# Evaluating Stand Density Measures for Regulating Mid-Rotation Loblolly Pine Plantation Density in the Western Gulf, USA

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## Abstract

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## 1. Introduction

^{2}${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$, 500 (45% of the maximum) SDI [3], 0.20 to 0.25 RS, or $\le $35% CR, it is the biological time to thin. Do these measures lead to a similar time to thin for a given stand? The results from the few published studies on this topic are varied, with some studies supporting consistencies [8,12,15] but others finding inconsistencies between the measures [14]. Operationally, foresters may employ simple criteria such as using a combination of stand age and market criteria that thin loblolly pine stands around the age of 15 years since by that age virtually all trees have reached a merchantable size (e.g., 15 cm in DBH and 12 m in height). Given the wide use of these measures in pine density management, in particular, in determining thin time, understanding the relationships between measures and their relevance to the operational criteria would greatly help foresters in understanding stand development post-thinning.

## 2. Data and Methods

^{−1}, ranging from 897 to 3367 trees ha

^{−1}. At a base age of 25 years, the site index (SI) ranged from 9 to 28 m, which was calculated following Coble and Lee [18]. Most plots were measured for about 8 cycles, with the longest ones being measured for 12 cycles (10 plots). For the last measurement, the plots averaged 26 years old, ranging from 12 to 45 years. Table 1 lists detailed summary statistics.

^{2}per ha). SDI (in trees per ha) [2] and RS were calculated as follows:

^{2}) and the root mean square error (RMSE) was selected. R

^{2}and RMSE were calculated as follows:

_{i}” is used to refer to model parameters (See Table 2). The use of the same “b

_{i}” in models throughout the paper does not represent the mathematical equivalence unless clearly specified. For example, while b

_{0}and b

_{1}are used in different equations, they do not imply the same biological interpretation.

_{0}× ${{\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}}^{{b}_{1}}$) which was selected to model the RS–${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$ relationship as an example (Table 2). b

_{0}and b

_{1}are model parameters, with b

_{0}being a scaling factor and b

_{1}representing the percentage change in RS resulting from a 1% increase in ${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$. IPT and SI were incorporated into the model via, respectively, expressing b

_{0}and b

_{1}as follows:

_{0}= b

_{00}+ b

_{01}× SI + b

_{02}× M

_{1}= b

_{10}+ b

_{11}× SI + b

_{12}× M

_{00}and b

_{10}are the fixed-effect parameters for the reference sites of N and SI = 0 m, b

_{02}and b

_{12}are the differences of the D sites with the N sites, and b

_{01}and b

_{11}represent the changes in b

_{00}and b

_{10}with increasing SI. Under the full model, we found that a random plot-to-plot variation in b

_{0}and a power variance function structure were suitable based on log-likelihood tests. Supplementary Table S1 provides details for the selected random structures by model. After selecting the random structures, the significance of b

_{01}, b

_{11}, b

_{02}, and b

_{12}(hypothesis: estimate differs significantly from zero) was tested using the partial F-test at an $\alpha $ = 0.05, unless otherwise specified. The insignificant variables were removed one by one, until all parameters were significant, which was reported as the final model (Table 2). Supplementary Table S2 provides the parameter estimates for adjusting significant SI and dense effects by model. The model performance was evaluated using R

^{2}and RMSE and also the relationships between observed and predicted values, which are included in Supplementary Figure S1. Model assumptions (e.g., normality, equal variance, and independence) were visually checked using residual plots and were adjusted using weight functions and autoregression order 1 when required (S1). Package R was used for data analysis [22]. To show the effects of SI and density, predictions for sites with an SI of 18.282 (=60 ft) and 24.384 m (=80 ft) (SI

_{18}and SI

_{24}, representing poor and good sites, respectively) paired with normal (N) and dense (D) densities by respective models were presented.

