Amaro A., Reed D., Soares P. (editors) Modelling Forest Systems

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24Amaro Forests - Chap 21 1/8/03 11:53 am Page 237

21 Using Process-dependent Groups

of Species to Model the Dynamics

of a Tropical Rainforest

Nicolas Picard,

Sylvie Gourlet-Fleury

and Plinio Sist

Abstract

The high tree species diversity in tropical forests is difﬁcult to take into account in models. The

usual solution consists of deﬁning groups of species and then adjusting a set of parameters for

each group. In this study, we address this issue by allowing a species to move from one species

group to another, depending on the biological process that is concerned. We developed this

approach with a matrix model of forest dynamics, for a tropical rainforest in French Guiana, at

Paracou, focusing on the methodological aspects. The forest dynamics is split into three com-

ponents: recruitment, growth and mortality. We then built ﬁve recruitment groups, ﬁve growth

groups and ﬁve mortality groups. One species is characterized by a combination of the three

groups, thus yielding in total 5  5  5 = 125 possibilities, out of which 43 are actually

observed. The resulting matrix model provides a better view of the ﬂoristic composition of the

forest, and does not have more parameters than it would have with ﬁve global species groups.

However, its predictions are no more precise than those of the matrix model based on ﬁve

global groups.

Introduction

When trying to model the dynamics of a tropical rainforest, one is confronted with

the huge diversity of tree species. Even though some authors treat every single

species separately (e.g. Shugart et al., 1980), the usual solution consists of building ad

hoc groups (generally by cluster analysis) from species characteristics that are in use

in the model, and then adjusting a set of parameters for each group (Favrichon,

1998; Köhler and Huth, 1998; Finegan et al., 1999; Köhler et al., 2000, 2001; Huth and

Ditzer, 2001). Groups may be built from ecological characteristics of the species, thus

providing so-called ‘functional groups’, but they may also take into consideration

extraneous information such as commercial categories (Wan Razali, 1986; Vanclay,

1989; Boscolo and Vincent, 1998). Still, deﬁning groups in relation to a model of for-

est dynamics remains difﬁcult, as two species may appear similar with respect to a

biological function, and at the same time different with respect to another function.

Cirad-forêt, Montpellier, France

Correspondence to: nicolas.picard@cirad.fr

24Amaro Forests - Chap 21 1/8/03 11:53 am Page 238

238 N. Picard et al.

For instance, species groups that are homogeneous with respect to growth may be

built, but it is likely that these groups will be heterogeneous with respect to recruit-

ment or mortality.

In this chapter, we address this issue by deﬁning three distinct species group-

ings: one is gr

owth-speciﬁc, that splits the species along their growth characteristics;

the second is recruitment-speciﬁc, that gathers species with similar recruitment

characteristics; and the third is mortality-speciﬁc. The key point is that the model of

forest dynamics allows a species to shift from one species group to another depend-

ing on the biological process (growth, recruitment or mortality) that is concerned. A

species may thus take its recruitment parameters from one species group, its growth

parameters from another and its mortality parameters from a third. The resulting

model thus has as many parameters as in the classical approach, but their combina-

tions give a much greater richness of modelled species.

We developed this approach with a matrix model of forest dynamics, for a trop-

ical rainfor

est in French Guiana, at Paracou. We have deliberately focused on the

methodological aspects in this chapter, leaving the study of the ecological relevance

for future work. We analysed the Paracou data to build ﬁve growth-speciﬁc groups,

ﬁve recruitment-speciﬁc groups, ﬁve mortality-speciﬁc groups and ﬁve comprehen-

sive species groups that simultaneously rely on growth, recruitment and mortality.

Two matrix models were then developed: one is classical and is based on the ﬁve

comprehensive species groups; the other one illustrates the proposed new approach

and makes use of the three process-speciﬁc groupings into ﬁve groups. The predic-

tions of the two matrix models are then compared.

