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A density-dependent matrix model was developed for spruce-balsam fir natural forest stands in New Brunswick, Canada. It predicted the number and basal area of trees for 5 species groups (spruce, balsam fir, other softwood, soft hardwood and hard

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A Matrix Growth Model of Natural Spruce-Balsam Fir forest in NewBrunswick, Canada
Xiangdong Lei
Institute of Forest Resource InformationTechniques, Chinese Academy of Forestry, Beijing, China xdlei@caf.ac.cn
Changhui Peng
Institute of Environment Science, Universityof Quebec at Montreal (UQAM), Montreal, Canada
peng.changhui@uqam.ca
Yuanchang Lu
Institute of Forest Resource InformationTechniques, Chinese Academy of Forestry, Beijing, China ylu@caf.ac.cn
Xiaopeng Zhang
Institute of Automation, Chinese Academy of Science, Beijing, China xpzhang@nlpr.ia.ac.cn
Abstract
A density-dependent matrix model was developed for spruce-balsam fir natural forest stands in New Brunswick, Canada. It predicted the number and basalarea of trees for 5 species groups (spruce, balsam fir,other softwood, soft hardwood and hard hardwood)and 10 diameter classes. Upgrowth, ingrowth and mortality models were established with explanatoryvariables representing tree size, stand density and stand structure. The model was based on 305 sample plots with inventory periods from 2 to 9 years. Themajority of the data (80%) was used for modeldevelopment, and the rest (20%) was used for modelvalidation. It was concluded that the model is areliable and fairly accurate tool for short-term prediction of growth of spruce-balsam fir forest inCanada. Future work on refinements of the model isdiscussed.
1. Introduction
Since Usher’s work in 1970s[10-11], matrix growthmodels have been widely used in the projection of stand evolution and the evaluation of managementregimes because of their simple structure, coarse datarequirements and easy incorporation in optimizationmodels with economic and ecological objectives [4-6,13]. The parameters in transition probabilityequations are either fixed or variable [3]. Theoretically,a matrix growth model with parameters dependent onstand state is superior because stand state is expectedto affect growth. Therefore, the density-dependentmulti-species nonlinear models were developed withoptimization of management regimes. Recently, theeffect of stand structural diversity on diameter growth,mortality and recruitment possibilities was taken intoaccounted [6]. The consistency of long-termpredictions of matrix models with individual modelswas also tested [12].Boreal forest is Canada's largest biome orenvironmental community and known to be very largestores of carbon. Forest growth models at differentscales were well developed or adapted such asFORECAST, FVS, TRIPLEX and SORTIEetc.[2,7,9,14]. However, matrix models are rarelyreported in boreal forest management, Canada. Thepurpose of this study was to develop a matrix growthmodel of spruce-balsam fir natural forest stands for itsdevelopment and management in New Brunswick,Canada.
2. Model structure
The model structure is similar to that in Buongiornoand Michie [4]. It is a density-dependent model withmultiple species and diversity effects added (Equation1).
y
t+1
=
Gy
t
+ r (1)
1i1i2i,n-1i,n-1
i,ni,n
u susu s
i
i
sG
⎡ ⎤
⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎦
12
m
r r r r
⎡ ⎤
⎣
⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
00
ii
Rr
⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦
12m
GG
GG
⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎦
⎣
Where
y
t
= [
y
ijt
] is a column vector representingthe number of live trees in species group
i
(
i
= 1, 2,3, …
m
) and diameter class
j
(
j
= 1, 2,…,
n
) at time
t
,per ha.
G
is a state-dependent matrix that describeshow the trees grow or die between
t
and
t+1
.
r
is astate-dependent vector representing the number of trees recruited in the smallest diameter class of eachspecies group, between
t
and
t+1
. As a result of thedependency of
G
and
r
on the stand state
y
t
, the modelis nonlinear.The annual transition rates equation by speciesgroups were estimated as upgrowth equation (Equation2), mortality equation (Equation 3), and ingrowthequation (Equation 4).(2)(3)(4)(5)
u
ijt
is the probability for a tree to grow from thediameter class
j
to the diameter class
j+1
betweenyears
t
to
t+1
,
g
ijt
is the yearly growth of a tree of species
i
in diameter class
j
,
d
is the width of adiameter class,
Xsize
,
Xdensity
and
Xstructure
arevariables representing tree size, stand density and standstructure respectively. The Shannon–Wiener diversityindex, was used to characterize the diversity of treespecies (
Hspecies
) and diameter distributions (
Hsize
).The formula is expressed as:(6)where
p
i
is the proportion of basal area in the ithspecies or diameter class.
