evolve, over a verylong time period. The parameters in Table 4have been computed with data from managedforests observed over thirty years only. Moreo-ver, since these forests have always been man-aged in the past, they are far from natural forestsin structure and composition. Thus, the mostappropriate application of the model should beto analyze only slightly different managementregimes over a relatively short time.

Nevertheless, it is useful to see if the implica-tions of the model under the extreme conditionsof no harvest for a very long time correspond, atleast in a general way, to existing knowledge offorest succession. Checking the model behaviorin such extreme conditions is a good test of

model robustness, since stands characteristicssuch as basal area, number of trees and speciescomposition should remain within limits con-sistent with prior biological knowledge. Thus,equation (1) was applied recursively with noharvest (ht = 0 for all t), for as long as 500 years.

The initial stand state for the simulation wasthe average of the stand states observed between1960 and 1980 (Year = 0 in Fig. 2). The modelpredicted that if the harvests stopped, the numberof trees in the intermediate classes would in-crease, over the years, and the number of largesttrees would rise as well. The increased densitywould cause a decrease in regeneration, and there-fore in the number of trees in the smallest dia-meter classes. So, between 50 and 100 years theforest would assume a structure similar to that ofan even-aged forest, with a high basal area.Around the year 200 the stand would consistmostly of a few very large spruces and firs andmany small beech. After that, a sufficient numberof trees would die to create openings large enoughfor new regeneration to take place, and in about300 years the forest would assume again a typicaluneven-aged distribution, but more of the largesttrees than in the current forest. This kind of dis-tribution is similar to that of virgin mixed forestsof fir, beech and spruce (Susmel 1980, p. 68).

Basal area would peak around year 75, whenthe medium and large size trees dominate, atabout 45 m2/ha, and it would bottom around year150, at some 20 m2/ha (Fig. 3). Basal area wouldstabilize in the long run at about 25 m2/ha. Theseranges of basal areas are possible for these kindof forests, even though basal areas of 45 m2/haare infrequent. Susmel (1951) found mixed for-ests of spruce and fir in another area of the Alpswith basal areas ranging from 10 to 50 m2/ha. Inthe compartments used to calibrate the model,the lowest basal area was 8.5 m2/ha and thehighest was 37 m2/ha.

According to the simulations, the number oftrees would increase during the first two decades,to a maximum of 290 trees/ha, then decreasegradually until the year 200 to a minimum of 125trees/ha, increasing then again (Fig. 3). In thelong run, the number of trees would convergetowards about 170 trees/ha. The number of treespredicted is lower than the numbers usually re-corded for these forests, though it is possible. In

34


Table 4. Matrix G and vector c for a time interval of 10 years

Speciesdiam.

Spruce 22.532.542.552.562.5

>67.5Fir 22.5

32.542.552.562.5

>67.5Larch 22.5

32.542.552.562.5

>67.5Beech 22.5

32.542.552.562.5

>67.5

Speciesdiam.