Materials and Methods

The Paracou plots

The matrix models were built from the data of the Paracou forest, in French Guiana,

40 km west of Kour

ou (5° 15 N, 52° 55 W). The climate is equatorial with an aver-

age annual rainfall of 3160 mm and an average temperature of 26°C. On this site

twelve 6.25-ha plots of natural rainforests were settled in 1984 by the Cirad-Forêt.

Nine plots underwent silvicultural treatments in 1987/88, and three plots were left

as controls. Each plot is divided into four subplots, which yields 48 subplots in total.

The girth of every tree greater than 10 cm diameter at breast height (DBH) was mea-

sured, and taxonomic information was also noted. As the botanical inventory did

not allow every tree to be identiﬁed at the species level, a tree was characterized

either by its species or by a group of species, which we shall refer as a pseudo-

species. Measurements have been made annually from 1984 to 1995, and every 2

years since. In total, more than 46,000 trees have been inventoried and 202 pseudo-

species have been identiﬁed. More details about the Paracou experimental station

can be found in Schmitt and Bariteau (1990).

Characterizing pseudo-species

Each pseudo-species was ﬁrst characterized by a set of parameters that describe its

owth, its recruitment and its mortality. On each of the 48 subplots and at years t =

1984, 1988, 1990 and 1992, we computed the average diameter increment, the mor-

tality rate and the recruitment ﬂux between year t and t + 2. Let us denote a the

nts

average diameter increment of the pseudo-species s (= 1 ... 202) between year t and

24Amaro Forests - Chap 21 1/8/03 11:53 am Page 239

239 Process-dependent Groups for Modelling

t + 2 in subplot n, y

nts

the number of trees of the pseudo-species s that are alive in

subplot n at year t, z

nts

the number of trees of the pseudo-species s that die in sub-

plot n between year t and t + 2, and r

nts

the number of trees of the pseudo-species s

that are recruited (i.e. whose DBH surpasses 10 cm) in subplot n between year t and

t + 2.

Following Favrichon (1998)’s work, growth and recruitment were related to the

stocking of the plot, wher

eas mortality was not. The stocking of a subplot at year t

was quantiﬁed by the ratio b of its basal area at year t over its basal area in 1984

(assuming that 1984 characterizes the steady state of the subplot):

n ,1984

(1)

where B

is the basal area of subplot n at year t. The average diameter growth of the

pseudo-species s was then characterized by the parameters

and

of the regres-

sion: a =

 α

b +

nts

, where

nts

are the residuals. Similarly the recruitment of

nts nt

the pseudo-species s was characterized by the parameters

and

of the regres-

sion: r =

 β

b +

. As the diameter increments or the recruitment ﬂuxes on

nts nt nts

the same subplot at 2 consecutive years are not independent, a standard linear

regression (which would assume that the

nts

are independent) is not appropriate. A

regression for repeated measurements (Diggle et al., 1996) is required. A preliminary

analysis showed that an exponential model (Diggle et al., 1996: 57) could adequately

ﬁt the variance–covariance structure of the residuals. The parameters

were then estimated by maximizing the log-likelihood (Diggle et al., 1996: 63).

As mortality was assumed to be density-independent, the mortality rate m of

the pseudo-species s was simply estimated as: m

= (

n,t

z /(

n,t

nts

). The computa-

nts

)

tions were actually achieved for all pseudo-species with at least 15 individuals (on

average between 1984 and 1992) in at least thr

ee subplots. At this step, we have thus

got the set of dynamics parameters (

, m) for a subset of pseudo-species,

which will be used to build species groups by cluster analysis.

Building species groups

In a comprehensive approach, the dissimilarity d

ss

between any two pseudo-species

s and s was deﬁned as the Euclidian distance between the standardized vector

* * * * * *

of their dynamics parameters: d = [(



0s

)

+ (



)

+ (



0s

)

ss s s s s

(



s

)

+ (m

 m



)

]

0.5

where for any parameter x, x

= (x  x¯ )/

x¯ is the

s s s , s s

empirical mean of x

over all pseudo-species and s

is the empirical variance of x . A

hierarchical cluster analysis using Ward’s minimum variance method was then used

to deﬁne ﬁve comprehensive species groups. Species groups that are speciﬁc to a

process of the forest dynamics were also deﬁned by restricting the parameters used

to compute the dissimilarity between pseudo-species. Five growth-speciﬁc groups

were thus obtained in the same way from the following dissimilarity: d

ss

= [(



* * *

0s

)