m
ijt
is the probability that alive tree of
j
diameter class in species
i
died betweenyears
t
and
t+1
,
m
ijT
is the probability that a tree diesbetween
t
and
t+T
,
T
is the inventory interval.
s
ijt
is theprobability that a live tree of species
i
and diameterclass
j
stays in the same class
j
between years
t
and
t+1
.
r
it
is the number of trees of species
i
that enters thesmallest diameter class between
t
and
t+1
,
r
iT
is thenumber of trees of species
i
that enters the smallestdiameter class between
t
and
t+T
.
3. Data
The data to estimate the parameters came from NewBrunswick Permanent Sample Plot Database [8]. 305SPBF (Spruce-balsam fir) plots were used which hadbeen measured at least two successive forest surveyswith inventory periods from 2 to 9 years. Trees weredivided into twelve 5cm d.b.h. classes with minimumdiameter 5cm and five species groups: sprucegroup(Group 1,
Picea mariana
,
Picea glauca
,
Picearubens
), balsam fir (Group 2,
Abies balsamea
), othersoftwood (Group 3,
Pinus strobus
,
Pinus banksiana
),soft hardwood (Group 4,
Acer rubrum
,
Fraxinusamericana
,
Betula papyrifera
,
Populus tremuloides
,
Populus grandidentata
,
Alnus spp
.,
Acer pensylvanicum
,
Betula populifolia
,
Populus balsamea
,and hard hardwood (Group5,
Acer saccharum
,
Betulaalleghaniensis
,
Fagus grandifolia
,
Ostrya virginiana
,
Quercus rubra
,
Acer spicatum
,
Sorbus Americana
) .Summary statistics for all plots was shown in table 1.
ijt ijt
g
4. Parameter estimations
The growth model was estimated with the majorityof the data (244 plots, 80%), and the rest (61 plots,20%) was used for model validation. To make theparameters from the various intervals commensurate,the transition rates were adjusted to yearly probabilities.Tables 2-4 showed the empirical equations forupgrowth, mortality and ingrowth.
5. Model validation
5.1. Individual equations
Figures 1, 2 showed the predicted mortality andupgrowth rates by the empirical equations and the 95%confidence intervals of the observations on the 61validation plots for spruce and balsam fir. With fewand unsystematic exceptions, the predicted values fellwithin the confidence interval of the observed means.
ud
=
gf
(,,)
ijtsizedensitystructure
XXX
=
ijT ij
mT
(,,)
ijtsizedensitystructure
XXX
=
t
m
=
mf
1
ijtijtijt
sum
−= −
iT it
r
1
(,)
iTdensitystructure
rfXX
=
isi
log
∑
=
r T
=
i
p p H
1
⋅−=
'
Figure 1. Mortality rates of predicted and observedvalues
0.000.010.020.030.045 10 15 20 25 30 35 40 45 50 55 60
Diameter class
M o r t a l i t y r a t e
0.000.050.100.150.205 10 15 20 25 30 35
Diameter class
M o r t a l i t y r a t e
SpruceBalsam fir
5.2. Whole model
Figure 3 showed how the predicted number of treeson the validation plots was close to the actual numberfor spruce and balsam fir. For each plot, the transitionmatrix
Gt
changed as a functions of stand variables ateach iteration. The average of the predicted number of trees of each species group was within the 95%confidence interval of average observed number. Ttests also confirmed that the predicted mean number of trees in each diameter class was not significantlydifferent from the observed means at the 5%significant level.