## 3. Results

#### 3.1. Density—Age Relationships

^{2}= 0.93, RMSE = 5.0 m

^{2}ha

^{−1}(21.7%)). Starting at age 8, which marked the initiating crown closure of most loblolly pine plots in this study, ${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$ initially increased rapidly with stand aging, gradually approached a maximum asymptote, and leveled off as the stands further grew older (Figure 1a). This relationship varied significantly with SI and IPT, yet the former had more influence than the latter. Both asymptote (b

_{0}) and rate (b

_{1}) increased with SI, leading to a larger ${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$ at any given age and sooner approach to the asymptote for the higher SI sites. The D sites had a larger b

_{1}than the N sites, resulting in a larger ${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$, but this difference reduced gradually with stand aging and eventually disappeared. At age 15, ${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$ varied greatly from 21 (N + SI

_{18}sites) to 33 m

^{2}(D + SI

_{24}sites). Similarly, the Chapman–Richards function was the best fit for the SDI–age relationship (Table 2; Figure 1b). Sites with higher SI or higher IPT greatly increased both asymptote (b

_{0}) and rate (b

_{1}), making the differences in SDI practically significant between SI18 and SI24 at a given IPT or between D and N sites at a given SI. At age 15, the SDI value ranged from 490 (N + SI

_{18}) to 750 (D + SI

_{24}). For both the ${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$–age and SDI–age relationships, SI and IPT did not affect the model parameter b

_{2}, the instantaneous rate of growth in the inflection point.

^{2}= 0.92, RMSE = 0.12 (28.50%)). Between ages 8 and 35 years, RS dropped rapidly with increasing stand age until about age 15, approached a low asymptote thereafter, and then remained relatively constant, forming an inverse “J” curve (Figure 1c). Sites with higher SI had a smaller b

_{1}(decreasing faster) while D sites, in addition to a smaller b

_{1}, had a larger b

_{0}than L sites. Therefore, sites with higher SI consistently had smaller RS and so were N sites compared to D sites. At age 15, the corresponding RS values were 0.11, 0.16, 0.20, and 0.28 for D + SI

_{24}, N + SI

_{24}, D + SI

_{18}, and N + SI

_{18}sites, respectively.

^{2}= 0.91 and RMSE = 0.08 (16.32%)). Starting at age 8, CR displayed a gradual and constant decline as age increased (Figure 1d). SI significantly affected model parameter b

_{1}, with the value being larger for sites of higher SI, while IPT did not show significant effects on either model parameter. Consequently, trees growing at higher SI sites had lower CR at any given age. At age 15, the CR were 0.55 and 0.49 for SI

_{18}and SI

_{24}, respectively.

#### 3.2. PAI_{D}–Density Relationships

^{2}= 0.89, RMSE = 0.23 cm year

^{−1}(28.05%)). Starting from the onset of crown closure (${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$ = 10 m

^{2}), the ${\mathrm{P}\mathrm{A}\mathrm{I}}_{\mathrm{D}}\text{}$decreased quickly with increasing ${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$ to around 20 m

^{2}and thereafter gradually approached a minimum asymptote (Figure 2a). Both b

_{0}and b

_{1}decreased significantly with increasing SI, resulting in larger ${\mathrm{P}\mathrm{A}\mathrm{I}}_{\mathrm{D}}$ for higher SI sites, although this difference decreased with increasing ${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$. Between D and L sites, no significant difference was found for each model parameter. At a ${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$ of 25 m

^{2}, the ${\mathrm{P}\mathrm{A}\mathrm{I}}_{\mathrm{D}}\text{}$was 0.50 for SI

_{18}sites but increased to over 0.80 cm year

^{−1}for the SI

_{24}sites. Similarly, the Farazdaghi function fitted the ${\mathrm{P}\mathrm{A}\mathrm{I}}_{\mathrm{D}}$–SDI relationship well (Table 2; R