+ (



1 s

)

]

0.5

. Five recruitment-speciﬁc groups were also obtained from the

* * * *

dissimilarity: d

ss

= [(



0s

)

+ (



1 s

)

]

0.5

, and ﬁve mortality-speciﬁc groups

s s

were obtained from the dissimilarity: d = |m

 m



ss s

, g

Let g

(s = 1 ... 5), g

and g

be respectively the comprehensive species

group to which the pseudo-species s belongs, the growth-speciﬁc group to which it

belongs, the recruitment-speciﬁc group to which it belongs, and the mortality-

speciﬁc group to which it belongs. The dynamic behaviour of a species can be char-

,acterized either by its comprehensive species group g

, or by the combination (g

24Amaro Forests - Chap 21 1/8/03 11:53 am Page 240

240 N. Picard et al.

, g

rc mt

) of its process-speciﬁc groups. The former characterization encompasses ﬁve

categories, whereas the latter potentially encompasses 5  5  5 = 125 categories.

Matrix model

The model is an Usher model with species groups and density-dependent coefﬁ-

cients. Its principles come fr

om Buongiorno and Michie (1980), Favrichon (1998) and

Usher (1969). The trees of the stand are broken down by diameter class and species

group. Time is discrete with a time step ∆t. Between time t and t + ∆t, a tree of

species group s and diameter class i has three possibilities: (i) it dies, with probabil-

ity q

(t); (ii) it stays alive and moves up to the next diameter class, with probability

(t); or (iii) it stays alive in the same diameter class, with probability p

(t) = 1



(t)



(t).q

Let N

(t) be the number of trees of species group s in diameter class i at time t.

Its equation of evolution is:

(

t +

∆

)

= p

()

t tN

si −1

() (

i1

)

(2)

()

+ u

si−1

()

(

t Nt tNt +

∆

)

= p

()

+ R

()

(3)

where R

(t) is the recruitment ﬂux. A linear relationship between the u

values and

the ratio b(t) of the subplot basal area at year t over its basal area in 1984 (see

Equation 1) was selected: u

(t) =

b(t). For recruitment, a linear relationship

0si

 δ

1si

between b(t) and either the recruitment ﬂux R

(t) or its log-transform was selected:

(t) =

γ  γ

b(t) or ln R

(t) =

 γ

b(t).

0s 1s 1s

Parameter estimation

Given two inventories at year t and t + ∆

t, the upgrowth transition probability u

can readily be estimated as the proportion of trees of species group s and diameter

class i that move up to diameter class i + 1. Similarly, the mortality rate q

can read-

ily be estimated as the proportion of trees of species group s and diameter class i

that die. Let u

sint

be the estimate of u

be the estimate of q

obtained from and q

sint

the subplot n (= 1 ... 48) and from the inventories t and t + ∆t.

To reduce the number of parameters of the model and to ensure the smoothness

of the transition probabilities, the following regressions were actually performed:

2 3

sint

2 s

−

sint

(4)

sint

0 s

2 s

sint

(5)

where D

is the average diameter of diameter class i, and b

is given by Equation 1.

As in the work of Favrichon (1998), we selected a time step ∆t = 2 years. W

then used the data from the 48 subplots at t = 1984, 1988, 1990, 1992. Each subplot

thus appears four times in Equations 4 or 5, so that the residuals

cannot be con-

sint

sidered as independent, and a longitudinal data analysis is again required. Again an

exponential model was selected for the variance–covariance structure:

Cov

(

′′

)

= 0if nn

′

or i ≠ i

′

, ∀

(

t t

′

)

(6)

≠ ,

sint si n

−

′

σρ

Cov

(

sint

′

)

s s

(7)

24Amaro Forests - Chap 21 1/8/03 11:53 am Page 241

241 Process-dependent Groups for Modelling

and the parameters were estimated by maximizing the log-likelihood (Diggle et al.,

1996).