6. Discussion and conclusion
As expected, the upgrowth rates were significanthigher in stands with lower basal area. Tree speciesdiversity also had positive effects on growth rate forspruce and balsam fir groups. This effect wasassociated with higher recruitment in stands withhigher tree-species diversity. But tree size diversitywas only significantly negative with growth rates of balsam fir. It may be explained by that recruitment waslower on plots of higher tree-size diversity. Similarresults were observed from Liang et al. [6]. Treespecies diversity had negative effects on mortality rate,which showed complicated inter-specific competitionrelationships. Greater diversity results in the lowerdensity of any single species, which could cause lowermortality. Small trees always have higher mortalityrate. Ingrowth rate was negatively related to standdensity but positively to the number of trees in thesmallest diameter class of the same tree species. LowR
2
in individual transition rates equations show thatmuch of the variations in transition rates were eithercaused by other variables, or inherently random andunpredictable. Options of tree species group is anotherpossible factor.Generally, the model can be used for short-termprediction. In Canada, silviculture methods based onnatural disturbance and uneven-aged forestmanagement have been developed [1, 15]. Climatechange and forest management for complexity havebeen more and more concerned. Accordingly, themodel needs to be improved by incorporating newvariables such as harvesting and natural disturbancewith optimization methods. It has the potential to beused in long-term prediction and harvest simulationwith economic and ecological goals for natural spruce-balsam fir forests.Matrix models have been proven as a powerful toolfor prediction of stand development and selection of management regimes. There are still some questions.For example: (1) how multi-criteria of sustainableforest management can be achieved by optimizationmodels with matrix models as constraints? (2) what aretree seedlings or saplings dynamics? (3) what are theeffects of environment or natural process on transitionprobability? (4) how to consider growthautocorrelation for the accuracy of matrix modelpredictions? Therefore, future work on refinements of the models is still needed.
Acknowledgements
The study was partly supported by Natural Science andEngineering Research Council of Canada (NSERCC),
00.020406080.15 10 15 20 25 30 35 40 45 50 55 60
Diameter class
U p g r o w t h r a t e
00.020.040.060.080.15 10 15 20 25 30 35
Diameter class
U p g r o w t h r a t e
Balsam fir
0.0.0.
SpruceFigure 2. Upgrowth rates of predicted and observed valuesFigure 3. The number of trees of predicted andobserved values
0501001502005 10 15 20 25 30 35
Diameter class
N u m b e r o f t r e e s ( s t e m s / h a )
0501001502002503003505 10 15 20 25 30 35 40 45 50 55 60
Diameter class
N u m b e r o f t r e e s ( s t e m s / h a )
SpruceBalsam fir
Canada Research Chair (CRC) programme, NationalNatural Science Foundation of China (Grant No.30371157, 60073007, 60473110). Special thanks toMr. Edwin Swift, at Atlantic Forestry Centre,Canadian Forest Service, for providing NewBrunswick PSP data used in this study.
References
[1]
A. Groot, S. Gauthier and Y. Bergeron, “Standdynamics modelling approaches for multicohort managementof eastern Canadian boreal forests”.
Silva Fennica
, 2004,38(4): pp.437-448.[2]
C. H. Peng, J. X. Liu, Q. L. Dang, M. J. Apps, and H.Jiang, “TRIPLEX: A generic hybrid model for predictingforest growth and carbon and nitrogen dynamics”.
Ecological Modellling
, 2002,153:pp. 109-130[3]
C.R. Lin and J. Buongiorno, “Fixed versus variable-parameter matrix models of forest growth: the case of maple-birch forests”.
Ecological modeling
, 1997, 99(2):pp.263-274.[4]
J. Buongiorno and B. Michie, “A matrix model of uneven-aged forest management”.
Forest Science
, 1980,26(4):pp. 609-625.[5]
J. Buongiorno, J. L. Peyron, F. Houllier and M.Bruciamacchie. “Growth and management of mixed-species,uneven-aged forests in the French Jura: Implications foreconomic returns and tree diversity”.
Forest Science
, 1995,41(3):pp.397-428.[6]
J. J. Liang, J. Buongiorno and R. A. Monserud,“Growth and yield of all-aged Douglas-fir – westernhemlock forest stands: a matrix model with stand diversityeffects”.
Canadian Journal of Forest Research
, 2005,35(10):pp.2368-2381.[7]
J. P. Kimmins, D. Mailly, and B.Seely, “Modellingforest ecosystem net primary production: the hybridsimulation approach used in FORECAST”, EcologicalModelling, 1999, 122: pp.195-224.[8]
K. B. Porter, D. A. MacLean, K. P. Beaton and J.Upshall,
New brunswick permanent sample plot database(PSPDB v1.0): User's guide and analysis
, Canadian ForestService-Atlantic Forestry Centre, Natural Resources Canada,2001.[9]
K. D. Coates, C.D. Canham, M. Beaudet, D. L. Sachsand C. Messier, “Use of a spatially explicit individual-treemodel (SORTIE/BC) to explore the implications of patchiness in structurally complex forests”.
