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Decomposing tree annual volume increments and constructing a system dynamic model of tree growth
E[OLOrHL mODE|LIIIlG ELSEVIER
Ecological Modelling 82 (1995) 299-312
Decomposing tree annual volume increments and constructing a system dynamic model of tree growth Yude Pan *, Dudley J. Raynal College of Ent.,ironmental Science and Forestry, State UnA,ersiO' of New York, Syracuse, NY 13210, US,4 Received 5 January 1994; accepted 25 May 1994
Abstract
We developed a dynamic system growth model of 55-year-old Pinus resinosa trees, compartmentalizing growth based on statistical approaches. Annual volume increment of trees was subdivided into several distinct components, each of which has some unique properties and responds to different factors that influence growth. The formulas for different growth components were estimated by statistical or numerical methods. The software STELLA was used to transfer a conceptual dynamic growth model into a practical computer model. Using STELLA, tree growth can be simulated over time and examined in relation to prevailing environmental events or conditions. The model can aid in evaluation of tree growth response to environmental stimuli and intrinsic factors, and provides a basis to predict future growth. Extrapolation of model results should be limited to conditions under which the model was dcvelopcd and not to novel environmental change or for older trees. Keywords." Growth, plant; Pine; STELLA
1. Introduction
Tree growth is a complicated process. In addition to intrinsic biological characteristics which regulate it, many environmental factors modify growth (Larson, 1963). Wilson (1984) presented process models including flow charts and computer programs, which are helpful for understanding the growth of a tree as a system of interrelated growth processes. In process model-
* Corresponding author. Present address: The Ecosystem Center, Marine Biological Laboratory, Woods Hole, MA 02543, USA.
ing the behavior of a system is represented by a set of component processes or compartments (Godfrey, 1983). Most process models are also dynamic, such that behavior is regulated by feedback relationships among the components (Blake et al., 1990). Tree growth can be compartmentalized into a set of fundamental functions representing the relationship between growth components and the environment. These functions or processes are quantitative expressions which describe tree growth. Process modeling can integrate responses of a whole-tree to environmental factors or stresses and aid in understanding and predicting tree growth. Process models developed for prediction of
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Y. Pan, D.J. Raynal / Ecological Modelling 82 (1995) 299-312
tree and stand growth under changing environmental conditions cover a broad range of approaches, including physiological processes and mechanisms related to growth and dynamics of the individual tree and forest stand, dynamics of ecosystems and regional patterns of growth and development (Dixon et al., 1990). For individual tree growth, typical models are those based on growth in three components: foliage, fine roots and wood tissue, and a "supply and demand" relationship reflecting the balance of carbon allocation within the tree (M~ikel~i, 1990). Stem-analysis data (Duff and Nolan, 1953) provide detailed information of annual growth increment along the tree-bole. Annual wood growth of the tree is an important component of whole-tree growth and is the most practical feature expressing tree growth. Construction of a stem analysis-based process model of a tree depends on decomposing or subdividing annual growth into a finite number of increments which represent the sum of influences on tree growth (Van Deusen, 1990; Cook, 1987). Subdivision of information contained in growth data is considered a general problem in time series analysis. The time series analysis method has been applied in dendrochronology to evaluate tree-ring data which are valuable for studying environmental changes (Graybill, 1982; Cook, 1985; Van Deusen, 1990). To extract signals or remove "noise" embodied in tree-rings, Cook (1987) developed a conceptual linear aggregate model (LAM) for a hypothetical ring-width series, which decomposed tree-ring series into five discrete types of signals indicative of biological growth trend, climate, endogenous disturbance, exogenous disturbance, and random error. In studying tree growth, the LAM is also a suitable model for determining how the growth of different component parts is controlled by various factors. Using a conceptual framework similar to the LAM model, we developed a dynamic system model of growth of Pinus resinosa, a native conifer of the northeastern U.S. commonly grown in plantations. In the model, annual volume increment was segregated into several growth components related to auto-correlative lag effect, and
the effects of age-related growth trend, climate, soil fertilization and random error. Because the characteristics of annual volume increment data are different from conventional tree-ring data, new methods for estimating growth components were developed. Since the model of annual volume growth of trees was designed as both a process and a dynamic model, a computer-run simulation model of growth changes over time was required. Such simulation study enables the modeler to observe integrated growth responses of a tree as it responds to both biotic and abiotic factors and to detect the outcome of such influences. For modeling purposes, the software STELLA (Richmond et al., 1987) provides a good tool to transfer a conceptual dynamic growth model into a practical computer model. STELLA was developed in the conceptual framework and methodology known as System Dynamics (Forrester, 1961; Richmond et al., 1987). The software of STELLA is compatible with the Macintosh computer and is icon-oriented. It provides great flexibility in modeling dynamic systems (Wu and Vankat, 1991). Once the user completes a structural diagram, STELLA can write the equations for all levels internally in the form of first-order difference equations, and provide a list of rate variables and auxiliary variables necessary for mathematical formulation. STELLA has been recommended as an excellent modeling tool and provides potential to break new ground for simulating biologically complex systems (Costanza, 1987). Through STELLA simulation, studies exploring response patterns of our tree growth model are facilitated.
2. A dynamic system of tree growth The structure of the dynamic system of tree growth is illustrated in Fig. 1. In the diagram, a single tree is expressed as a system, its annual volume increment (AVI) being based on two types of changes over time: changes of inputs outside the system and changes within the system itself. Annual growth is affected by lag growth of the last few years of climate (Fritts, 1976) and slowly changing, age-related growth trends (Cook, 1987).
Y. Pan, D.J. Raynal / Ecological Modelling 82 (1995) 299-312
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sources causing fluctuations of yearly growth. Because of geometric changes in the tree-bole which affect the magnitude of annual volume data, there is a phenomenon of growth signal amplification reflecting climate influence. A function expressing the changeable amplitudes was designed to adjust the signals related to climate. Using adjusted signals leads to a more accurate estimation of growth-climate relationships, which are modeled by multi-variate regression methods. A linear equation is used to describe the dynamic system of tree growth and is expressed as: 5
V ( t ) (dm3/yr) = Y'~ R ( t , t - i ) X ( t - i )
+ G(t)
i-I
+ c A ( t ) C ( t ) + F(t) + E(t) ×:It-5)
V(t)-~Rlt,t-il*Xlt-i)*G(t)*cA(tIC(t)÷F(t)*e(t) dynamic system of tree growth Fig. I. Diagram of the tree growth system. The symbols in the diagram are described in Table 2.
The linkage between the current year's growth and growth in previous years is defined as autocorrelation and is a feedback to the system. Auto-correlated growth also contains intrinsic growth trend inherited continually from the previous growth. The age-related trend of growth is a physiological process of the tree and reflects the vigor of bole cambium activity that changes with tree age. The age-related intrinsic growth trend is expressed as a time function. Annual volume increment also is influenced by environmental variables which are extrinsic inputs or control signals to the system. In the context of this study, extrinsic inputs include influences of soil fertilization and climat:'c variations. Fertilization can be treated as a time-response function of a disturbance pulse acting on growth. Broad-scale meteorological variables are signals that directly or indirectly limit the annual growth of trees. Climatic conditions are major
(1) where V(t) is annual volume increment of tree; R ( t , t - i) is the auto-correlation between growth of year t - i and t; here the lag effect of the previous five years is considered as the sum of auto-correlation effect on current year's growth. X ( t - i ) is the optimal estimate of V ( t - i ) at year t - i; G(t) is the intrinsic growth trend; F(t) is the fertilization effect on growth; C(t) is the signal growth due to climatic variable at year t, A(t) is defined as amplitude which reflects amplified growth signals with time due to magnitude effect of annual volume data, and c is a constant used to adjust the signal growth to a reasonable scale (this will be discussed later). E(t) is random error which represents unexplained variance in the growth series after other growth influences have been taken into account (Cook, 1987). Some likely sources of E(t) are microsite differences within stand and measurement errors. E(t) thexefore is the variance of annual growth of trees in relation to their mean-value growth in the stand. In this model, the influence of endogenous disturbance which appeared in the LAM model (Cook, 1987) was not considered. Such disturbances are frequently a consequence of gap-phase stand development in which individual trees are removed from the canopy by processes that do not affect the stand as a whole (White, 1979). Even though mortality occurred with the develop-
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Y Pan, D.J. Raynal / Ecological Modelling 82 (1995) 299-312
ment of stands at Pack Forest (White et al., 1990), we assumed that tree growth in even-aged plantations is less affected by gap-phase stand dynamics than in natural mixed-age forests.
3. Tree growth data for model construction The growth data used for constructing the model were measurements of annual growth of Pinus resinosa collected from stands at Pack Demonstration Forest in the southeastern Adirondack Mountains of New York (LeBlanc et al., 1987). The stands were 55-year-old plantations growing in sandy, potassium-deficient soil on abandoned agricultural land. Fifteen sample trees were felled in each of two unthinned plots from a long-term comprehensive investigation of conifer establishment. One plot was under control conditions and the other one a fertilization trial. Fertilization was performed in 1939 with combined NPK fertilizer at 560 kg ha -1. Each plot was about 0.2 ha in area and with tree spacing 1.2 m × 1.2 m. The entire stem bole of each sample tree was used for analysis (LeBlanc et al., 1987). Cross sectional disks were cut from the center of each internode of a tree and annual rings were counted for each disk. Ring widths were measured on four radii in each disk and then averaged. These detailed stem-analysis data together with height growth of trees were used to calculate annual volume increments of trees over growth years (Pan, 1993).
Growth Trend Without Leg Effect ( red pine at 1.2m spacing, in control plot) G(t)=1,0503*exp (0.008629t)- 1
1.6
R^2=0.255
<
1.4
N
1.2
~ -~
1 0,8 0.6
"~
0.4
~
0.2 o
-0,2 ~ -0.4 -o.6
"i' I' l'Ib'~l' I' I ~g~ I' I' I hN6 I' I' l'Ib'N I ' i' I'i19~I' 1-1935
1945
1955
1965
1975
1985
year •
AVl without lag
--
growth trend
Fig. 2. Annual volume increments (AVI) without lag effect over 55 years (which had been removed) were fitted by an exponential function, which reflects an intrinsic growth trend.
is different from growth trend which usually reflects non-random variations in growth that last longer than the entire growth series. It is usually considered in a time span of several years. In dendrochronological analysis, growth data are first detrended to remove long-term systematic
< E
4. Estimations of growth terms in the dynamic model
1.8
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2
Symmetric Curv~ of Amplitude Functiona ( red pinein 1.2m control plot ) A(t) =0.030331 *exp(0.055483t) R^2=0'524
"
o.~ -0.1 -0.2 .0.3
4.1. Estimating autocorrelation An autocorrelation coefficient statistically describes serial dependence or association in a time series with previous conditions or states (Fritts, 1976). Autocorrelation in growth series is related to several influences such as growth cycle length, long-term variations in microsite conditions lasting for several years or decades, or other trends in the data (Fritts, 1976). Autocorrelation growth
-0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1
'1 '1' J i~l' 1935
I'1 ib'~ I' I' f :d~d J'l' 1'1~-~6 I'1' I ibW I '1
1945
1955
1965
1975
1985
year --
growth fltc'tuations
....
amplitude functions
Fig. 3. Growth fluctuations derived from the annual volume increments after removing lag growth and intrinsic growth. An exponential function, defined as amplitude function, was used to adjust increased amplitudes of growth fluctuations.