^{2}= 0.89, RMSE = 0.27 (32.92%)). SI negatively affected b

_{0}and b

_{2}and D sites had a smaller b

_{2}than N sites (Table 2), leading to greater ${\mathrm{P}\mathrm{A}\mathrm{I}}_{\mathrm{D}}$ at higher SI sites and also in D sites compared to N sites. Even so, the actual difference in ${\mathrm{P}\mathrm{A}\mathrm{I}}_{\mathrm{D}}$ between D and N sites was small regardless of the SDI level, having no practical significance. The difference in ${\mathrm{P}\mathrm{A}\mathrm{I}}_{\mathrm{D}}$ between SI24 and SI18 was large when SDI was low (e.g., =300) but decreased quickly with increasing SDI (Figure 2b). At an $\mathrm{S}\mathrm{D}\mathrm{I}$ of 500, the ${\mathrm{P}\mathrm{A}\mathrm{I}}_{\mathrm{D}}\text{}$was above 0.71 for SI

_{18}sites but increased to over 1.20 cm year

^{−1}for the SI

_{24}sites.

^{2}= 0.66, RMSE = 0.32 cm year

^{−1}(36.58%)). Increasing SI significantly reduced intercept (b

_{0}) but enhanced slope (b

_{1}) estimates while the D sites had a significantly larger b

_{0}than the N sites. The differences between SI18 and SI24 sites were small when the RS was low, but increased quickly with increasing RS. An RS of 0.2 corresponded to a ${\mathrm{P}\mathrm{A}\mathrm{I}}_{\mathrm{D}}$ of 0.48, 0.54, 0.78, and 0.84 cm yr

^{−1}for the N + SI18, D + SI18, N + SI24 and D + SI24, respectively (Figure 2c).

^{2}= 0.94, RMSE=0.23 cm year

^{−1}(28.05%)). With CR decreasing from 0.70 (around age 8) to 0.20 (around age 30), the ${\mathrm{P}\mathrm{A}\mathrm{I}}_{\mathrm{D}}\text{}$gradually decreased in a nonlinear pattern (Figure 2d). Increasing SI significantly increased b

_{0}only while IPT did not show a significant impact on model parameters. Therefore, at a given CR, the sites with higher SI had larger ${\mathrm{P}\mathrm{A}\mathrm{I}}_{\mathrm{D}}$, but this advantage decreased quickly with reducing CR. At a CR of 0.4, the ${\mathrm{P}\mathrm{A}\mathrm{I}}_{\mathrm{D}}$ was 0.50 for SI

_{18}sites and 0.65 cm year

^{−1}for SI

_{24}sites.

#### 3.3. Between-Measures Relationships

^{2}= 0.95, RMSE = 0.08 (=4.88%)). Both SI and IPT significantly affected the intercept (b

_{0}, which increased with increasing SI or was greater for D sites than N sites). SI negatively affected the slope (b

_{1}) (Figure 3a), and, therefore, SDI increased slower with increasing ${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$ at higher SI sites. Sites with greater SI had larger SDI at a given$\text{}{\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$, as did the D sites compared to the L sites, although the differences were negligible from a practical viewpoint. At a ${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$ of 25 m

^{2}, the corresponding SDI ranged from 551 to 593. Given the strong linear relationship of SDI–${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$, the relationships of SDI with CR and RS are not presented here.

^{2}= 0.94, RMSE = 0.05 (18.52%)). RS decreased rapidly when ${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$ increased from 5 to 15 m

^{2}and then decreased gradually with further increasing ${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$ to 40 m

^{2}(Figure 3b). SI and IPT were found only to affect model b

_{0}, which increased with increasing SI and was smaller for D sites than N sites, leading to slightly smaller RS for sites with higher IPT and greater SI. Regardless of statistical significance, the practical influence of changing SI or planting density was negligible. At a ${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$ of 25 m

^{2}, the RS was similar, being around 0.21 for the combinations of IPT and SI.

^{2}= 0.93, RMSE = 0.09 (18.00%)). When ${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$ increased from 5 to 40 m

^{2}, CR decreased in an approximately linear pattern (Figure 3c). The model parameters varied significantly with SI in that higher SI sites had larger b

_{0}but smaller b

_{1}, leading to greater CR at any given ${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$ for higher SI sites. No significant differences in both parameters were found between D and N sites (Figure 3c). At a ${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$ of 25 m

^{2}, the CR were 0.46 and 0.57 for the SI

_{18}and SI

_{24}sites, respectively.