Spruce 22.532.542.552.562.5

>67.5Fir 22.5

32.542.552.562.5

>67.5Larch 22.5

32.542.552.562.5

>67.5Beech 22.5

32.542.552.562.5

>67.5

22.5

0.680.35

0000

-0.0200000

-0.0100000000000

22.5

-0.0600000

-0.0200000

0.610.40000000000

32.5

0.030.570.35

000

-0.0300000

-0.0200000000000

32.5

-0.1200000

-0.0300000

0.060.550.37

000000000

Spruce42.5

-0.060

0.560.35

00

-0.0600000

-0.0300000000000

52.5

-0.1700

0.540.35

0-0.09

00000

-0.0500000000000

Larch42.5

-0.2100000

-0.0600000

0.040

0.580.33

00000000

52.5

-0.3200000

-0.0900000

0.0300

0.610.28

0000000

62.5

-0.30000

0.530.35

-0.1200000

-0.0700000000000

62.5

-0.4500000

-0.1200000

0.01000

0.620.26

000000

>67.5

-0.500000

0.86-0.18

00000

-0.1000000000000

>67.5

-0.6500000

-0.1800000

-0.020000

0.86000000

22.5

-0.0600000

0.740.31

0000

-0.0100000000000

22.5

-0.0600000

-0.0200000

-0.0100000

0.570.35

0000

32.5

-0.1200000

0.090.620.31

000

-0.0200000000000

32.5

-0.1200000

-0.0300000

-0.02000000

0.490.35

000

Fir42.5

-0.2100000

0.070

0.60.31

00

-0.0300000000000

52.5

-0.3200000

0.0400

0.580.31

0-0.05

00000000000

Beech42.5

-0.2100000

-0.0600000

-0.030000000

0.410.35

00

52.5

-0.3200000

-0.0900000

-0.0500000000

0.330.35

0

62.5

-0.4500000

0.00000

0.560.31

-0.0700000000000

62.5

-0.4500000

-0.1200000

-0.07000000000

0.260.35

>67.5

-0.6500000

-0.050000

0.85-0.10

00000000000

>67.5

-0.6500000

-0.1800000

-0.100000000000

0.51

c

35.2700000

10.7500000

6.0700000

12.8300000

35


Spruce F i r

diameter (cm) diameter (cm)

Larch Beech80

60

40

20

0

diameter (cm) diameter (cm)

200

160

120

Total

diameter (cm)

Fig. 1. Predicted and observed diameter distribu-tion of trees after 30 years. P = predicted, Hand L = high and low bounds of 0.95 confi-dence interval of observed mean.

the compartments that gave the data, the numberof trees ranged between 115 trees/ha and 422trees/ha. But, the simulations describe an unman-aged situation, where there are more trees in thelargest diameter classes than in typical managedforests. As a result basal area remains alwaysquite high, which tends to limit regeneration, andthus the number of trees in the stand.

This general pattern, of long, damped oscilla-tions, both in basal area and number of trees, issimilar to results of previous studies, for differ-ent forest types (Borman and Likens 1979, Buon-giorno and Michie 1980), and for similar forestsin other regions (Buongiorno et al. 1995).

Other data provided by this simulation con-cern the dynamics of the species composition in

36


120

100

Year = 0

n V r .

120

100

80

60

40

20

22.5 32.5 42.5 52.5 62.5 >67.5

diameter (cm)

Year = 50

22.5 32.5 42.5 52.5 62.5 >67.5

diameter (cm)

120

100

80

60

40

20

Year = 100

120

100

eo 80

260

20

Year = 200

422.5

120

100

«80c260o

32.5 42.5diameter

Year

52.5 62.5 >67.5(cm)

= 300

20 llfTflTfTff',lOlrniixrn

o —22.5 32.5 42.5 52.5 62.5 >67.5

diameter (cm)

iibeechK! larch11 f i rn spruce

[Mil22.5 32.5 42.5 52.5 62.5 >67.5

diameter (cm)Fig. 2. Predicted long-term growth of an undis-

turbed stand.

a naturally growing stand (Fig. 3). Spruce which, 500 years. Fir maintained its share of basal area,at present, dominates the stands, both as number and even increased it to near 50 % by the yearof trees and basal area, had large oscillations, 500. Larch always stayed a secondary species,with a decreasing presence over 500 years. The never more than 10 % of the trees. Beech in-species never reached the current percentage in creased very quickly in the first one hundred

37


beechlarch

111111 f i r

sprucenumber of trees

300

250

200

150 %

100

50

o o o o o o o o o o o o o o o o oD C M n O O ^ I O C O C D O C J O O O

yearFig. 3. Predicted basal area and total number of trees in an undisturbed stand.

years, then it remained more or less constant ataround 30 % of the basal area.

These results should not be viewed as an exactprediction of future stand composition. At best,given the limitations of the model pointed outabove, they describe a general tendency. As such,they still have management implications. In thepast, silviculture favored spruce over fir, and con-ifers over broadleaves because of their relativeeconomic value. This caused spruce forests tosupplant, in many cases, the original mixed for-ests dominated by fir (Susmel 1980, p. 245-251).So, while in the high Alps, spruce and larch forestsrepresent, at least in composition, the natural for-est, at lower altitudes, between 700 and 1200meters, the current forests are very often deeplyaltered both in structure and in composition.