Y. Pan, D.J. Raynal / Ecological Modelling 82 (1995) 299-312 Weighted growth signals without effect of tree-size ( red pine in 1.2m control plot ) 0.9 0.8
¢n
0.7 0.6 0.5 • 0.4
]~ ,
0.3
•
o, o.!f
-0.2 ! -0.3 '-~ -0.4 -05 -0.6
-0.7 f -0.8 -0.9 -1
I I t~l 1935
I'li~l
1945
1'l~6t
1955
I'l~b'~61'l'l~b'd61 ~
1965
1975
1985
year
Fig. 4. Growth signals derived from annual volume increment after removing lag growth, intrinsic growth and tree-size effect.
variation and to develop growth index series. This step in analysis is called standardization (Fritts, 1976). Since the data in this study consist of annual volume increment measurements which have dif-
303
ferent characteristics compared to tree-ring data, we first calculated and removed autocorrelation growth and subsequently removed growth trends from the data. This was necessary because increased annual volume due to both tree-size and increased photosynthate assimilation controls the change of tree geometry to a greater extent than the cambium vigor as a function of tree age. If we assume that growth trend within the data is agerelated and reflects particularly changes of cambium activity, removing autocorrelation growth first would be more rational. The growth trend from data after removal of autocorrelative growth can be considered more likely to reflect the state of cambium activity apart from the tree volume effect. To determine the autocorrelation among each year's growth, the correlation coefficients between each two successive years were calculated using 15 sampled red pine trees in both plots. Then the m e a n value of these correlation coefficients was calculated. The mean value is considered as average correlation coefficient between any two successive years. This average coefficient is also the sum of different order auto-correla-
Table 1 Results of stepwise regression between annual volume increment and 13 principal components of climate R-square = 0.54962729, C(p) = 7.73009812 DF
Sum of squares
Mean square
F
Prob > F
Regression Error Total
13 40 53
2.79123025 2.28717525 5.07840550
0.21471002 0.05717938
3.76
0.0006
Variable
P a r a m e t e r estimate
Standard error
Type II sum of squares
F
Prob > F
INTERCEP P1 P3 P4 P5 P6 P7 P8 P9 P10 Pll P17 P18 P19
0.02047711 -0.05831572 0.02513936 0.02367905 0.08198715 0.04525579 -0.06289433 -0.03610099 - 0.07946068 0.03293147 -0.06731607 - 0.08190571 - 0.05200777 -0.07568856
0.03254040 0.02077444 0.02240839 0.02389068 0.02471989 0.02570697 0.02695294 0.02822107 0.03068309 0.03218310 0.03361301 0.04794422 0.05219009 0.05583817
0.02264285 0.45055957 0.07196588 0.05617087 0.62898152 0.17720907 0.31135085 0.09356873 0.38348261 0.05986951 0.22933065 0.16687655 0.05678058 0.10506003
0.40 7.88 1.26 0.98 11.00 3.10 5.45 1.64 6.71 1.05 4.01 2.92 0.99 1.84
0.5327 0.0077 0.2686 0.3276 0.0019 0.0860 0.0247 0.2082 0.0133 0.3123 0.0520 0.0953 0.3250 0.1829
The above model is the best 13-variable model found.
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Y. Pan, D.J. Raynal / Ecological Modelling 82 (1995) 299-312
tion, because the average correlation coefficient (ACC) represents the average effects of last year's growth on the current year's growth and accumulates all the lag effect of past years. The relation is expressed as:
Reeponaa Function of Annual Growth to Weather Conditions (rod pin• at 1.2rn spacing, control plot) 0.2 Temperature
Precipitation
0.15 0.1 t' ,.
ACC = r 1 + r 2 + r 3 + r4 + r5 + r6 • • •
(2)
.c
H e r e rl is first-order autocorrelation (n = 1). The autocorrelation for high order lags (n > 1) will decrease in the fashion as Fritts (1976) indicated:
0.05 0
&
-0.05
(3)
-0.1
where rn is the autocorrelation between the growth at the specified lag n. We only considered 5 years' lag effect on the current year growth even though there may have longer lag. The effect of lags longer than 5th order is usually very weak. Based on Eqs. 2 and 3,
.0.15
Fig. 5. Response f u n c t i o n o f a n n u a l v o l u m e i n c r e m e n t to m o n t h l y mean t e m p e r a t u r e and total precipitation. T h e dashed lines are a p p r o x i m a t e 9 5 % confidence intervals. T h e
ACE = R + R 2 + R 3 + R 4 + R 5
response indices were transferred from the coefficients of regression between annual volume increment and 13 principal
r, = r ~
-0.2
(4)
where R is first-order autocorrelation, and is defined as R ( t , t - 1 ) , i.e. autocorrelation between growth at year t and growth at year t - 1. So, R i is expressed as R ( t , t - i) and is the autocorrelation between growth at year t and growth at year t - i. Eq. 4 is a one-variable fifth-order equation which can be solved by the numerical method. After obtaining the first-order autocorrelation, the other order autocorrelation can be produced. The time series analysis method (filter modeling) for estimating autocorrelation was not used in our model (Cook, 1985,1987; Box and Jenkins, 1976); the method employed here is more straightforward and more easily interpretable biologically.
13 PCs (cumulative= 71.37%) O~tlDogJFe~).lAprlJ,',nlA.~l O~lDeclFeblkRrlJt~nlA~Q~-Nov ,Jan Mar MAy dUl ~ e p Nov Jan Mar May Jul Sap
components of climatic factors. tion growth from annual volume increment, the growth series were plotted to detect whether a growth trend was evident. Then a exponential equation was used to fit the growth series (Fig. 2). The format of the equation is:
Y ( t ) = a *e bt - 1
(5)
Fertilization Effect on Annual G r o w t h AVl(fertilized tree)-AVl(control tree) F(t) =0.084486"t ^ 1.546281*exp(-O.096803t) a R^2=0.530g •
1.8 1.6
(
1.4
r.
1.2
E
0.8
1
4.2. Estimating growth trend
0.6
Growth trend in data may result from such p h e n o m e n a as the changing growth potential of the tree that results from increasing age, successional alterations in the forest, or gradual variations in climate. T h e r e are different methods for estimating the growth trend, G(t). The approaches fall into two categories: deterministic functions and statistic methods. In our problem, an exponential function was applied to estimate the growth trend. After removing auto-correla-
,<
0.4 0.2 0
~
-0.2
-0'4!
' I' I' Iig,~dl' I' I'1~'~1'I' I~t~dI' I' Ii~M61' I' Ii ~ T I 1935
1945
•
1955 year AVI differences - -
1965
1975
1985
fitting curve
Fig. 6. Fertilization effect on annual volume increments. A
general exponential function was used to fit the growth in relation to fertilization effect.
Y. Pan, D.J. Raynal / Ecological Modelling 82 (1995) 299-312
The fitted curve shows a slightly increasing trend as the tree ages, possibly indicating that cambium activity of the whole-tree had been increased with tree development. The coefficient of determination for the fitted curve is 0.255, the correlation coefficient (R) is 0.505 and is significant at a = 0.005 (R > R0005 = 0.364).
4.3. Estimating climatic environmental inputs
305
tude of annual volume increments with tree development require adjustment. A pair of symmetric curves which are exponential functions were used to fit this increased amplitude of climatic growth signals (Fig. 3). One positive curve of the pair was designed as an changeable amplitude function:
A ( t ) = a * e b'
(6)
A(t)* C(t) are those residual series where C(t) is After the intrinsic growth trend was determined and then removed from the growth series, the residuals of the series were used to determine the possible effect of climatic environmental inputs. These residuals fluctuate around the mean of zero, and show gradually enlarged amplitudes of vibration. This phenomenon results from scale changes of annual volume data that accompany tree enlargement, and cannot be removed by autocorrelation and detrending processes. In order to build a climate-growth model using those residuals that reflect climatic effects, the distorted growth signals due to changes on magni-
adjusted climatic growth signal which fluctuates around a mean of zero in a more uniform amplitude (Fig. 4). To stabilize the variance of C(t), a constant c was used as a factor weighting C(t). This weighting makes climatic signals fall into the actual data range. Weighted C(t) is considered as the signals created by the current year's climate. This growth signal due to climate is called "white noise" and consists of serially random inputs that drive the tree growth system (Cook, 1987). It can be estimated by climatic variables. In our study, the mean monthly temperature and total monthly
Table 2 T h e d y n a m i c s y s t e m e q u a t i o n s for t r e e g r o w t h and e s t i m a t e s of f o r m u l a s for g r o w t h c o m p o n e n t s (red pine growing at 1.2-m spacing, control \ fertilized plots)
V(t ) = F~5=iR(t,t - i ) X ( t - i) + G ( t ) + c A ( t ) C ( t ) + F ( t ) + e(t) R ( t , t - 1) = 0.4794 R ( t , t - i) = R ( t , t - 1) i X ( t ) = V(t)* = V ( t ) - e(t) G ( t ) = 1.0503 * e 0"008629t - 1 A ( t ) = 0.030331 * e 0.055483t c=5 F ( t ) = 0.0844486 * t 1.546281 . e -0.096803t
C ( t ) = a o + a l P l ( t ) + a2P2(t) + . . . +a12P12(t) + b l T l ( t ) + b2T2(t) + . . . +b12T12(t) ( R 2 = 0.56, r e g r e s s i o n is significant w h e n F > F a ( 1 8 , 3 5 ) = 2.47 at a = 0.01) E[e(t)] = 0, V a r [ e ( t ) ] = V a r [ V ( t ) - V(t)* ] V(t) (dm3): a n n u a l v o l u m e i n c r e m e n t . R ( t , t - i): a u t o c o r r e l a t i o n b e t w e e n g r o w t h in y e a r t - i and y e a r t. X ( t ) , V(t)* (dm3): the o p t i m a l e s t i m a t e of a n n u a l v o l u m e i n c r e m e n t in y e a r t. G ( t ) (dm3): intrinsic g r o w t h trend. A ( t ) : c h a n g e a b l e a m p l i t u d e s e m b o d i e d in g r o w t h signals, b r o u g h t u p by tree-size c h a n g e . F ( t ) (din3): function of fertilization effect. c: w e i g h t i n g index. C ( t ) (dm3): a d j u s t e d g r o w t h signals d u e to c l i m a t i c effect a n d d e p e n d e n t v a r i a b l e of a r e g r e s s i o n m o d e l . Pl(t), •. • ,P12(t) (mm): total m o n t h l y p r e c i p i t a t i o n from J a n u a r y to D e c e m b e r . Tl(t), • ' • ,T12(t) (° C): m e a n m o n t h l y t e m p e r a t u r e from J a n u a r y to D e c e m b e r . a o, a l, • • • ,a12, b 1, • • • ,b~2: coefficients of a r e s p o n s e function, c o n v e r t e d from coefficients of a stepwise r e g r e s s i o n b e t w e e n C ( t ) and p r i n c i p a l c o m p o n e n t s of c l i m a t i c variables. e(t): r a n d o m e r r o r s w i t h z e r o m e a n a n d v a r i a n c e [ V ( t ) - V(t)* ].