^{2}= 0.93, RMSE = 0.08 (16.00%)). CR decreased slowly when RS changed from 0.70 to around 0.50, thereafter decreasing quickly to 0.15 at RS = 0.10. Both SI and IPT affected b

_{0}significantly in that b

_{0}decreased with increasing SI and planting density, leading to a substantially smaller CR for D + SI

_{18}than the other combinations (Figure 3d). It could also be seen that the differences between the combinations were small when RS was either low (e.g., RS = 0.1) or high (e.g., RS = 0.7; Figure 3d). When RS = 0.2, the CR ranged from 0.41 to 0.52 other than N + SI

_{18}, which had 0.24.

## 4. Discussion

^{2}${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$ [23], 45% of maximum SDI [3], or 0.20–0.25 RS [24]). We focus on this period in the following discussion.

_{D}, as shown in Figure 2. This finding further supports the linear relationship between these two variables. The consistent trend highlights how as one increases, the other tends to follow suit, reinforcing the interconnectedness of these metrics in the context of forest management. SDI was calculated using a “b” value of 1.605, although some studies suggest that a value of 1.505 may be more appropriate for loblolly pine in the Western Gulf region [13]. However, changing the “b” value to 1.505 did not affect the SDI-related relationships observed in this study. The relationship between ${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$ and RS was found in a reverse J shape, which is parallel to that reported by [11]. While CR cannot be easily converted to ${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$ or RS by a simple transformation since they indicate density differently, CR showed an approximately linear relationship with ${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$ and a decreasing concave curve with RS. Similar relationships between CR and SDI [14] and between CR and RS [8] to those of this study were reported in loblolly pine elsewhere. Based on the relationships observed in this study, a loblolly pine plantation with a ${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$ of 25 m

^{2}corresponded with an SDI from 557 to 590, an RS of around 0.21, and a CR from 0.46 to 0.57, depending on SI. When the plantations reached an average CR of 0.40, the values of RS ranged from 0.12 to 0.21 [8], which are roughly comparable to the corresponding values of RS in this study, ranging from 0.15 to 0.25.

^{2}, SDI = 900, CR = 0.1 or RS = 0.1), SI effects on ${\mathrm{P}\mathrm{A}\mathrm{I}}_{\mathrm{D}}$ became practically insignificant. SI affected the between-measure relationships in more complex formats (Figure 3). While SI impacts were statistically significant on all the between-measure relationships, its practical significance on the relationships of RS–${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$ and SDI–${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}\text{}$were negligible. The effects of SI on the CR–RS relationship varied with IPT and RS levels. The difference in CR at a given RS could be substantial between N sites and D sites if the sites were of poor quality (SI18) and also when RS was >0.20. In supporting our results, Zhao et al. [8] found that the CR–RS relationship varied with SI and the SI effect was RS level dependent. The SI effect was substantial when RS > 0.20 but became negligible when RS approached 0.20. Note that in this study, site quality was simply expressed as SI, without considering other effects that may alter responses, including soil and climate conditions and genetics. Incorporating SI effects into stand density development is essential for making informed decisions regarding stand management to maximize productivity and ensure sustainable forest management.

_{18}(Figure 3d). Observationally, the significant IPT effects on the SDI–age (Figure 1b) and CR–RS (Figure 3d) relationships became more evident when the stands entered the self-thinning stage. In parallel to our findings, significant effects of planting density on the CR–RS relationship were found in the SE loblolly pine plantations [8]. In this study, the observed effect of IPT on the ${\mathrm{P}\mathrm{A}\mathrm{I}}_{\mathrm{D}}$–SDI relationship was even contradictory to the theoretical prediction. Typically, trees at N sites would have a greater ${\mathrm{P}\mathrm{A}\mathrm{I}}_{\mathrm{D}}$ than those at D sites at a given SDI. Nonetheless, our results supported a reverse pattern, even if the difference was negligible (Figure 2b). Overall, our understanding of the relationships between tree growth and IPT is still limited. Further research or a reevaluation of the data may be needed to better understand the role of IPT in stand density development.