Forest Ecologyand Management
, 2003, 186: pp.297-310.[10]
M.B. Usher, “A matrix approach matrix approach tothe management of renewable resources, with specialreference to selection forests”.
Journal of Applied Ecology
,1966, 3:pp. 355-67.[11]
M.B. Usher, “A matrix model for forest management”.
Biometrics
, 1969, 25:pp.309-315.[12]
S. Gourlet-Fleury, G. Cornu, S. Jésel, H. Dessard, J.-G.Jourget, L. Blanc and N. Picard, “Using models forpredicting recovery and assessing tree species vulnerabilityin logged tropical forests: a case study from French Guiana”.
Forest Ecology and Management
, 2005, 209: pp.69-86.[13]
V. Favrichon, “Modeling the dynamics and speciescomposition of a tropical mixed-Species uneven-aged naturalforest: effects of elternative cutting regimes”.
Forest Science
,1998, 44(1):pp.113-124.[14]
V. Lacerte, G. R. Larocque, M. Woods, W. J. Patonand M. Penner, “Calibration of the forest vegetationsimulator (FVS) model for the main forest species of Ontario,Canada”.
Ecological Modellling
, 2006, 199: pp.336-349.[15]
Y. B. Bergeron, Leduc A. Harvey and S. Gauthier,“Forest management guidelines based on natural disturbancedynamics: stand and forest-level considerations”.
TheForestry Chronicle
, 1999, 75(1):pp.49-54.
Table 1. Summary statistics for all plots (N=305)
SpeciesgroupDensity(trees
﹒
ha
-1
)Basal area(m
2
﹒
ha
-1
)growth rate(cm
﹒
year
-1
)Recruitment(trees
﹒
ha
-1
﹒
year
-1
)1
1031(672) 23.04(7.8) 0.128(0.120) 31.7(80.6)
2
274(254) 3.75(4.07) 0.136(0.128) 28.3(79.1)
3
209(188) 4.49(4.63) 0.172(0.158) 6.4(22.8)
4
272(259) 4.38(4.05) 0.168(0.157) 9.3(38.1)5 246(255) 4.10(4.04) 0.182(0.174) 11.0(30.8)
Values are means with standard deviation in parentheses
.
Table 2. one year upgrowth probability equations by species groups
Variable Group 1 Group 2 Group 3 Group 4 Group 5
Constant 0.0142*** 0.0514*** 0.0766*** 0.132*** 0.0388***BA -0.000772*** -0.000453*** -0.00089***D
ij
0.00169*** 0.000845*** 0.000927*** 0.000733*** 0.00101***D
ij2
-0.0000206***Hspecies 0.0211*** 0.0170***Hsize -0.0197***BA
i2
0.00000953*Ln(N) -0.0164*** 0.00000944*N
i
0.0000285***Ln(N
i
) -0.00859***R
2
0.19 0.12 0.23 0.12 0.08BA: Stand basal area; D
ij
: diameter of i species and j diameter class; Hspecies: tree species diversity index; Hsize:tree size diversity index; BA
i
: basal area of i species; N: stand density; N
i
: the number of trees of i species; N
i1
: thenumber of trees in the smallest diameter of i species; D
i
: diameter of j diameter class. R
2
: coefficient of determinant.***p<0.001, **p<0.01, *p<0.05. Same in Tables 3 and 4.
Table 3. Ingrowth equations by species groups
Variable Group 1 Group 2 Group 3 Group 4 Group 5
Constant 70.1*** 38.5*** 21.2*** 9.364*** 4.092***N -0.0376*** -0.0212* -0.00317*** -0.0203*N
i1
0.104*** 0.139*** 0.134***Hspecies -30.267*BA
i2
-0.0302 -0.0385*R
2
0.20 0.09 0.18 0.04 0.05
Table 4. Mortality rates equations by species groups
Variable Group 1 Group 2 Group 3 Group 4 Group 5
Constant 0.0464*** 0.0321*** 0.0224*** 0.06285*** 0.0521***Hspecies -0.0206*** -0.0148** -0.0234* -0.0255* -0.0119**N
i
-0.0000092***D
j
-0.00338* -0.00165*** -0.00225*** -0.00121*** -0.00265***R
2
0.13 0.09 0.04 0.18 0.05

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