Y. Pan, D.J. Raynal / Ecological Modelling 82 (1995) 299-312
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precipitation from 1930 to 1985, collected from eight weather stations in the Pack Forest area were used to study climate-growth relationship. Principal components analysis was first applied to treat 24 climatic variables (including monthly temperature and precipitation). With 97% of the cumulative proportion variance accounted for, 21 principal components (PCs) were chosen from 24 PCs and used in stepwise regression to determine their relationship with growth signals. The regression model with 13 PCs was adopted to simulate the growth due to climate at significance level a = 0.0006 (Table 1) coefficients of the regression model were converted by calculating the principal component scores to an explicable set of coefficients which express growth responses to variations in particular climatic variables. In this study,
the correlation represents relative effect of variations in temperature and precipitation on tree growth for each month of growth period (Fig. 5). The correlation is defined as the growth response function (Fritts, 1976). 4.4. E s t i m a t i n g fertilization effect
The effect of soil fertilization was determined by comparing growth in control and fertilized plots. The mean annual volume increment in the fertilized plot minus that in the control plot year by year was used to study fertilization effects. A general exponential function in the format F(t)
(7)
= a * t b * e (-ct)
was used to fit growth affected by fertilization
Annual growth
A°no, grow
Tree Volume
-X.
............
Temperature
Oct
~.j
Precipitation
Sep
Aug
startFt
Jul
Oct2
Sep2
error2
Aug2
Ct
Jul2
Fig. 7. The structural diagram of S T E L I ~ (Richmond ct al., 1987) for system dynamic model of tree growth. Symbols and variables used in the diagram are interpreted in Table 3.
307
Y. Pan, D.J. Raynal / Ecological Modelling 82 (1995) 299-312 Table 3 Variables in the system dynamic model of tree growth (red pine trees growing in 1.2-m spacing sites at the Pack Forest, NY) State variables Annual growth (dm3): Tree Volume (dm3): FerTreeVL (dm3): Rate variables AV_ increment (dm3/yr): LastYrAVI (dm3/yr): AVI (dm3/yr): Auxiliary variables Vt_ 1 (dm3): Signals (din3): Ct (din3): cAt: Temperature (dm3): Precipitation (din3): error 1: Vt_2 (dm3): Age trend (dm3): lag_growth (dm3): AVIfer (dm3): startFT (dm3): Ft (din3): error2: Table cariables (e.g.) Jan (° C): Jan2 (ram):
annual volume increment of control trees cumulative volume growth of control trees cumulative volume growth of fertilized trees each year's new volume increment of control trees last year's volume increment of control trees each year's new volume increment of fertilized trees annual growth due to extrinsic factors growth signals due to climatic effect adjusted growth signals related to climate without tree-size effect weighted amplitude function reflecting tree-size effect on growth signals growth due to temperature effect growth due to precipitation effect random errors due to measurements annual growth due to intrinsic factors age-related, intrinsic growth trend auto-correlative growth, i.e. lag efffect on current year growth annual volume increment of fertilized trees fertilization effect on growth started at treatment year function of fertilization effect random error due to interactive effect of fertilization and climate monthly mean temperature from 1932-1985 in January monthly total precipitation from 1932-1985 in January
(Fig. 6). U s i n g the n u m e r i c a l a p p r o a c h m e t h o d , the o p t i m a l curve of a g e n e r a l e x p o n e n t i a l function was f o u n d to fit the data. T h e fertilization effect can b e expressed as a t i m e - r e s p o n s i v e patt e r n using a d e t e r m i n i s t i c function. T h e P e a r s o n ' s c o r r e l a t i o n coefficient ( R ) of the fitted curve is 0.73 which indicates a significant c o r r e l a t i o n at the level a = 0.005 ( R > R0.005 = 0.364).
s a m p l e d tree in the stand. T a b l e 2 p r e s e n t s the g e n e r a l e q u a t i o n s for e s t i m a t i n g each t e r m in the tree growth model.
1 vt 1
2:Vt2
3: stadFt
il
4.5. Estimating random errors E ( t ) is the r a n d o m v a r i a n c e in growth data series d u e to m i c r o - e n v i r o n m e n t a l factors a n d m e a s u r e m e n t errors etc. u n i q u e to each tree. It can be e s t i m a t e d simply by s u b t r a c t i n g averaged a n n u a l v o l u m e growth from each s a m p l e d a n n u a l v o l u m e growth series in the stand. So w h e n acc o u n t i n g for the c o n t r i b u t i o n s of each growth t e r m except E ( t ) in the aggregate growth model, the m e a n values of a n n u a l v o l u m e i n c r e m e n t were t a k e n in the e s t i m a t i o n p r o c e d u r e for each
1 oo
14!2s
2;so
40%
s,.o~
Time
Fig. 8. Behavior pattern of the model system. In the figure, Vtl (curve 1) represents extrinsic inputs of the system including growth due to climate and random disturbances. Vt2 (curve 2) shows intrinsic growth including age-related growth and lag growth. The StartFt (curve 3) reflects growth due to fertilization effect.
Y. Pan, D.J. Raynal / Ecological Modelling 82 (1995) 299-312
308 5. S i m u l a t i o n
study using
a n c y is a t t r i b u t a b l e
STELLA
to growth due
to climatic
e f f e c t s , w h i c h is p r o d u c e d b y a r e g r e s s i o n m o d e l . The structural diagram of the STELLA
simu-
The
r e g r e s s i o n m o d e l w e u s e d is b a s e d o n 13
T h e v a r i a b l e s in t h e d i a g r a m a r e i n t e r p r e t e d i n
principal components which account for 71.4% of t h e c u m u l a t i v e v a r i a n c e . S o m e i n f o r m a t i o n is l o s t
T a b l e 3. T h e c o r r e s p o n d i n g m a t h e m a t i c a l m o d e l
by using s t e p w i s e r e g r e s s i o n results. E v e n with a
l a t i o n m o d e l o f t r e e g r o w t h is s h o w n in Fig. 7.
written
in
STELLA
T a b l e 4. T h e
language
STELLA
model
in
regression model combining more principal com-
simulates annual
ponents, the result of the model cannot simulate
is p r e s e n t e d
volume tree growth based on the underlying rela-
exactly the
t i o n s h i p s a n d f u n c t i o n s of t h e variables. Fig. 8
t h e r e m a y exist n o n l i n e a r r e l a t i o n s h i p s b e t w e e n
e x h i b i t s t h e b e h a v i o r p a t t e r n o f t h e m o d e l syst e m . I n Fig. 9, s i m u l a t i o n r e s u l t s w e r e c o m p a r e d
growth
with real annual volume increment data of both control trees and fertilized trees. The simulated
The differences between simulation and real growth of fertilized trees are obvious during the
g r o w t h f o r t r e e s in b o t h s t a n d s is c l o s e t o a c t u a l
effective period of fertilization, because interac-
growth. However, growth fluctuations do not re-
tion effects of fertilization with other factors were
flect closely the actual growth data. The discrep-
i g n o r e d in t h e a g g r e g a t e m o d e l .
and
actual
growth fluctuations because
monthly weather
conditions which
are not expressed by the linear regression model.
Table 4 Equations of STELLA for system dynamic model of tree growth []
[]
D
O O O O O O O O O
O O O
O O
Annual_growth(t) = Annual_growth(t - dt) + (AV_increment - LastYrAVI)* dt INIT Annual_ growth = 0 INFLOWS: AV_increment = Vt_ 1 + Vt_2 OUTFLOWS: LastYrAVI = DELAY(AV_ increment,DT,0) FerTreeVL(t) = FerTreeVL(t - dt) + (AVI) * dt INIT FerTreeVL = 0 INFLOWS: AVI = AVIfer Tree_ Volume(t) = Tree_Volume(t - dt) + (LastYrAVI)* dt INIT Tree_Volume = 0 INFLOWS: LastYrAVI = DELAY(AV_ increment,DT,0) Age _ trend = 1.0503 * EXP(0.008629 * TIME) - 1 AVIfer = AV_increment + startFt + error2 cAt = 5 * 0.030331 * EXP(0.055483 * TIME) Ct = Precipitation + Temperature errorl = (RANDOM(0,1) - 0.5) * 2 * Annual - growth * 0.01 error2 = 0.2 * Ct * RANDOM(0,1) Ft = 0.0844486 * (TIME "1_.546281)* EXP( - 0.096803 * TIME) lag_ growth = 0.835 * Annual_ growth Precipitation = - 0.008724 * Oct2 + 0.004619 * Nov2 - 0.01409 * Dec2 - 0.0478 * Jan2 - 0.02344 * Feb2 - 0.000855 * Mar2 + 0.02098 * Apr2 + 0.0435 * May2 + 0.04407 * Jun2 + 0.04245 * Jul2 - 0.04355 * A u g 2 - 0.01628 * Sep2 Signals = cAt * Ct startFt = DELAY(Ft,7,0) Temperature = 3.726 - 0.008369 * Jan - 0.00965 * Feb + 0.00448 * Mar + 0.005307 * Apr - 0.01659 * May - 0.005882. Jun - 0.01666. Jul - 0.01276 * Aug - 0.009422 * Sep - 0.005844 * Oct + 0.005639 * Nov + 0.009625 * Dec Vt_ 1 = Signals + errorl Vt_2 = A g e t r e n d + lag_growth
Y. Pan, D.J. Raynal / Ecological Modelling 82 (1995) 299-312
6.00" i: AV recreant
~]
Ia)
2: realAVl
3o0
o.oo 1oo
1: AVlfer
14.25
27.50
40.75
27I~)
405S
5400
2: FerDala
2
(b)
100
14~5
S4.d
Time
Fig. 9. Simulated annual volume increment in contrast to real annual volume increment of red pine for: (a) trees in control plot, AV increment (curve 1) is simulated growth and realAVI (curve 2) is actual growth; (b) trees in fertilized plot, AVIfer (curve 1) is simulated growth and FerData (curve 2) is actual growth.
