Deriving tree growth models from stand models based on the self-thinning rule of Chinese fir plantations

Abstract

Self-thinning due to density-dependent mortality usually occurs during the forest development. To improve predictions of such processes during forest successions under climate change, reliable stand-level models are needed. In this study, we developed an integrated system of tree- and stand-level models by deriving tree diameter and survival models from stand growth and survival models based on climate-sensitive self-thinning rule of Chinese fir plantations in subtropical China. The resulting integrated system, having a unified mathematical structure, should provide consistent estimates at both tree and stand levels. Predictions were reasonable at both stand and tree levels. Because stand-level values aggregated from the tree model outputs are different from those predicted directly from the stand models, the disaggregation approach was applied to provide numerical consistency between models of different resolutions. Compared to the unadjusted approach, predictions from the disaggregation approach were slightly worse for tree survival but slightly better for tree diameter. Because the stand models were developed under the climate-sensitive self-thinning trajectory, the integrated system could offer reasonable predictions that could aid in managing Chinese fir plantations under climate change.

Introduction

The self-thinning rule, which describes the relationship between stand density and the average tree size in a situation of density-dependent mortality, continues to attract foresters’ attention ([6]). Variables such as tree biomass ([42]), quadratic mean diameter ([45]), and average height ([7]) have been used to describe tree size in the self-thinning rule. Reineke ([31]) proposed the slope in the self-thinning line (or maximum size-density line) was a constant with value of -1.605. However, the existence of a constant slope has been widely debated ([29], [17]). Several reports argued that the slope may not be universally invariant, changing due to tree species ([12]), initial spacing ([38]), site quality ([5]), and climate conditions ([46]).

Because the self-thinning rule refers to density-dependent mortality, it has been used as the basis for developing stand management diagrams ([38], [35]), constructing density indices ([39]), and especially for predicting forest growth and stand survival ([25]). Note that in this study, stand survival is defined as number of surviving trees per ha.

Forest growth and yield models provide an important basis for managing forest reasonably. These models are categorized into four kinds of models: whole-stand models, size-class models, diameter distribution models, and individual tree-models ([8]). Whole-stand models offer information on the whole stand such as stand basal area ([21]), stand survival ([34]), and stand volume ([20]). Size-class models deal with trees classified into diameter classes. These models include stand-table projection models, which predict the frequency in each diameter class ([26], [1]), and diameter-distribution models, which use a probability density function to model the diameter distributions ([11]). Individual-tree models, on the other hand, provide detailed tree information such as tree diameter growth ([36]), tree survival ([24]), or both ([23]).

In general, for unthinned stands whole-stand models provide more accurate stand-level predictions than individual-tree models, because stand-level predictions aggregated from tree models usually lead to accumulating errors in plantations ([30]). For the same site, a forest manager might prefer stand-level models to predict stand attributes, but might need tree models to aid management decisions that require detailed tree-level information. Because stand-level attributes predicted from tree and stand-level models are numerically inconsistent with each other, disaggregation has been employed as a method for maintaining compatibility between the two types of models ([32]). This method attempts to adjust stand-level variable predictions obtained from individual-tree models such that the aggregated predictions would match the outputs of whole-stand model ([32]).

Chinese fir (Cunninghamia lanceolata (Lamb.) Hook.) is a native coniferous tree species widespread in southern China. It is commonly used for timber production because its straight stem and good resistance to bending and cracking ([47]). The planting area for this species is about 8.95 million hectares, about 30% of all afforestation in China ([22]). Zhang et al. ([45]) used the segmented regression technique to develop the self-thinning model of Chinese fir plantations. The slopes in the self-thinning lines were found to be variable from site to site, and could be predicted by the use of several climate factors ([46]). Based on this work, Zhang et al. ([47]) developed a system of stand basal area, quadratic mean diameter growth, and survival models according to the self-thinning rule. They found that the stand models under the climate-sensitive self-thinning trajectories provided reasonable predictions under climate change.

Daniels & Burkhart ([13]) put forward a concept of the integrated system of forest growth models, all of which having a unified mathematical structure, so as to provide consistent estimates at different level models. This concept has been applied by Cao ([9]) to develop an integrated system to predict stand survival at both tree and stand levels. Cao ([10]) later extended this technique to derive a tree survival model from any existing stand survival models. Thus, it is necessary to derive the tree growth and survival models from the stand models which will improve model predictions and compatibility between stand- and tree-level models.