The simulation indicates how, in an unman-aged situation, the presence of fir and beechwould increase, suggesting that the current com-position is not a climax for this site and climate,but mainly the result of human action. The evo-lution of beech is particularly difficult to project,since it is now present as coppice, but over along period regeneration by seeds should startand substitute the coppice. Therefore, an increasein this species is probable, as predicted by themodel.

The simulation shows that competition betweenspruce and fir is very complex, and that the currentcomposition is far from what would result fromnatural growth. The oscillation in the relativepresence of fir and spruce are typical in the dy-namics of mixed fir and spruce forests (Del Fave-ro and Lasen 1993, p. 215). Previous ecologicalstudies mention that, for alpine forests, it is verydifficult, if not impossible, to state what the natu-ral composition was, even when it is clear that thecurrent composition is far from the original (Sus-mel 1980, p. 247). The simulation illustrates thefact that the natural state itself is hard to define,since it changes constantly. Indeed, in the fivehundred year simulation, the stand never reachesa steady state. Still, it is clear that it approachessuch a climax state in which stand growth justreplaces mortality, a state which can be computeddirectly with the model.

4.3 Steady State

Since the growth model is linear, its steady state,if it exists, can be computed exactly. At equilib-rium, regardless of the value of t, one must have:

yt+i = yt = y*

38


rsssss* larchfir

'spruceSusmel's modelvirgin forest

22.5 32.5 42.5 52.5

diameter(cm)

62.5 >67.5

Fig. 4. Steady state distribution predicted by the matrixmodel, distribution of virgin forest, and Susmelscultural model distribution.

where y* is the equilibrium distribution. Thiscondition and ht = 0 (no harvest), substituted inequation (1) leads to:

y* = (I - G)-1 c (2)

where I is the identity matrix of order n. Theequilibrium distribution depends only on thegrowth potential of the stand as defined by Gand c, and it is independent of the initial standcondition (Buongiorno and Michie 1980). Withthe parameters in Table 4, equation (2) has aunique solution, graphed in Fig. 4.

At equilibrium the model predicts that therewould be 174 trees/ha and a basal area of 27.3m2/ha. The trees would be distributed accordingto an inverse J shaped curve, with the number oftrees decreasing with the larger diameter classes,except in the last one, where the number of treesis relatively large. The steady-state stand wouldbe composed mostly of spruce and beech, whilefir and larch would be secondary.

This state is mostly of theoretical interest. Itsupports the plausibility of the model, even atthe limit. But, as noted above, even in the simu-lation of 500 years the forest never got to thissteady state. Over such a long time it is unrealis-tic to assume that the forest growth parameters

would remain the same. Furthermore, catastroph-ic disturbances alone would insure that a steadystate would never be reached by any one stand,though it could be observed as an average over alarge forest area.

The fact that in the steady state spruce woulddominate while in the simulations fir is alwaysan important component of the forest is notableand not explained easily. Mixed forests wherespruces and beeches dominate do exist, but theyare not considered climax, rather transition for-ests. They are often the result of past humanactions (Del Favero and Lasen 1993, p. 203).

Ignoring species composition, the distributionof all trees by size is similar to the one observedin natural virgin forests that are presumably nearthe climax state (Fig. 4, Susmel 1980, p. 68),even though the model predicts fewer trees. Themodel supports the hypothesis that, for theseforests, an uneven-aged structure is what naturalgrowth would tend towards, so that it should bestable in the long run. The fewer trees can beexplained by the fact that in the studied forestsfertility is lower than in the virgin forests of Fig.4. Their average current dominant height isaround 30 meters, while the virgin forests towhich they are compared have a dominant heightof 41 meters.

4.4 Comparison with Susmels CulturalModel

The average potential height computed by theForest Administration for the 30 compartments ofthis study is 33.4 m. Therefore, the stand diameterdistribution proposed by Susmels cultural modelwould have the following parameters:

qNBV

= 1.334= 300= 32.43 m2/ha= 368.79 nvVha

Dn

But, since the current average dominant height is30 m, an intermediate model would be used formanagement, based on a height of 31.7 m. Fig. 4shows the corresponding tree distribution, andhow it compares with the steady state predicted

39


by the matrix growth model. The cultural modelhas more trees in all size classes, except thelargest. The implication is that the steady stateassumed by the cultural model may not be reach-able given the growth potential of the forestsunder study. The potential is embedded in theparameters of the matrix model, calibrated withdata of the forests under consideration.