In order to test the effect of extreme weather conditions on tree growth, the actual weather was replaced by hypothetical weather data. Precipitation of May, June and July over 54 years was set with low precipitation (0.5 inch) in every 10 years. The annual volume increments in the end year 1985 is 0.96 dm 3 less and cumulative volume is 16.0 dm 3 less than that of trees under actual precipitation (Fig. 10). The model verifies that drought during the growing season can affect total tree volume. Even though the hypothetical weather conditions do not change annual volume increment of the tree very much, the dynamic system model no doubt can serve effectively as a simulation tool to evaluate different types of changes in weather variables and their effects on tree growth.
6. Discussion
The system dynamic model of tree growth provides both a conceptual framework and means to
309
characterize and predict tree growth responses to environmental stimuli. An assumption in the model is that growth can be subdivided into several distinct components, each of which has some unique properties and responds to different factors that influence growth. The quantitative expression of the system links processes together in a predictive and testable format and allows the investigator to track growth behavior through time. With knowledge of growth in an initial year and climatic conditions over time, growth in subsequent years can be simulated based on system feedback, intrinsic biological characteristics and climate-growth relationships. Although the dynamic model is based on determinate relationships and feedback between components of the system, all parameters of the model were estimated using available experimental data and statistical methods. The approach taken involves a procedure starting with the estimation and removal of autocorrelation growth
119.78
1: A V tncrernenl
2: T o e V o l u ~
/' 59 89
(a)
ooc
,1 1 co
10371
, 27.50
1425
1: AV increment
2: T ~
i 4075
54.0~
Vo~urr~
A/ 2,5
v ~ ~
(b)
2
.J'~, I
51 86
,-
0(30 1 0 0
~
1425
,
2750
,
4075
,
5400
Time
Fig. 10. Simulated annual volume increment and cumulative volume of red pine trees (a) under normal climate; (b) under hypothetical low precipitation (0.5 inch) during growing season at time = 10, 20, 30, 40, 50. AV increment (curve 1) is annual volume increment and Tree Volume (curve 2) is cumulative volume.
310
Y. Pan, D.J. Raynal / Ecological Modelling 82 (1995) 299-312
within the data, followed by estimations of growth trend, climatic signals, random error, and finally fertilization effects. The relationships determined by statistical methods will restrict application of the model to some extent. For example, there are limits in stand conditions such as spacing of plantation trees, land use history and soil nutrient availability, all of which affect growth of trees in the stand. Time limitation is another problem with the model. The behavior of age-related trend changes in different stages of tree development. In our problem, the sample trees were only 55 years old, so the age trend of annual volume increment is presented as a positive exponential function. This case will affect the accuracy of growth prediction for tree development stage beyond the behavior described by the age trend, when the trend would be more likely logistic than exponential. This problem could be solved by obtaining data of older trees and evaluating the pattern of growth behavior over the whole lifecycle of the species. In addition, tree age may also mirror the sensitivity of growth to climate reflected by the response function. Variable growth behavior of different tree species is obviously another factor limiting model application. The term response function (Fig. 5) refers to coefficients used to estimate growth based on climate variables (Fritts, 1976). The set of coefficients are elements of the response function which can be interpreted directly in terms of monthly temperature or monthly precipitation. With approximate 95% confidence intervals estimated for coefficients of response function, we can be confident that the true coefficients between annual growth and monthly climatic variables lie within the intervals. The response function can be interpreted as showing a significant direct relationship between growth and precipitation during May to July and inverse relationship between growth and precipitation in January and August. Temperature in the previous January and August of the current season have an inverse relationship with growth, especially in August. The results indicate that for annual volume growth the precipitation from May to July is most important probably because during the time cambium division and
cell enlargement are most active. The cause of the possible negative effect on growth from precipitation in August is not certain. January precipitation always accompanies snow storms and could damage shoots and affect tree growth in the following year (Kozlowski, 1971). Temperature in August has an inverse relationship with growth, implying that high temperature causes a higher rate of respiration and carbohydrate catabolism. Actually, high temperature during the growing season (May-August) has an inverse effect on growth, but is most significant in August. High temperature in January inversely affects growth, and may result from stimulation of stored carbohydrate use, thus affecting subsequent growth. The inverse relationship between growth and January and February temperature also im4..5
fertilization vs, control growth
• fertilization * control
3.5 3
'~
2.5 2 1.5
(a)
I 1.g 1,8
ferffilzatlon function
1.7 1.8 1.5 1.4 1,3 1.2
~
(b)
~
y = d + a * X ^ b*exp(-c*X)
o.g 0.8 0.7 O.e 0.5 0.4 0.3 0.2
o.~
......................
i_i=.i_
'l'l'lb~'l'l'l'lb~'l+l'l'lb~+l'l'l'lb~'l'l'l'lb'as
1940
lg50
lg00
years
lg70
lg80
Fig. 11. Fertilization effect on tree ring growth (at DBH) of red pine. (a) Tree-ring growth in fertilized plot in contrast to that in control plot. (b) A general exponential function used to fit the effect of fertilization on ring growth. The function shows that growth response of ring width to fertilization is sharper and shorter than that of annual volume increments.
Y Pan, D.J. Raynal/Ecological Modelling 82 (1995) 299-312
311
plies that low t e m p e r a t u r e injury in winter may occur before January. Growth due to fertilization effects shows marked fluctuations (Fig. 6). This indicates there exists some interactive effect of fertilization with climatic factors. For annual volume increment, the fertilization effect on growth is not like that reflected in indices of tree rings or specific volume (Fig. 11), where the stimulation of growth was very strong in the immediate years after the treatment but declined quickly afterwards. Fertilization effects on growth can be divided into direct effects, which reflect annual ability of fertilized trees to produce wood per unit area of growth surface; and residual effects, which are due to the fact that larger trees grow faster (Comerford et al., 1980). Growth indices such as ring widths are measures of the direct effect of fertilization. Annual volume increment, on other hand, is a measure combining both the direct and residual effects of fertilization since this index depends on both tree size and growth per unit cambium. If fertilization increases the size of tree relative to an unfertilized tree, subsequent equivalent radial growth on both trees will result in a greater amount of absolute growth on the larger tree. The fertilization effect on annual volume increment achieved its maximum value at about 15 years after fertilization which is much later than that in tree rings. The response arises from the residual effect of fertilization.
or conditions. Such a model can aid in evaluation of tree growth and provide a basis to predict future growth. The simulation results from the dynamic model closely reflect actual growth and show that the model is very reliable for estimating growth of red pine trees ( < 55 year-old). However, since the limits of p a r a m e t e r s of the system are determined by experimental data and statistics, the model will require further development if it is to be used to characterize the growth of older trees, trees growing in different site conditions, and for different tree species.
7. Conclusion
References
The dynamic system model and its related aggregate equation developed in this study provide a useful approach for reconstructing and estimating tree growth. The method of reducing growth into several components and then quantifying these components not only provides a concrete approach for studying growth, but also reveals insight into the biological process of tree growth. The S T E L L A dynamic model serves as a good tool for simulation trials. Using the model tree growth can be simulated over time and examined in relation to prevailing environmental conditions and to extreme environmental events
Blake, J.l., Somers, G.L. and Ruark, G.A., 1990. Perspectives on process modeling of forest growth response to environmental stress. In: R.K. Dixon, R.S. Medahl, G.A. Ruark and W.G. Warren (Editors), Process Modeling of Forest Growth Response to Environmental Stress. Timber Press, Portland, OR, pp. 9-17. Box, G.E.P. and Jenkins, G.M., 1976. Time Series Analysis: Forecasting and Control. Holden-Day, Oakland. Comerford, N.B., Lamson, N.I. and Leaf, A.L., 1980. Measurement and interpretation of growth response of Pinus resinosa Ait. to K-fertilization. For. Ecol. Manage., 2: 353-367. Cook, E.R., 1985. A time series analysis approach to tree-ring standardization. Ph.D. Thesis, Univ. of Arizona, Tucson. Cook, E.R., 1987. The decomposition of tree-ring series for environmental studies. Tree-Ring Bull., 47: 37-59.
Acknowledgements We are grateful to Drs. C. Hall, C. Davis and H. T e p p e r in S U N Y - E S F for their reviews and helpful comments on this study, and to Dr. G. Jacoby for providing facilities at the Tree-Ring Lab, Lamont-Doherty Earth Observatory, Columbia University. This research was part of the Ph.D. thesis research of the senior author at College of Environmental Science and Forestry, State University of New York at Syracuse and was supported in part by the McIntire-Stennis Cooperative Forestry Research Program, New York State Energy Research and Development Authority, and Empire State Electrical Energy Research Corporation, and the New York State Electric and Gas Corporation.
312
Y. Pan, D..I. Raynal / Ecological Modelling 82 (1995) 299-312
Costanza, R., 1987. Simulation modeling on Macintosh using STELLA. BioScienee, 37: 129-132. Dixon, R.K., Medahl, R.S., Ruark, G.A. and Warren, W.G. (Editors), 1990. Process Modeling of Forest Growth Response to Environmental Stress. Timber Press, Portland, OR, 441 pp. Duff, G.H. and Nolan, N.J., 1953. Growth and morphogenesis in Canadian forest species. I: the controls of cambial and apical activity in Pinus resinosa Ait. Can. J. Bot., 31: 471-513. Forrester, J.W., 1961. Industrial Dynamics. MIT Press, Cambridge, MA, 464 pp. Fritts, H.C., 1976. Tree Rings and Climate. Academic Press, London, 567 pp. Godfrey, K., 1983. Compartmental Models and Their Application. Academic Press, New York, NY, 291 pp. Graybill, D.A., 1982. Chronology development and analysis. In: M.K. Hughes, P.M. Kelly, J.R. Pilcher and V.V. Lamarche Jr. (Editors), Climate from Tree Rings. Cambridge University Press, Cambridge, pp. 21-28. Kozlowski, T.T., 1971. Growth and Development of Trees, Vol. I. Academic Press, New York, NY, 443 pp. Larson, P.R., 1963. Stem form development of forest trees. For. Sci. Monogr. 5, 42 pp. LeBlanc, D.C., Raynal, D.J. and White, E.H., 1987. Acidic deposition and tree growth: I. The use of stem analysis to study historical growth patterns. J. Environ. Qual., 16: 325-333.