Similarly for recruitment: let R

snt

be the number of trees of species group s that

are recruited between years t and t + ∆t on subplot n (= 1 ... 48). The following

regressions were performed: R =

 γ

b +

or ln R =

 γ

b +

. As

snt nt snt snt nt snt

each subplot appears four times, a regression for repeated measurements using the

same variance–covariance structure as before was achieved.

In regressions 4 and 5, all the subplots have the same weight. This can lead to

some distortions, as a subplot with a low number of tr

ees (and thus imprecise esti-

mates of u

sint

and q

sint

) will have the same weight as a subplot with a high number of

trees (and thus accurate estimates of u

sint

and q

sint

). For a few species groups contain-

ing few species and few individuals, we indeed obtained very unrealistic mortality

rates. In those cases we used in place of Equation 5 an a posteriori estimate of the

mortality rates that is derived from Equation 2 (Houde and Ledoux, 1995).

A set of parameters (

)

s=1 ... 5

is associated

with each species grouping. The complete parameter set was estimated for the com-

prehensive species grouping. The parameters (

)

s=1 ... 5

were then

estimated for the growth-speciﬁc grouping, (

)

s=1 ... 5

were estimated for the

mortality-speciﬁc grouping, and (

)

s=1 ... 5

were estimated for the recruitment-

speciﬁc grouping.

Results

Species groups

Of the 202 pseudo-species present in the database, 152 (75%) had enough data for

their parameters to be estimated. The ﬁve compr

ehensive groups that are derived

from all parameters may be described as follows (Table 21.1): Group 1: low

medium

s; this group thus includes slow-growing species that are rather insensi-

tive to a change in the stand stocking; Group 2: medium

s and medium

s; this

group thus includes species with an intermediate behaviour in every respect; Group

3: high

s, medium

s; this group thus includes fast-growing trees that are sensitive

to a change in the stand stocking; Group 4: low

s, medium

s and high mortality

rate; this group thus includes the species that require a stand stocking large enough

to recruit young trees; Group 5: very high

s and very high

s; this group thus

includes the species that grow very fast and have a high recruitment when the stand

is open (pioneer species). Groups 4 and 5 include species with outlying characteris-

tics and thus encompass few species.

The growth-speciﬁc groups discriminate the species along a gradient from

slow-gr

owing, density-independent species (Group 1 with low

s) to fast-growing

density-dependent species (Group 5 with high

s), see Table 21.2. Similarly the

Table 21.1. Mean values of the growth parameters

, the recruitment parameters

, and the

mortality rates m in the ﬁve comprehensive species groups. S is the number of species in the groups.

m S

1 0.64 0.40 0.70 0.56 0.024 46

2 1.33 1.04 0.28 0.20 0.010 75

3 2.76 2.24 0.90 0.84 0.016 25

4 0.95 0.76 0.54 1.05 0.184 4

5 3.15 1.57 24.29 25.09 0.027 2

24Amaro Forests - Chap 21 1/8/03 11:53 am Page 242

242 N. Picard et al.

Table 21.2. Mean values of the growth parameters

in the ﬁve growth-speciﬁc groups, the

recruitment parameters

in the ﬁve recruitment-speciﬁc groups, and the mortality rates m in the ﬁve

mortality-speciﬁc groups. S is the number of species.

S g

m S

1 0.42 0.17 35 1 0.15 0.08 117 1 0.014 34

2 1.67 1.31 46 2 1.01 0.81 23 2 0.005 71

3 1.04 0.80 51 3 2.83 2.43 9 3 0.025 32

4 2.73 2.19 17 4 7.80 8.03 1 4 0.060 11

5 5.67 4.90 3 5 24.29 25.09 2 5 0.184 4

recruitment-speciﬁc groups discriminate the species along a gradient from low to

high recruitment (Table 21.2), and the mortality-speciﬁc groups discriminate the

species along a gradient from low to high mortality. Each pseudo-species is then

characterized by a triplet (g

, g

Forty-three distinct triplets are observed; that is, one-third of the theoretical

maximum (125). This simply follows fr

om biological trade-offs between the dynam-

ics traits. For instance, no tree species can be at the same time fast-growing with a

high recruitment and low mortality. Indeed, the triplet (5, 5, 1) does not correspond

to any pseudo-species. In other words, the process-speciﬁc groups are not indepen-

dent. For instance, a

test reveals a relationship between the growth-speciﬁc and

the recruitment-speciﬁc groups (P = 0.04).