The objective of this study was to establish an integrated system by deriving individual-tree diameter growth and survival models from stand-level growth and survival models for Chinese fir plantations. The benefits of this integrated system consist of stand-level models that follow the self-thinning rule, and correspond to tree-level models that are flexible for detailed tree growth projection, tree merchandizing, and also for simulation purposes. Because the stand models were climate-sensitive, the integrated system with the above properties should provide reasonable growth and yield predictions that could aid future management of Chinese fir under climate change.

Materials and methods

Study sites and data measurement

The experimental sites were situated in Fujian, Jiangxi, Guangxi, and Sichuan provinces in southern China. Fujian, Jiangxi, and Sichuan provinces belong to middle-subtropical climate zones, whereas Guangxi has southern-subtropical climate (Tab. 1, Fig. 1). The landforms at the four sites are low mountains and high hills, with elevation ranging from 300 to 500 m a.s.l. Parent rocks in Fujian and Guangxi are Granite. In Jiangxi and Sichuan, parent rocks are sandy shale and shale, respectively. The soil type is mainly Laterite for Fujian, Guangxi, and Sichuan, and is mainly yellow-brown for Jiangxi.

Tab. 1 - Mean annual temperature (MAT), annual precipitation (AP), degree-days below 0° C (DD0), summer mean maximum temperature (SMMT), and winter mean minimum temperature (WMMT) of study period, by site.

Site	MAT (°C)	AP (mm)	DD0	SMMT (°C)	WMMT (°C)
Fujian	18.90	1768	1	32.18	4.75
Jiangxi	18.04	1572	2	32.01	4.23
Guangxi	22.27	1494	0	31.79	12.24
Sichuan	18.31	1179	1	30.69	7.09

Fig. 1 - Study sites located in subtropical climate zones in this study.

The plots were planted in 1982 in Fujian, Guangxi and Sichuan, and in the spring of 1981 in Jiangxi. Each site was established with four planting densities: 2 × 1.5 m (3333 trees ha^-1), 2 × 1 m (5000 trees ha^-1), 1 × 1.5 m (6667 trees ha^-1), and 1 × 1 m (10,000 trees ha^-1). The experiments were installed in a random block arrangement with three replications for each level. Four plots were distributed in each block (one plot for each spacing trial). In each site, twelve plots (20 × 30 m each) were established. Surrounding each plot there were two rows of trees with the same spacing, which formed a buffer zone.

In each plot, diameters at a height of 1.3 m (dbh) of all trees were measured, and over 50 trees were randomly chosen for measuring height. Dominant height was computed as the average height of the six tallest trees in the sample. In 1998, a snowstorm in Jiangxi killed some trees, and therefore the data after 1999 for this site were dropped from this study. The study period ranged from 1985 to 2010 in Fujian, 1985 to 1999 in Jiangxi, 1990 to 2009 in Guangxi, and 1985 to 2013 in Sichuan. Field measurement was done in the winter every 1 to 3 years. Tab. 2 showed the summary statistics of stand and tree factors by replication. Fig. 2 showed the number of trees per ha over time.

Tab. 2 - Summary statistics (mean ± standard deviation) of stand- and tree-level variables of Chinese fir plantations from 1985 to 2013, by replication.

Variable	Replication 1	Replication 2	Replication 3
A: stand age (years)	13 ± 6.2	12 ± 6.0	12 ± 6.0
N: number of trees ha-1	6159 ± 2473	6120 ± 2495	6080 ± 2452
B: stand basal area (m2 ha-1)	36.12 ± 16.32	33.89 ± 16.81	36.94 ± 15.34
Q: quadratic mean diameter (cm)	9.0 ± 3.0	8.7 ± 3.2	9.2 ± 3.2
H: stand dominant height (m)	11.1 ± 3.9	11.2 ± 4.5	11.4 ± 4.1
d: tree diameter at breast height (cm)	8.7 ± 3.8	8.4 ± 3.9	8.9 ± 4.0

Fig. 2 - Number of trees per ha (N) over time by replication. Fujian: every year from 1985 to 1990; every two years from 1990 to 2010; Jiangxi: every year from 1985 to 1989; every two years from 1989 to 1999; Guangxi: every year from 1990 to 1995; every two years from age 1995 to 2009; Sichuan: every year from 1985 to 1995; every two years from 1995 to 1999 and from 2002 to 2010; every three years from 1999 to 2002 and 2010 to 2013.