M§kel~i, A., 1990. Modeling structural-functional relationship in whole-tree growth: resource allocation. In: R.IC Dixon, R.S. Medahl, G.A. Ruark and W.G. Warren (Editors), Process Modeling of Forest Growth Response to Environmental Stress. Timber Press, Portland, OR, pp. 81-95. Pan, Y., 1993. Growth response of conifers in Adirondack plantations to changing environment: Model approaches based on stem-analysis. Ph.D. Thesis, ESF, State Univ. of New York, Syracuse, 272 pp. Richmond, B., Peterson, S. and Vescuso, P., 1987. An academic user's guide to STELLA. High Performance System, Lyme, NH, 392 pp. Van Deusen, P.C., 1990. A simultaneous approach to tree-ring analysis. In: R.K. Dixon, R.S. Medahl, G.A. Ruark and W.G. Warren (Editors), Process Modeling of Forest Growth Response to Environmental Stress. Timber Press, Portland, OR, pp. 357-367. White, E.H., Raynal, D.J. and Mitchell, M.J., 1990. Acidic deposition and Adirondack forest soil fertility: an appraisal. New York State Energy Research and Development Authority, Rep. 90-5. White, P.S., 1979. Pattern, process, and natural disturbance in vegetation. Bot. Rev., 45: 229-299. Wilson, F.B., 1984. The Growing Tree. The Univ. of Massachusetts Press, Amherst, 134 pp. Wu, J. and Vankat, J.L., 1991. An area-based model of species richness dynamics of forest islands. Ecol. Model., 58: 249-271.
Ecological Modelling 82 (1995) 299-312
Decomposing tree annual volume increments and constructing a system dynamic model of tree growth Yude Pan *, Dudley J. Raynal College of Ent.,ironmental Science and Forestry, State UnA,ersiO' of New York, Syracuse, NY 13210, US,4 Received 5 January 1994; accepted 25 May 1994
Abstract
We developed a dynamic system growth model of 55-year-old Pinus resinosa trees, compartmentalizing growth based on statistical approaches. Annual volume increment of trees was subdivided into several distinct components, each of which has some unique properties and responds to different factors that influence growth. The formulas for different growth components were estimated by statistical or numerical methods. The software STELLA was used to transfer a conceptual dynamic growth model into a practical computer model. Using STELLA, tree growth can be simulated over time and examined in relation to prevailing environmental events or conditions. The model can aid in evaluation of tree growth response to environmental stimuli and intrinsic factors, and provides a basis to predict future growth. Extrapolation of model results should be limited to conditions under which the model was dcvelopcd and not to novel environmental change or for older trees. Keywords." Growth, plant; Pine; STELLA
1. Introduction
Tree growth is a complicated process. In addition to intrinsic biological characteristics which regulate it, many environmental factors modify growth (Larson, 1963). Wilson (1984) presented process models including flow charts and computer programs, which are helpful for understanding the growth of a tree as a system of interrelated growth processes. In process model-
* Corresponding author. Present address: The Ecosystem Center, Marine Biological Laboratory, Woods Hole, MA 02543, USA.
ing the behavior of a system is represented by a set of component processes or compartments (Godfrey, 1983). Most process models are also dynamic, such that behavior is regulated by feedback relationships among the components (Blake et al., 1990). Tree growth can be compartmentalized into a set of fundamental functions representing the relationship between growth components and the environment. These functions or processes are quantitative expressions which describe tree growth. Process modeling can integrate responses of a whole-tree to environmental factors or stresses and aid in understanding and predicting tree growth. Process models developed for prediction of
0304-3800/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 0 4 - 3 8 0 0 ( 9 4 ) 0 0 0 9 6 - Z
300
Y. Pan, D.J. Raynal / Ecological Modelling 82 (1995) 299-312
tree and stand growth under changing environmental conditions cover a broad range of approaches, including physiological processes and mechanisms related to growth and dynamics of the individual tree and forest stand, dynamics of ecosystems and regional patterns of growth and development (Dixon et al., 1990). For individual tree growth, typical models are those based on growth in three components: foliage, fine roots and wood tissue, and a "supply and demand" relationship reflecting the balance of carbon allocation within the tree (M~ikel~i, 1990). Stem-analysis data (Duff and Nolan, 1953) provide detailed information of annual growth increment along the tree-bole. Annual wood growth of the tree is an important component of whole-tree growth and is the most practical feature expressing tree growth. Construction of a stem analysis-based process model of a tree depends on decomposing or subdividing annual growth into a finite number of increments which represent the sum of influences on tree growth (Van Deusen, 1990; Cook, 1987). Subdivision of information contained in growth data is considered a general problem in time series analysis. The time series analysis method has been applied in dendrochronology to evaluate tree-ring data which are valuable for studying environmental changes (Graybill, 1982; Cook, 1985; Van Deusen, 1990). To extract signals or remove "noise" embodied in tree-rings, Cook (1987) developed a conceptual linear aggregate model (LAM) for a hypothetical ring-width series, which decomposed tree-ring series into five discrete types of signals indicative of biological growth trend, climate, endogenous disturbance, exogenous disturbance, and random error. In studying tree growth, the LAM is also a suitable model for determining how the growth of different component parts is controlled by various factors. Using a conceptual framework similar to the LAM model, we developed a dynamic system model of growth of Pinus resinosa, a native conifer of the northeastern U.S. commonly grown in plantations. In the model, annual volume increment was segregated into several growth components related to auto-correlative lag effect, and
the effects of age-related growth trend, climate, soil fertilization and random error. Because the characteristics of annual volume increment data are different from conventional tree-ring data, new methods for estimating growth components were developed. Since the model of annual volume growth of trees was designed as both a process and a dynamic model, a computer-run simulation model of growth changes over time was required. Such simulation study enables the modeler to observe integrated growth responses of a tree as it responds to both biotic and abiotic factors and to detect the outcome of such influences. For modeling purposes, the software STELLA (Richmond et al., 1987) provides a good tool to transfer a conceptual dynamic growth model into a practical computer model. STELLA was developed in the conceptual framework and methodology known as System Dynamics (Forrester, 1961; Richmond et al., 1987). The software of STELLA is compatible with the Macintosh computer and is icon-oriented. It provides great flexibility in modeling dynamic systems (Wu and Vankat, 1991). Once the user completes a structural diagram, STELLA can write the equations for all levels internally in the form of first-order difference equations, and provide a list of rate variables and auxiliary variables necessary for mathematical formulation. STELLA has been recommended as an excellent modeling tool and provides potential to break new ground for simulating biologically complex systems (Costanza, 1987). Through STELLA simulation, studies exploring response patterns of our tree growth model are facilitated.
2. A dynamic system of tree growth The structure of the dynamic system of tree growth is illustrated in Fig. 1. In the diagram, a single tree is expressed as a system, its annual volume increment (AVI) being based on two types of changes over time: changes of inputs outside the system and changes within the system itself. Annual growth is affected by lag growth of the last few years of climate (Fritts, 1976) and slowly changing, age-related growth trends (Cook, 1987).
Y. Pan, D.J. Raynal / Ecological Modelling 82 (1995) 299-312
301
sources causing fluctuations of yearly growth. Because of geometric changes in the tree-bole which affect the magnitude of annual volume data, there is a phenomenon of growth signal amplification reflecting climate influence. A function expressing the changeable amplitudes was designed to adjust the signals related to climate. Using adjusted signals leads to a more accurate estimation of growth-climate relationships, which are modeled by multi-variate regression methods. A linear equation is used to describe the dynamic system of tree growth and is expressed as: 5
V ( t ) (dm3/yr) = Y'~ R ( t , t - i ) X ( t - i )
+ G(t)
i-I
+ c A ( t ) C ( t ) + F(t) + E(t) ×:It-5)
V(t)-~Rlt,t-il*Xlt-i)*G(t)*cA(tIC(t)÷F(t)*e(t) dynamic system of tree growth Fig. I. Diagram of the tree growth system. The symbols in the diagram are described in Table 2.
The linkage between the current year's growth and growth in previous years is defined as autocorrelation and is a feedback to the system. Auto-correlated growth also contains intrinsic growth trend inherited continually from the previous growth. The age-related trend of growth is a physiological process of the tree and reflects the vigor of bole cambium activity that changes with tree age. The age-related intrinsic growth trend is expressed as a time function. Annual volume increment also is influenced by environmental variables which are extrinsic inputs or control signals to the system. In the context of this study, extrinsic inputs include influences of soil fertilization and climat:'c variations. Fertilization can be treated as a time-response function of a disturbance pulse acting on growth. Broad-scale meteorological variables are signals that directly or indirectly limit the annual growth of trees. Climatic conditions are major
(1) where V(t) is annual volume increment of tree; R ( t , t - i) is the auto-correlation between growth of year t - i and t; here the lag effect of the previous five years is considered as the sum of auto-correlation effect on current year's growth. X ( t - i ) is the optimal estimate of V ( t - i ) at year t - i; G(t) is the intrinsic growth trend; F(t) is the fertilization effect on growth; C(t) is the signal growth due to climatic variable at year t, A(t) is defined as amplitude which reflects amplified growth signals with time due to magnitude effect of annual volume data, and c is a constant used to adjust the signal growth to a reasonable scale (this will be discussed later). E(t) is random error which represents unexplained variance in the growth series after other growth influences have been taken into account (Cook, 1987). Some likely sources of E(t) are microsite differences within stand and measurement errors. E(t) thexefore is the variance of annual growth of trees in relation to their mean-value growth in the stand. In this model, the influence of endogenous disturbance which appeared in the LAM model (Cook, 1987) was not considered. Such disturbances are frequently a consequence of gap-phase stand development in which individual trees are removed from the canopy by processes that do not affect the stand as a whole (White, 1979). Even though mortality occurred with the develop-
302
Y Pan, D.J. Raynal / Ecological Modelling 82 (1995) 299-312
ment of stands at Pack Forest (White et al., 1990), we assumed that tree growth in even-aged plantations is less affected by gap-phase stand dynamics than in natural mixed-age forests.