Model predictions

As in the work of Favrichon (1998), 11 diameter classes were deﬁned, ranging from 10

to 60

cm with a constant width of 5 cm, the last diameter class grouping all the trees

greater than 60 cm DBH. The parameters for the classical matrix model based on com-

prehensive species groups are given in Table 21.3. A posteriori estimates of the mortal-

ity rates were used for species groups 4 and 5: for species group 4, q

= 0.0224, ∀i; for

species group 5, q

= 0.2381, q

= 0.0505, q

= 0.1258 and q

= 1, ∀i ≥ 4.

The parameters for the matrix model based on growth-, recruitment- or mortality-

speciﬁc gr

oups are given in Table 21.4. Each of the 43 species groups that result from

the crossing of these three groupings is speciﬁed by a triplet (g

, g

), so that the

species group (g

, g

) takes its upgrowth transition parameters from group g

for growth (Table 21.4), its recruitment parameters from group g

for recruitment,

and its mortality parameters from group g

for mortality. The resulting matrix

model thus behaves like a matrix model with 43 species groups.

Both matrix models were used to project each of the 48 subplots from 1988 (at

the end of the silvicultural tr

eatments) to 1994. Evaluation is made at the end of the

growing period by comparing the observed and predicted values of number of

trees. This procedure does not validate the models as it does not rely on indepen-

dent data. Let N

sin

be the observed and N

sin

the predicted number of trees in the ith

diameter class of species group s in subplot n in 1994. The quality of the prediction

for the ith diameter class of species group s is quantiﬁed by the percentage of

explained variance: PEV

= 1



MSS

/S(N

), where S(N

) is the empirical variance

of N

over the 48 subplots and MSS

= 



sin

)

/ 48 is the mean sum of

n =1

sin

squares. The higher the PEV is, the better the predictions are.

able 21.5 shows the PEV values for each diameter class and each of the ﬁve

comprehensive groups, according to the classical matrix model. As MSS



= S(N

¯ ¯ ¯

) + (N



)

, where N is the empirical mean of the N

and N

is the empirical

24Amaro Forests - Chap 21 1/8/03 11:53 am Page 243

243

Table 21.3. Selected regression equations and parameter values of the matrix model for the comprehensive species grouping.

g Growth

(%) Recruitment

(%) Mortality

(%)