Results and discussions

Integrated system of tree- and stand-level models

Tab. 3showed the parameter estimates of the growth and survival models in both stand and tree levels, which showed that all parameters of these models were significant at α = 0.05. Outputs from the stand-level models yielded consistently higher values of R² and smaller values of MAE as compared to aggregated outputs from the unadjusted tree-level models (Tab. 4). Direct prediction of stand variables apparently was preferable to aggregating tree-level outputs. This result is consistent with the findings by Qin & Cao ([30]) who predicted stand basal area, survival, and volume of loblolly pine (Pinus taeda Linn.), Zhang et al. ([43]) who predicted stand basal area and survival of Chinese pine (Pinus tabuliformis Carr.) in China, and Hevia et al. ([19]) who predicted stand survival and basal area of birch-dominated stands in Spain. García ([16]) reported that problems of accumulation of errors tend to be more serious when aggregating tree-level outputs.

Tab. 3 - Parameter estimates (± standard errors) of the stand- and tree-level growth and survival models by group. All parameters were significant at the 0.05 level. (Group 1): not containing replication 1; (Group 2): not containing replication 2; (Group 3): not containing replication 3.

Models	Parameter	Group 1	Group 2	Group 3
Stand-level models	α1	5.8134 ± 0.6049	5.5446 ± 0.5935	5.5420 ± 0.5735
	α2	-0.4423 ± 0.0686	-0.4052 ± 0.0672	-0.4158 ± 0.0653
	α3	-1.9376 ± 0.2896	-2.7626 ± 0.4705	-1.8974 ± 0.2831
	χ0	5.1654 ± 0.9406	5.1133 ± 0.8885	6.8275 ± 0.8285
	χ 1	0.0654 ± 0.0102	0.0539 ± 0.0096	0.0618 ± 0.0091
	χ 2	0.0469 ± 0.014	0.0664 ± 0.0156	0.0843 ± 0.0153
	χ 3	-0.0809 ± 0.0303	-0.0939 ± 0.0311	-0.1506 ± 0.0315
	χ 4	-0.3693 ± 0.0929	-0.3701 ± 0.0855	-0.5384 ± 0.0816
Tree-level models	δ1	-0.5005 ± 0.0068	-0.5028 ± 0.0071	-0.4742 ± 0.0068
	δ2	0.9963 ± 0.0309	1.2017 ± 0.0434	0.9410 ± 0.0337
	δ3	0.9138 ± 0.0123	0.8329 ± 0.0145	0.9341 ± 0.0144
	χ 5	0.9989 ± 0.0008	1.0099 ± 0.0008	1.0011 ± 0.0008

Tab. 4 - Evaluation statistics for prediction of stand variables from stand-level models and tree-level models (before adjustment). Numbers in italic denote the best statistics for that variable.

Variable	Type of model	Mean error (ME)	Mean absolute error (MAE)	Fit index (R2)
N: Number of trees ha-1	Stand-level	3.5358	181.646	0.9836
	Tree-level	-56.1225	206.089	0.9762
B: Stand basal area (m2 ha-1)	Stand-level	1.7019	3.7775	0.8915
	Tree-level	0.5531	4.3241	0.8171
Q: Quadratic mean diameter (cm)	Stand-level	0.2503	0.4835	0.9633
	Tree-level	0.2437	0.4854	0.9632

A logical step would be to adjust the tree-level outputs to match the aggregated values with outputs from stand-level models. For many years, the disaggregation method has been used to link tree- and stand-level models. It not only ensured compatibility regarding stand-level outputs, but also yielded better tree-level predictions than the unadjusted approach in former studies ([19]). Qin & Cao ([30]) found that the success of the disaggregated tree-level predictions depends largely on the precision of the stand-level model.