3. Tree growth data for model construction The growth data used for constructing the model were measurements of annual growth of Pinus resinosa collected from stands at Pack Demonstration Forest in the southeastern Adirondack Mountains of New York (LeBlanc et al., 1987). The stands were 55-year-old plantations growing in sandy, potassium-deficient soil on abandoned agricultural land. Fifteen sample trees were felled in each of two unthinned plots from a long-term comprehensive investigation of conifer establishment. One plot was under control conditions and the other one a fertilization trial. Fertilization was performed in 1939 with combined NPK fertilizer at 560 kg ha -1. Each plot was about 0.2 ha in area and with tree spacing 1.2 m × 1.2 m. The entire stem bole of each sample tree was used for analysis (LeBlanc et al., 1987). Cross sectional disks were cut from the center of each internode of a tree and annual rings were counted for each disk. Ring widths were measured on four radii in each disk and then averaged. These detailed stem-analysis data together with height growth of trees were used to calculate annual volume increments of trees over growth years (Pan, 1993).
Growth Trend Without Leg Effect ( red pine at 1.2m spacing, in control plot) G(t)=1,0503*exp (0.008629t)- 1
1.6
R^2=0.255
<
1.4
N
1.2
~ -~
1 0,8 0.6
"~
0.4
~
0.2 o
-0,2 ~ -0.4 -o.6
"i' I' l'Ib'~l' I' I ~g~ I' I' I hN6 I' I' l'Ib'N I ' i' I'i19~I' 1-1935
1945
1955
1965
1975
1985
year •
AVl without lag
--
growth trend
Fig. 2. Annual volume increments (AVI) without lag effect over 55 years (which had been removed) were fitted by an exponential function, which reflects an intrinsic growth trend.
is different from growth trend which usually reflects non-random variations in growth that last longer than the entire growth series. It is usually considered in a time span of several years. In dendrochronological analysis, growth data are first detrended to remove long-term systematic
< E
4. Estimations of growth terms in the dynamic model
1.8
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2
Symmetric Curv~ of Amplitude Functiona ( red pinein 1.2m control plot ) A(t) =0.030331 *exp(0.055483t) R^2=0'524
"
o.~ -0.1 -0.2 .0.3
4.1. Estimating autocorrelation An autocorrelation coefficient statistically describes serial dependence or association in a time series with previous conditions or states (Fritts, 1976). Autocorrelation in growth series is related to several influences such as growth cycle length, long-term variations in microsite conditions lasting for several years or decades, or other trends in the data (Fritts, 1976). Autocorrelation growth
-0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1
'1 '1' J i~l' 1935
I'1 ib'~ I' I' f :d~d J'l' 1'1~-~6 I'1' I ibW I '1
1945
1955
1965
1975
1985
year --
growth fltc'tuations
....
amplitude functions
Fig. 3. Growth fluctuations derived from the annual volume increments after removing lag growth and intrinsic growth. An exponential function, defined as amplitude function, was used to adjust increased amplitudes of growth fluctuations.
Y. Pan, D.J. Raynal / Ecological Modelling 82 (1995) 299-312 Weighted growth signals without effect of tree-size ( red pine in 1.2m control plot ) 0.9 0.8
¢n
0.7 0.6 0.5 • 0.4
]~ ,
0.3
•
o, o.!f
-0.2 ! -0.3 '-~ -0.4 -05 -0.6
-0.7 f -0.8 -0.9 -1
I I t~l 1935
I'li~l
1945
1'l~6t
1955
I'l~b'~61'l'l~b'd61 ~
1965
1975
1985
year
Fig. 4. Growth signals derived from annual volume increment after removing lag growth, intrinsic growth and tree-size effect.
variation and to develop growth index series. This step in analysis is called standardization (Fritts, 1976). Since the data in this study consist of annual volume increment measurements which have dif-
303
ferent characteristics compared to tree-ring data, we first calculated and removed autocorrelation growth and subsequently removed growth trends from the data. This was necessary because increased annual volume due to both tree-size and increased photosynthate assimilation controls the change of tree geometry to a greater extent than the cambium vigor as a function of tree age. If we assume that growth trend within the data is agerelated and reflects particularly changes of cambium activity, removing autocorrelation growth first would be more rational. The growth trend from data after removal of autocorrelative growth can be considered more likely to reflect the state of cambium activity apart from the tree volume effect. To determine the autocorrelation among each year's growth, the correlation coefficients between each two successive years were calculated using 15 sampled red pine trees in both plots. Then the m e a n value of these correlation coefficients was calculated. The mean value is considered as average correlation coefficient between any two successive years. This average coefficient is also the sum of different order auto-correla-
Table 1 Results of stepwise regression between annual volume increment and 13 principal components of climate R-square = 0.54962729, C(p) = 7.73009812 DF
Sum of squares
Mean square
F
Prob > F
Regression Error Total
13 40 53
2.79123025 2.28717525 5.07840550
0.21471002 0.05717938
3.76
0.0006
Variable
P a r a m e t e r estimate
Standard error
Type II sum of squares
F
Prob > F
INTERCEP P1 P3 P4 P5 P6 P7 P8 P9 P10 Pll P17 P18 P19
0.02047711 -0.05831572 0.02513936 0.02367905 0.08198715 0.04525579 -0.06289433 -0.03610099 - 0.07946068 0.03293147 -0.06731607 - 0.08190571 - 0.05200777 -0.07568856
0.03254040 0.02077444 0.02240839 0.02389068 0.02471989 0.02570697 0.02695294 0.02822107 0.03068309 0.03218310 0.03361301 0.04794422 0.05219009 0.05583817
0.02264285 0.45055957 0.07196588 0.05617087 0.62898152 0.17720907 0.31135085 0.09356873 0.38348261 0.05986951 0.22933065 0.16687655 0.05678058 0.10506003
0.40 7.88 1.26 0.98 11.00 3.10 5.45 1.64 6.71 1.05 4.01 2.92 0.99 1.84
0.5327 0.0077 0.2686 0.3276 0.0019 0.0860 0.0247 0.2082 0.0133 0.3123 0.0520 0.0953 0.3250 0.1829
The above model is the best 13-variable model found.
304
Y. Pan, D.J. Raynal / Ecological Modelling 82 (1995) 299-312
tion, because the average correlation coefficient (ACC) represents the average effects of last year's growth on the current year's growth and accumulates all the lag effect of past years. The relation is expressed as:
Reeponaa Function of Annual Growth to Weather Conditions (rod pin• at 1.2rn spacing, control plot) 0.2 Temperature
Precipitation
0.15 0.1 t' ,.
ACC = r 1 + r 2 + r 3 + r4 + r5 + r6 • • •
(2)
.c
H e r e rl is first-order autocorrelation (n = 1). The autocorrelation for high order lags (n > 1) will decrease in the fashion as Fritts (1976) indicated:
0.05 0
&
-0.05
(3)
-0.1
where rn is the autocorrelation between the growth at the specified lag n. We only considered 5 years' lag effect on the current year growth even though there may have longer lag. The effect of lags longer than 5th order is usually very weak. Based on Eqs. 2 and 3,
.0.15
Fig. 5. Response f u n c t i o n o f a n n u a l v o l u m e i n c r e m e n t to m o n t h l y mean t e m p e r a t u r e and total precipitation. T h e dashed lines are a p p r o x i m a t e 9 5 % confidence intervals. T h e
ACE = R + R 2 + R 3 + R 4 + R 5
response indices were transferred from the coefficients of regression between annual volume increment and 13 principal
r, = r ~
-0.2
(4)
where R is first-order autocorrelation, and is defined as R ( t , t - 1 ) , i.e. autocorrelation between growth at year t and growth at year t - 1. So, R i is expressed as R ( t , t - i) and is the autocorrelation between growth at year t and growth at year t - i. Eq. 4 is a one-variable fifth-order equation which can be solved by the numerical method. After obtaining the first-order autocorrelation, the other order autocorrelation can be produced. The time series analysis method (filter modeling) for estimating autocorrelation was not used in our model (Cook, 1985,1987; Box and Jenkins, 1976); the method employed here is more straightforward and more easily interpretable biologically.
13 PCs (cumulative= 71.37%) O~tlDogJFe~).lAprlJ,',nlA.~l O~lDeclFeblkRrlJt~nlA~Q~-Nov ,Jan Mar MAy dUl ~ e p Nov Jan Mar May Jul Sap
components of climatic factors. tion growth from annual volume increment, the growth series were plotted to detect whether a growth trend was evident. Then a exponential equation was used to fit the growth series (Fig. 2). The format of the equation is:
Y ( t ) = a *e bt - 1
(5)
Fertilization Effect on Annual G r o w t h AVl(fertilized tree)-AVl(control tree) F(t) =0.084486"t ^ 1.546281*exp(-O.096803t) a R^2=0.530g •
1.8 1.6
(
1.4
r.
1.2
E
0.8
1
4.2. Estimating growth trend
0.6
Growth trend in data may result from such p h e n o m e n a as the changing growth potential of the tree that results from increasing age, successional alterations in the forest, or gradual variations in climate. T h e r e are different methods for estimating the growth trend, G(t). The approaches fall into two categories: deterministic functions and statistic methods. In our problem, an exponential function was applied to estimate the growth trend. After removing auto-correla-
,<
0.4 0.2 0
~
-0.2
-0'4!
' I' I' Iig,~dl' I' I'1~'~1'I' I~t~dI' I' Ii~M61' I' Ii ~ T I 1935
1945
•
1955 year AVI differences - -
1965
1975
1985
fitting curve
Fig. 6. Fertilization effect on annual volume increments. A
general exponential function was used to fit the growth in relation to fertilization effect.
Y. Pan, D.J. Raynal / Ecological Modelling 82 (1995) 299-312
The fitted curve shows a slightly increasing trend as the tree ages, possibly indicating that cambium activity of the whole-tree had been increased with tree development. The coefficient of determination for the fitted curve is 0.255, the correlation coefficient (R) is 0.505 and is significant at a = 0.005 (R > R0005 = 0.364).