1 u =

D +

i i i

2 u =

D +



i i i i

3 u =

D +



i i i i

4 u =

D +

i i i i

5 u =

D +



i i i

0.17

0.02

0.09



0.04

0.13

3.5

12.4

7.9

2.1

14.6

R =



R =



ln R =



ln R = 

ln R =



0.03

0.01

0.24

0.44

0.12

41.5

30.6

40.2

8.3

43.7

D 0.13

D 0.11

D 0.05

D 0.00

D +

0.03

i i

7.1

0.5

0.2

1.3

3.7

Group 1 Group 2 Group 3

Estimate

SE P Estimate SE P Estimate SE P

–

3.347 10

3

–5.358 10

5

–

32.169

25.680

–4.840 10

2

3.361 10

3

–

1.908 10

4

3.955 10

6

–

1.795

2.161

1.125 10

2

2.968 10

4

–

0.001***

< 0.001***

–

0.001***

< 0.001***

–

2.702 10

1

–9.369 10

3

3.458 10

4

–3.671 10

6

1.415 10

1

21.265

15.207

5.783 10

3

2.789 10

4

–

2.843 10

2

2.667 10

3

7.754 10

5

6.870 10

7

1.054 10

2

1.364

1.640

3.223 10

3

8.038 10

5

–

< 0.001***

0.001***

< 0.001***

0.036*

0.001***

–

5.292 10

1

–1.976 10

2

7.265 10

4

–7.470 10

6

2.900 10

1

3.844

3.140

1.295 10

2

2.854 10

4

–

6.922 10

2

6.573 10

3

1.957 10

4

1.767 10

6

2.782 10

2

0.271

0.330

5.782 10

1

1.588 10

4

–

< 0.001***

0.001**

0.001***

< 0.001***

0.013*

0.036*

–

Group 4 Group 5

Estimate

SE P Estimate SE P

1.288 10

1

–1.286 10

2

4.514 10

4

–4.475 10

6

–

–0.822

4.907 10

1

–3.112 10

3

–

3.846 10

2

4.018 10

3

1.230 10

4

1.129 10

6

–

7.717 10

2

2.805 10

2

8.453 10

4

–

< 0.001***

0.001***

< 0.001***

–

0.001***

< 0.001***

–

1.350

–4.595 10

2

5.197 10

4

–

3.509 10

1

5.549

5.276

1.479 10

1

–1.410 10

2

3.441 10

4

1.796 10

1

1.103 10

2

1.681 10

4

–

1.605 10

1

0.422

0.610

5.214 10

2

5.847 10

3

1.555 10

4

< 0.001***

–

0.014**

0.001***

< 0.001****

0.002**

0.008**

0.013*

Levels of signiﬁcance: *** 0.1%, ** 1%, * 5%.

is the correlation coefﬁcient between any two successive estimates on the same subplot, as deﬁned in Equation 7.

Process-dependent Groups for Modelling

24Amaro Forests - Chap 21 1/8/03 11:53 am Page 244

Table 21.4. Selected regression equations and parameter values of the matrix model for the growth-, recruitment- and mortality-speciﬁc groupings.

g Growth

(%) Recruitment

(%) Mortality

(%)

1 u =



b 0.14 1.5 ln R =



b 0.18 16.5 q

– –

2 u =

D +



b 0.02 10.4 ln R =



b 0.18 41.2 q

 –

i i i i

3 u =

D +



b 0.01 9.0 ln R =



b 0.18 43.2 q

D 0.04 1.3

i i i i

4 u =

D +



b 0.02 7.1 ln R =



b 0.01 41.8 q

D 0.11 3.3

i i i i

5 u =

D +



b 0.01 14.8 ln R =



b 0.12 43.7 q

0.00 1.3

i i i i

Group 1 Group 2 Group 3

Estimate SE P Estimate SE P Estimate SE P

9.589 10

2

–

6.555 10

2

3.186

1.343

1.522 10

2

–

1.26010

2

–

1.480 10

2

2.042 10

1

2.420 10

1

5.428 10

2

–

< 0.001***

–

< 0.001***

–

3.419 10

1

–1.059 10

2

3.969 10

4

–4.309 10

6

1.803 10

1

3.965

2.644

9.293 10

3

–

4.076 10

2

3.831 10

3

1.116 10

4

9.913 10

7

1.521 10

2

2.076 10

1

2.463 10

1

5.617 10

2

–

< 0.001***

0.003**

< 0.001***

–

8.228 10

2

2.354 10

3

–4.374 10

5

–

5.945 10

2

4.148

3.030

–

8.867 10

4

1.094 10

2

5.210 10

4

6.885 10

6

–

8.111 10

3

2.271 10

1

2.706 10

1

–

6.810 10

5

< 0.001***

–

< 0.001***

–

< 0.001***

Group 4 Group 5

Estimate SE P Estimate SE P

3.120 10

1

8.030 10

3

–1.169 10

4

–

3.213 10

1

3.785

4.138

–

2.932 10

3

4.360 10

2

2.050 10

3

2.815 10

5

–

3.337 10

2

4.612 10

1

7.215 10

1

–

1.438 10

4

< 0.001***

–

< 0.001***

–

< 0.001***

2.219

–1.643 10

1

5.335 10

3

–5.044 10

5

5.043 10

1

5.549

5.276

4.904 10

1

–3.113 10

3

2.744 10

1

2.905 10

2

9.716 10

4

9.547 10

6

1.206 10

1

4.223 10

1

6.105 10

1

2.805 10

2

8.453 10

4

< 0.001***

< 0.001****

< 0.001***

Levels of signiﬁcance: *** 0.1%, ** 1%, * 5%.

is the correlation coefﬁcient between any two successive estimates on the same subplot, as deﬁned in Equation 7.