Tab. 5shows that the tree-level diameter and survival models derived from the stand growth and survival models performed well. The disaggregated method for tree diameter produced a larger value of R² (0.9468 vs. 0.9434) and a lower value of MAE (0.5528 vs. 0.6017) than did the unadjusted method. This result was supported by Qin & Cao ([30]) for tree diameter prediction and Hevia et al. ([19]) for tree basal area prediction. Conversely, the disaggregation method for tree survival yielded lower AUC value (0.8736 vs. 0.8907) and larger MAE value (0.0791 vs. 0.0745) as compared to the unadjusted method (Tab. 5). The result of tree survival prediction was contrary to findings from previous studies, which reported superior performance of the disaggregation method ([44]). The different findings may be due to the ecological differences among the studied species. However, the above results show similar overall performances between the adjusted and unadjusted tree models in terms of predicting tree attributes. The advantage of disaggregation in this study lies in achieving numerical consistency between tree- and stand-level models in predicting stand attributes.

Tab. 5 - Model evaluation statistics of tree diameter growth and survival probabilities models using the three-fold cross-validation by method.

Method	Tree diameter			Tree survival probability
	ME	MAE	R2	ME	MAE	AUC
Unadjusted	0.1521	0.6017	0.9434	-0.0011	0.0745	0.8907
Disaggregated	0.3097	0.5528	0.9468	0.0068	0.0791	0.8736

Daniels & Burkhart ([13]) put forward the concept of an integrated system of unified mathematical structure, which can predict forest growth at any level of resolution. The system developed in this study can consistently predict growth and survival at both tree and stand levels. In addition, because the stand models were based on the climate-sensitive self-thinning rule, the tree-level diameter and survival models derived from these stand models should provide reasonable predictions under climate change.

Model predictions under the self-thinning rule

Fig. 3displays the change of tree survival probability over time for various diameter percentiles. This figure is based on the plot located in the Fujian province. In this plot, a tree at each 5% diameter percentile at age 5 was grown to age 30. The graph shows that stand survival rate is equal to tree survival probability at approximately the 20% diameter percentile. A similar graph was drawn for tree survival basal area in Fig. 4. Change in stand basal area over time is therefore a result of two opposing trends: quadratic mean diameter always increases with time, whereas number of trees is unchanged at first, then begins to decrease with time, and finally follows the self-thinning line. Stand basal area, which is a function of the product of N and Q², increases with time, then stabilizes, and finally decreases (Fig. 4). Surviving tree basal area (computed as survival probability × tree basal area) behaves differently for different diameter sizes; it increases with time for large diameters, or increases, stabilizes, then decreases with time for small diameters.

Fig. 3 - Change of tree survival probability over time (thin curve) for various diameter percentiles, from 0% to 100%, by 5% intervals for a plot in the Fujian province. The lowest thin curve corresponds to the 0% diameter percentile, and the top curve corresponds to the 100% diameter percentile. The dark curve denotes stand survival rate.

Fig. 4 - Change of tree survival basal area over time (thin curve) for various diameter percentiles, from 0% to 100%, by 5% intervals for a plot. The lowest curve corresponds to the 0% diameter percentile. Tree survival basal area is the product of survival probability and basal area of that tree. The dark curve shows change of stand basal area over time.

Conclusions

In this study, we developed an integrated system of tree- and stand-level models by deriving tree diameter and survival models from stand growth and survival models. Predictions were reasonable at both stand and tree levels. In addition, the disaggregation approach was applied to provide numerical consistency between models of different resolutions. Compared to the unadjusted approach, predictions from the disaggregation approach were slightly worse for tree survival but slightly better for tree diameter. The advantage of disaggregation in this study lies in achieving numerical consistency between tree- and stand-level models in predicting stand attributes. The stand-level models, based on the self-thinning rule, were compatible with the underlying biological principles. Because they also included climate variables, these stand models and their derived tree-level models formed an integrated system that could offer reasonable predictions for scenarios beyond the data range. Future applications, including those which simulate various climate scenarios, that take advantage of this integrated system could play an important role in managing Chinese fir plantations under climate change.

Acknowledgements

XQ analyzed, conceived, and wrote the draft; QV analyzed, reviewed, and improved the draft; YC analyzed the data; JG conceived the study.

Funding

The study was supported by the National Key Research and Development Program of China (2021YFD2201304) and the National Natural Science Foundation of China (no. 31971645). Partial support for data analysis was received from the National Institute of Food and Agriculture, U.S. Department of Agriculture, Mclntire-Stennis project LAB94379. The authors thank Dr. Aiguo Duan for the field work.

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