4.3. Estimating climatic environmental inputs
305
tude of annual volume increments with tree development require adjustment. A pair of symmetric curves which are exponential functions were used to fit this increased amplitude of climatic growth signals (Fig. 3). One positive curve of the pair was designed as an changeable amplitude function:
A ( t ) = a * e b'
(6)
A(t)* C(t) are those residual series where C(t) is After the intrinsic growth trend was determined and then removed from the growth series, the residuals of the series were used to determine the possible effect of climatic environmental inputs. These residuals fluctuate around the mean of zero, and show gradually enlarged amplitudes of vibration. This phenomenon results from scale changes of annual volume data that accompany tree enlargement, and cannot be removed by autocorrelation and detrending processes. In order to build a climate-growth model using those residuals that reflect climatic effects, the distorted growth signals due to changes on magni-
adjusted climatic growth signal which fluctuates around a mean of zero in a more uniform amplitude (Fig. 4). To stabilize the variance of C(t), a constant c was used as a factor weighting C(t). This weighting makes climatic signals fall into the actual data range. Weighted C(t) is considered as the signals created by the current year's climate. This growth signal due to climate is called "white noise" and consists of serially random inputs that drive the tree growth system (Cook, 1987). It can be estimated by climatic variables. In our study, the mean monthly temperature and total monthly
Table 2 T h e d y n a m i c s y s t e m e q u a t i o n s for t r e e g r o w t h and e s t i m a t e s of f o r m u l a s for g r o w t h c o m p o n e n t s (red pine growing at 1.2-m spacing, control \ fertilized plots)
V(t ) = F~5=iR(t,t - i ) X ( t - i) + G ( t ) + c A ( t ) C ( t ) + F ( t ) + e(t) R ( t , t - 1) = 0.4794 R ( t , t - i) = R ( t , t - 1) i X ( t ) = V(t)* = V ( t ) - e(t) G ( t ) = 1.0503 * e 0"008629t - 1 A ( t ) = 0.030331 * e 0.055483t c=5 F ( t ) = 0.0844486 * t 1.546281 . e -0.096803t
C ( t ) = a o + a l P l ( t ) + a2P2(t) + . . . +a12P12(t) + b l T l ( t ) + b2T2(t) + . . . +b12T12(t) ( R 2 = 0.56, r e g r e s s i o n is significant w h e n F > F a ( 1 8 , 3 5 ) = 2.47 at a = 0.01) E[e(t)] = 0, V a r [ e ( t ) ] = V a r [ V ( t ) - V(t)* ] V(t) (dm3): a n n u a l v o l u m e i n c r e m e n t . R ( t , t - i): a u t o c o r r e l a t i o n b e t w e e n g r o w t h in y e a r t - i and y e a r t. X ( t ) , V(t)* (dm3): the o p t i m a l e s t i m a t e of a n n u a l v o l u m e i n c r e m e n t in y e a r t. G ( t ) (dm3): intrinsic g r o w t h trend. A ( t ) : c h a n g e a b l e a m p l i t u d e s e m b o d i e d in g r o w t h signals, b r o u g h t u p by tree-size c h a n g e . F ( t ) (din3): function of fertilization effect. c: w e i g h t i n g index. C ( t ) (dm3): a d j u s t e d g r o w t h signals d u e to c l i m a t i c effect a n d d e p e n d e n t v a r i a b l e of a r e g r e s s i o n m o d e l . Pl(t), •. • ,P12(t) (mm): total m o n t h l y p r e c i p i t a t i o n from J a n u a r y to D e c e m b e r . Tl(t), • ' • ,T12(t) (° C): m e a n m o n t h l y t e m p e r a t u r e from J a n u a r y to D e c e m b e r . a o, a l, • • • ,a12, b 1, • • • ,b~2: coefficients of a r e s p o n s e function, c o n v e r t e d from coefficients of a stepwise r e g r e s s i o n b e t w e e n C ( t ) and p r i n c i p a l c o m p o n e n t s of c l i m a t i c variables. e(t): r a n d o m e r r o r s w i t h z e r o m e a n a n d v a r i a n c e [ V ( t ) - V(t)* ].
Y. Pan, D.J. Raynal / Ecological Modelling 82 (1995) 299-312
306
precipitation from 1930 to 1985, collected from eight weather stations in the Pack Forest area were used to study climate-growth relationship. Principal components analysis was first applied to treat 24 climatic variables (including monthly temperature and precipitation). With 97% of the cumulative proportion variance accounted for, 21 principal components (PCs) were chosen from 24 PCs and used in stepwise regression to determine their relationship with growth signals. The regression model with 13 PCs was adopted to simulate the growth due to climate at significance level a = 0.0006 (Table 1) coefficients of the regression model were converted by calculating the principal component scores to an explicable set of coefficients which express growth responses to variations in particular climatic variables. In this study,
the correlation represents relative effect of variations in temperature and precipitation on tree growth for each month of growth period (Fig. 5). The correlation is defined as the growth response function (Fritts, 1976). 4.4. E s t i m a t i n g fertilization effect
The effect of soil fertilization was determined by comparing growth in control and fertilized plots. The mean annual volume increment in the fertilized plot minus that in the control plot year by year was used to study fertilization effects. A general exponential function in the format F(t)
(7)
= a * t b * e (-ct)
was used to fit growth affected by fertilization
Annual growth
A°no, grow
Tree Volume
-X.
............
Temperature
Oct
~.j
Precipitation
Sep
Aug
startFt
Jul
Oct2
Sep2
error2
Aug2
Ct
Jul2
Fig. 7. The structural diagram of S T E L I ~ (Richmond ct al., 1987) for system dynamic model of tree growth. Symbols and variables used in the diagram are interpreted in Table 3.
307
Y. Pan, D.J. Raynal / Ecological Modelling 82 (1995) 299-312 Table 3 Variables in the system dynamic model of tree growth (red pine trees growing in 1.2-m spacing sites at the Pack Forest, NY) State variables Annual growth (dm3): Tree Volume (dm3): FerTreeVL (dm3): Rate variables AV_ increment (dm3/yr): LastYrAVI (dm3/yr): AVI (dm3/yr): Auxiliary variables Vt_ 1 (dm3): Signals (din3): Ct (din3): cAt: Temperature (dm3): Precipitation (din3): error 1: Vt_2 (dm3): Age trend (dm3): lag_growth (dm3): AVIfer (dm3): startFT (dm3): Ft (din3): error2: Table cariables (e.g.) Jan (° C): Jan2 (ram):
annual volume increment of control trees cumulative volume growth of control trees cumulative volume growth of fertilized trees each year's new volume increment of control trees last year's volume increment of control trees each year's new volume increment of fertilized trees annual growth due to extrinsic factors growth signals due to climatic effect adjusted growth signals related to climate without tree-size effect weighted amplitude function reflecting tree-size effect on growth signals growth due to temperature effect growth due to precipitation effect random errors due to measurements annual growth due to intrinsic factors age-related, intrinsic growth trend auto-correlative growth, i.e. lag efffect on current year growth annual volume increment of fertilized trees fertilization effect on growth started at treatment year function of fertilization effect random error due to interactive effect of fertilization and climate monthly mean temperature from 1932-1985 in January monthly total precipitation from 1932-1985 in January
(Fig. 6). U s i n g the n u m e r i c a l a p p r o a c h m e t h o d , the o p t i m a l curve of a g e n e r a l e x p o n e n t i a l function was f o u n d to fit the data. T h e fertilization effect can b e expressed as a t i m e - r e s p o n s i v e patt e r n using a d e t e r m i n i s t i c function. T h e P e a r s o n ' s c o r r e l a t i o n coefficient ( R ) of the fitted curve is 0.73 which indicates a significant c o r r e l a t i o n at the level a = 0.005 ( R > R0.005 = 0.364).
s a m p l e d tree in the stand. T a b l e 2 p r e s e n t s the g e n e r a l e q u a t i o n s for e s t i m a t i n g each t e r m in the tree growth model.
1 vt 1
2:Vt2
3: stadFt
il
4.5. Estimating random errors E ( t ) is the r a n d o m v a r i a n c e in growth data series d u e to m i c r o - e n v i r o n m e n t a l factors a n d m e a s u r e m e n t errors etc. u n i q u e to each tree. It can be e s t i m a t e d simply by s u b t r a c t i n g averaged a n n u a l v o l u m e growth from each s a m p l e d a n n u a l v o l u m e growth series in the stand. So w h e n acc o u n t i n g for the c o n t r i b u t i o n s of each growth t e r m except E ( t ) in the aggregate growth model, the m e a n values of a n n u a l v o l u m e i n c r e m e n t were t a k e n in the e s t i m a t i o n p r o c e d u r e for each
1 oo
14!2s
2;so
40%
s,.o~
Time
Fig. 8. Behavior pattern of the model system. In the figure, Vtl (curve 1) represents extrinsic inputs of the system including growth due to climate and random disturbances. Vt2 (curve 2) shows intrinsic growth including age-related growth and lag growth. The StartFt (curve 3) reflects growth due to fertilization effect.
Y. Pan, D.J. Raynal / Ecological Modelling 82 (1995) 299-312
308 5. S i m u l a t i o n
study using
a n c y is a t t r i b u t a b l e
STELLA
to growth due
to climatic
e f f e c t s , w h i c h is p r o d u c e d b y a r e g r e s s i o n m o d e l . The structural diagram of the STELLA
simu-
The
r e g r e s s i o n m o d e l w e u s e d is b a s e d o n 13
T h e v a r i a b l e s in t h e d i a g r a m a r e i n t e r p r e t e d i n
principal components which account for 71.4% of t h e c u m u l a t i v e v a r i a n c e . S o m e i n f o r m a t i o n is l o s t
T a b l e 3. T h e c o r r e s p o n d i n g m a t h e m a t i c a l m o d e l
by using s t e p w i s e r e g r e s s i o n results. E v e n with a
l a t i o n m o d e l o f t r e e g r o w t h is s h o w n in Fig. 7.
written
in
STELLA
T a b l e 4. T h e
language
STELLA
model
in
regression model combining more principal com-
simulates annual
ponents, the result of the model cannot simulate
is p r e s e n t e d
volume tree growth based on the underlying rela-
exactly the
t i o n s h i p s a n d f u n c t i o n s of t h e variables. Fig. 8
t h e r e m a y exist n o n l i n e a r r e l a t i o n s h i p s b e t w e e n
e x h i b i t s t h e b e h a v i o r p a t t e r n o f t h e m o d e l syst e m . I n Fig. 9, s i m u l a t i o n r e s u l t s w e r e c o m p a r e d
growth
with real annual volume increment data of both control trees and fertilized trees. The simulated
The differences between simulation and real growth of fertilized trees are obvious during the
g r o w t h f o r t r e e s in b o t h s t a n d s is c l o s e t o a c t u a l
effective period of fertilization, because interac-
growth. However, growth fluctuations do not re-
tion effects of fertilization with other factors were
flect closely the actual growth data. The discrep-
i g n o r e d in t h e a g g r e g a t e m o d e l .
and
actual
growth fluctuations because
monthly weather
conditions which
are not expressed by the linear regression model.