244 N. Picard et al.

24Amaro Forests - Chap 21 1/8/03 11:53 am Page 245

245 Process-dependent Groups for Modelling

mean of the N

, a systematic bias of the model predictions may lead to high values

of MSS, and thus negative values of PEV. We indeed got negative values of PEV,

mainly for species groups 4 and 5. The number of trees in these groups was thus

badly predicted by the classical matrix model on a short run after disturbance. On

the other hand, the number of trees in the other species groups was quite well pre-

dicted, with PEV values ranging from 28 to 92% (excluding negative values).

Table 21.5 also shows the PEV values according to the matrix model based on

ocess-speciﬁc groups, after aggregating the 43 groups into the ﬁve growth-speciﬁc

groups. Fewer negative values were obtained than with the classical matrix model,

but the positive values were on average smaller than with the former model. Similar

results were obtained if the 43 groups were aggregated into the recruitment-speciﬁc

or the mortality-speciﬁc groups.

Note that a comparison of the PEV values according to the two matrix models

in T

able 21.5 is not relevant, as they use different deﬁnitions of groups. As the 43

process-speciﬁc groups cannot be aggregated into the ﬁve comprehensive species

groups (the former are not a partition of the latter), a direct comparison of the two

matrix models is only possible at a higher level of aggregation, namely the stand

level. The PEV of the total number of trees per hectare is 77% for the classical matrix

model and 68% for the matrix model based on process-speciﬁc groups. The PEV of

the total basal area per hectare is 62% for the classical matrix model and 66% for the

matrix model based on process-speciﬁc groups.

Discussion and Conclusion

Allowing the species to shift from one species grouping to another, depending on

the biological process of interest, presents a better description of the species rich-

ness. With each species grouping is a set of parameters, and this approach may

also be seen as a crossing over of parameter sets. The resulting model has as many

parameters as in the classical approach where species belong to a unique species

Table 21.5. Percentage of explained variance

(PEV) of the number of trees in each diameter class and

each species group in 1994. The projections are made using either the classical matrix model (clas.) or

the matrix model based on process-speciﬁc groups (proc.); for the former, the ﬁve groups (g) are the

comprehensive groups whereas for the latter, the ﬁve groups (g

) are the growth-speciﬁc groups. PEV

cannot be estimated when N

sin

= 0 for all n (dashed cells).

Diameter class

Model g or g

1 2 3 4 5 6 7 8 9 10 11

Clas. 1 83 75 74 70 62 56 60 66 51 57 < 0

Clas. 2 92 75 82 80 61 70 78 85 70 73 < 0

Clas. 3 < 0 34 59 59 36 40 28 46 49 36 37

Clas. 4 < 0 < 0 < 0 < 0 – < 0 < 0 – – – < 0

Clas. 5 < 0 < 0 < 0 < 0 < 0 – – – – – < 0

Proc. 1 34 17 61 72 83 81 24 77 35 67 < 0

Proc. 2 73 62 69 55 41 68 49 70 60 67 40

Proc. 3 70 26 43 63 25 62 48 68 50 40 19

Proc. 4 < 0 19 46 40 32 36 < 0 38 33 < 0 61

Proc. 5 37 9 < 0 < 0 < 0 – < 0 < 0 6 < 0 < 0

For species group s and diameter class i, PEV

= 1



MSS

/S(N

), where S(N

) is the empirical

variance of N

over the 48 subplots, MSS

=∑

/48 is its mean sum of squares, and N

sin

n=1

sin



sin

)

the number of trees in species group s, diameter class i and subplot n.