Still, this does not mean that the cultural mod-el, as a goal, is without interest, even if it couldnot be reached. Indeed, other research on forestsof the Dolomites suggests that a policy that triesto approach the cultural model (which could notbe reached there, either) was very attractive fromthe point of view of both ecological and eco-nomic criteria (Volin and Buongiorno 1996). Butsuch an inference would not be possible withouta growth model to predict the full dynamic con-sequences of policies based on the cultural mod-el, or other silvicultural principles.

5 Summary and Conclusion

The matrix model applied in this paper appearedsuitable for the Trentino situation. It can be cali-brated with the data generally collected by theForest Administration for normal management,and it offers the possibility of predicting forestgrowth with a degree of detail sufficient for man-agement purposes.

A specific matrix model was built with the man-agement data collected on 30 stands during the past30 years from three uneven-aged forests located inthe Valsugana valley of the Trentino. With thesedata, regression equations could be developed topredict ingrowth and upgrowth, but mortality rateshad to be estimated from other forests.

The model describes average stand growth overthe entire forests from which the data were orig-inated. Thus, it cannot predict accurately thegrowth of each individual stand. Predictions thatdeviate considerably from the current stand struc-tures should also be viewed cautiously, sincethey are the results of extrapolations beyond therange of observations. For the same reason, themodel should not be applied to other forestswithout ascertaining that sites and ecological con-ditions are similar.

Within these limits, the model seemed able tomake reliable predictions in the short term, asdemonstrated by the thirty year validation. Forforest management this may be enough, sinceharvesting plans are usually made for horizonsof 10 to 15 years.

It is also useful to know the consequences ofdecisions in the long run and the model appearsto behave plausibly in that respect, even thoughforecasts cannot be expected to be as accurate asin the short term. The model was applied here tosimulate forest evolution without harvest. Thisis an extreme application, since the data comefrom forests managed intensively, and probablyaltered as to structure and composition. Still, theresults of these simulations do not contradictpresent knowledge about forest dynamics.

One of the most difficult variable to predictwas the species composition of the stand. Thecomposition predicted by the model in the steadystate was different from that observed in the 500year simulation, when the steady-state composi-tion was never reached. Nevertheless, the modelpredictions support the view that the current for-ests are deeply altered as to composition, rela-tive to what would obtain from natural growth.The model also supports the concept of longcyclic changes in natural stands, gradually ap-proaching a climax state.

Since the model seems able to make accurateshort term forecasts and it gives reasonable re-sults in long term simulations, it can be used tostudy the effects of management guidelines. Al-though the model simplifies a reality that is verycomplex and cannot pretend to represent the fullrichness of forest succession, it has sufficientdetail to help understand the dynamics of naturaland managed forests. Recognition of four spe-cies of trees should help also in economic as-sessments, an aspect of forestry that remainscritical for Alpine communities (Merlo 1995).

In sum, it is possible to build practical matrixmodels with the management records availablefor the uneven-aged forests in the Italian Alps.This is a cheap source of data, available now,that can lead quickly to a workable model, adapt-ed to a specific forest. However, these data haveclear shortcomings: they do not come from arigorous experiment design and they are not al-ways accurate. In particular, the salvage data do

40


not seem to give good estimates of mortality.Formal biometric research, including the estab-lishment of permanent plots is needed to acquirebetter knowledge. But this will take time andmoney. In the meantime, much can be done withthe available records to develop first generationmodels that, though imperfect, can help greatlyin current management decisions.

Acknowledgments

This research was supported in part by Fondazi-one Aido Gini scholarship, by Me Intire-StennisGrant M 2855 and by the School of NaturalResources, University of Wisconsin. We thankValeria C. Volin, Maurizio Merlo and Luca Ce-saro for their collaboration, and Servizi Forestaliof Trentino Province that helped to collect thedata.

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