Table 4 Equations of STELLA for system dynamic model of tree growth []
[]
D
O O O O O O O O O
O O O
O O
Annual_growth(t) = Annual_growth(t - dt) + (AV_increment - LastYrAVI)* dt INIT Annual_ growth = 0 INFLOWS: AV_increment = Vt_ 1 + Vt_2 OUTFLOWS: LastYrAVI = DELAY(AV_ increment,DT,0) FerTreeVL(t) = FerTreeVL(t - dt) + (AVI) * dt INIT FerTreeVL = 0 INFLOWS: AVI = AVIfer Tree_ Volume(t) = Tree_Volume(t - dt) + (LastYrAVI)* dt INIT Tree_Volume = 0 INFLOWS: LastYrAVI = DELAY(AV_ increment,DT,0) Age _ trend = 1.0503 * EXP(0.008629 * TIME) - 1 AVIfer = AV_increment + startFt + error2 cAt = 5 * 0.030331 * EXP(0.055483 * TIME) Ct = Precipitation + Temperature errorl = (RANDOM(0,1) - 0.5) * 2 * Annual - growth * 0.01 error2 = 0.2 * Ct * RANDOM(0,1) Ft = 0.0844486 * (TIME "1_.546281)* EXP( - 0.096803 * TIME) lag_ growth = 0.835 * Annual_ growth Precipitation = - 0.008724 * Oct2 + 0.004619 * Nov2 - 0.01409 * Dec2 - 0.0478 * Jan2 - 0.02344 * Feb2 - 0.000855 * Mar2 + 0.02098 * Apr2 + 0.0435 * May2 + 0.04407 * Jun2 + 0.04245 * Jul2 - 0.04355 * A u g 2 - 0.01628 * Sep2 Signals = cAt * Ct startFt = DELAY(Ft,7,0) Temperature = 3.726 - 0.008369 * Jan - 0.00965 * Feb + 0.00448 * Mar + 0.005307 * Apr - 0.01659 * May - 0.005882. Jun - 0.01666. Jul - 0.01276 * Aug - 0.009422 * Sep - 0.005844 * Oct + 0.005639 * Nov + 0.009625 * Dec Vt_ 1 = Signals + errorl Vt_2 = A g e t r e n d + lag_growth
Y. Pan, D.J. Raynal / Ecological Modelling 82 (1995) 299-312
6.00" i: AV recreant
~]
Ia)
2: realAVl
3o0
o.oo 1oo
1: AVlfer
14.25
27.50
40.75
27I~)
405S
5400
2: FerDala
2
(b)
100
14~5
S4.d
Time
Fig. 9. Simulated annual volume increment in contrast to real annual volume increment of red pine for: (a) trees in control plot, AV increment (curve 1) is simulated growth and realAVI (curve 2) is actual growth; (b) trees in fertilized plot, AVIfer (curve 1) is simulated growth and FerData (curve 2) is actual growth.
In order to test the effect of extreme weather conditions on tree growth, the actual weather was replaced by hypothetical weather data. Precipitation of May, June and July over 54 years was set with low precipitation (0.5 inch) in every 10 years. The annual volume increments in the end year 1985 is 0.96 dm 3 less and cumulative volume is 16.0 dm 3 less than that of trees under actual precipitation (Fig. 10). The model verifies that drought during the growing season can affect total tree volume. Even though the hypothetical weather conditions do not change annual volume increment of the tree very much, the dynamic system model no doubt can serve effectively as a simulation tool to evaluate different types of changes in weather variables and their effects on tree growth.
6. Discussion
The system dynamic model of tree growth provides both a conceptual framework and means to
309
characterize and predict tree growth responses to environmental stimuli. An assumption in the model is that growth can be subdivided into several distinct components, each of which has some unique properties and responds to different factors that influence growth. The quantitative expression of the system links processes together in a predictive and testable format and allows the investigator to track growth behavior through time. With knowledge of growth in an initial year and climatic conditions over time, growth in subsequent years can be simulated based on system feedback, intrinsic biological characteristics and climate-growth relationships. Although the dynamic model is based on determinate relationships and feedback between components of the system, all parameters of the model were estimated using available experimental data and statistical methods. The approach taken involves a procedure starting with the estimation and removal of autocorrelation growth
119.78
1: A V tncrernenl
2: T o e V o l u ~
/' 59 89
(a)
ooc
,1 1 co
10371
, 27.50
1425
1: AV increment
2: T ~
i 4075
54.0~
Vo~urr~
A/ 2,5
v ~ ~
(b)
2
.J'~, I
51 86
,-
0(30 1 0 0
~
1425
,
2750
,
4075
,
5400
Time
Fig. 10. Simulated annual volume increment and cumulative volume of red pine trees (a) under normal climate; (b) under hypothetical low precipitation (0.5 inch) during growing season at time = 10, 20, 30, 40, 50. AV increment (curve 1) is annual volume increment and Tree Volume (curve 2) is cumulative volume.
310
Y. Pan, D.J. Raynal / Ecological Modelling 82 (1995) 299-312
within the data, followed by estimations of growth trend, climatic signals, random error, and finally fertilization effects. The relationships determined by statistical methods will restrict application of the model to some extent. For example, there are limits in stand conditions such as spacing of plantation trees, land use history and soil nutrient availability, all of which affect growth of trees in the stand. Time limitation is another problem with the model. The behavior of age-related trend changes in different stages of tree development. In our problem, the sample trees were only 55 years old, so the age trend of annual volume increment is presented as a positive exponential function. This case will affect the accuracy of growth prediction for tree development stage beyond the behavior described by the age trend, when the trend would be more likely logistic than exponential. This problem could be solved by obtaining data of older trees and evaluating the pattern of growth behavior over the whole lifecycle of the species. In addition, tree age may also mirror the sensitivity of growth to climate reflected by the response function. Variable growth behavior of different tree species is obviously another factor limiting model application. The term response function (Fig. 5) refers to coefficients used to estimate growth based on climate variables (Fritts, 1976). The set of coefficients are elements of the response function which can be interpreted directly in terms of monthly temperature or monthly precipitation. With approximate 95% confidence intervals estimated for coefficients of response function, we can be confident that the true coefficients between annual growth and monthly climatic variables lie within the intervals. The response function can be interpreted as showing a significant direct relationship between growth and precipitation during May to July and inverse relationship between growth and precipitation in January and August. Temperature in the previous January and August of the current season have an inverse relationship with growth, especially in August. The results indicate that for annual volume growth the precipitation from May to July is most important probably because during the time cambium division and
cell enlargement are most active. The cause of the possible negative effect on growth from precipitation in August is not certain. January precipitation always accompanies snow storms and could damage shoots and affect tree growth in the following year (Kozlowski, 1971). Temperature in August has an inverse relationship with growth, implying that high temperature causes a higher rate of respiration and carbohydrate catabolism. Actually, high temperature during the growing season (May-August) has an inverse effect on growth, but is most significant in August. High temperature in January inversely affects growth, and may result from stimulation of stored carbohydrate use, thus affecting subsequent growth. The inverse relationship between growth and January and February temperature also im4..5
fertilization vs, control growth
• fertilization * control
3.5 3
'~
2.5 2 1.5
(a)
I 1.g 1,8
ferffilzatlon function
1.7 1.8 1.5 1.4 1,3 1.2
~
(b)
~
y = d + a * X ^ b*exp(-c*X)
o.g 0.8 0.7 O.e 0.5 0.4 0.3 0.2
o.~
......................
i_i=.i_
'l'l'lb~'l'l'l'lb~'l+l'l'lb~+l'l'l'lb~'l'l'l'lb'as
1940
lg50
lg00
years
lg70
lg80
Fig. 11. Fertilization effect on tree ring growth (at DBH) of red pine. (a) Tree-ring growth in fertilized plot in contrast to that in control plot. (b) A general exponential function used to fit the effect of fertilization on ring growth. The function shows that growth response of ring width to fertilization is sharper and shorter than that of annual volume increments.
Y Pan, D.J. Raynal/Ecological Modelling 82 (1995) 299-312
311
plies that low t e m p e r a t u r e injury in winter may occur before January. Growth due to fertilization effects shows marked fluctuations (Fig. 6). This indicates there exists some interactive effect of fertilization with climatic factors. For annual volume increment, the fertilization effect on growth is not like that reflected in indices of tree rings or specific volume (Fig. 11), where the stimulation of growth was very strong in the immediate years after the treatment but declined quickly afterwards. Fertilization effects on growth can be divided into direct effects, which reflect annual ability of fertilized trees to produce wood per unit area of growth surface; and residual effects, which are due to the fact that larger trees grow faster (Comerford et al., 1980). Growth indices such as ring widths are measures of the direct effect of fertilization. Annual volume increment, on other hand, is a measure combining both the direct and residual effects of fertilization since this index depends on both tree size and growth per unit cambium. If fertilization increases the size of tree relative to an unfertilized tree, subsequent equivalent radial growth on both trees will result in a greater amount of absolute growth on the larger tree. The fertilization effect on annual volume increment achieved its maximum value at about 15 years after fertilization which is much later than that in tree rings. The response arises from the residual effect of fertilization.
or conditions. Such a model can aid in evaluation of tree growth and provide a basis to predict future growth. The simulation results from the dynamic model closely reflect actual growth and show that the model is very reliable for estimating growth of red pine trees ( < 55 year-old). However, since the limits of p a r a m e t e r s of the system are determined by experimental data and statistics, the model will require further development if it is to be used to characterize the growth of older trees, trees growing in different site conditions, and for different tree species.
7. Conclusion
References
The dynamic system model and its related aggregate equation developed in this study provide a useful approach for reconstructing and estimating tree growth. The method of reducing growth into several components and then quantifying these components not only provides a concrete approach for studying growth, but also reveals insight into the biological process of tree growth. The S T E L L A dynamic model serves as a good tool for simulation trials. Using the model tree growth can be simulated over time and examined in relation to prevailing environmental conditions and to extreme environmental events
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Acknowledgements We are grateful to Drs. C. Hall, C. Davis and H. T e p p e r in S U N Y - E S F for their reviews and helpful comments on this study, and to Dr. G. Jacoby for providing facilities at the Tree-Ring Lab, Lamont-Doherty Earth Observatory, Columbia University. This research was part of the Ph.D. thesis research of the senior author at College of Environmental Science and Forestry, State University of New York at Syracuse and was supported in part by the McIntire-Stennis Cooperative Forestry Research Program, New York State Energy Research and Development Authority, and Empire State Electrical Energy Research Corporation, and the New York State Electric and Gas Corporation.
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