^{2}${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$, 500 SDI, 0.20 to 0.25 RS, or $\le $35% CR), the time judged by ${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$ agreed with that by RS, but was later than that by SDI and earlier than that by CR. Results in such comparisons for loblolly pine populations outside the WG varied. In a study in the SE US, an RS of 0.20 corresponded to a ${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$ of 20~30 m

^{2}[11], supporting our results. A CR of 0.40 corresponded to an SDI of 710 in loblolly pine populations in the WG [14], which agrees with this study (SDI = 765) but corresponded approximately only to 555 in an SE population [15]. It is also of great interest to compare the biological thin times by these measures with the regional operational criterion, which often is determined by market and stand age, e.g., the first thin occurs around age 15 years. The results are promising in that the ranges at age 15 of ${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$ (21 to 32 m

^{2}) and RS (0.15 to 0.26), depending mainly on SI, cover the respective criterion values, suggesting that the operational criterion concurs approximately with the biological thin times by these measures. However, the biological criterion of SDI (500) was located at the low end of the range (490 to 750) at age 15, and that of CR (0.35) was outside the range from 0.48 to 0.55, suggesting the disparity between these measures and the operational criterion. Factors other than SI and IPT such as physiographic region also influenced thin time [6]. At age 15 and for SE loblolly pine populations, the ${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$ reached 35 m

^{2}, ranging from 32 m

^{2}of 741 trees ha

^{−1}to 40 m

^{2}of 3706 trees ha

^{−1}, and the SDI was 800, ranging from 650 to 1000 [7], with both exceeding the respective values of this study. Even higher SDI (>1000 at age 15) was reported for the fertilized sites in the Atlantic coastal plain [28]. At age 11, the CR averaged 0.26 (at a density of 4445 trees ha

^{−1}) to 0.34 (at a density of 1111 trees ha

^{−1}) based on Poudel et al. [30], much lower than those observed in this study (CR = 0.63). Likely, the commercial thinning of loblolly pine should be practiced later in the WG than in the SE US.

^{−1}[31]. Our results show that thinning loblolly pine stands by the current biological criteria would result in 0.50 cm year

^{−1}or more ${\mathrm{P}\mathrm{A}\mathrm{I}}_{\mathrm{D}}$ even at poor sites such as those with an SI of 18 m by all the measures other than by CR, which had a rate of 0.42 cm year

^{−1}. Determining the timing of thinning by SDI would achieve a higher ${\mathrm{P}\mathrm{A}\mathrm{I}}_{\mathrm{D}}$, which, however, is a result of the thinning age by SDI being younger than those by other measures for a given stand.

^{−1}, similar to the normal density sites in this study) in some areas. The resulting changes in stand survival and tree growth from these activities are expected to affect the models’ parameter estimates but are unlikely to change the trends found in this study. Ideally, these relationships should be investigated using data collected from intensively managed loblolly pine plantations. Before that information becomes available, however, the results of N + SI80 sites in this study can be treated as an interim guide for intensively managed loblolly pine plantations.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Relationships between stand density measures and stand age of loblolly pine plantations planted at sites of two different SI (18 m and 24 m at age 25) and at an initial density of $\ge $1500 trees per hectare (D) or $<$1500 trees per hectare (N): (

**a**) basal area per ha (BA); (

**b**) stand density index (SDI); (

**c**) relative spacing (RS); and (

**d**) crown length ratio (CR).

**Figure 2.**Relationship between stand density measures and period annual increment in DBH (${\mathrm{P}\mathrm{A}\mathrm{I}}_{\mathrm{D}}$) of loblolly pine plantations planted at sites of two different SI (18 m and 24 m) and at an initial density of $\ge $1500 trees per hectare (D) or $<$1500 trees per hectare (N): (

**a**) basal area per ha (BA); (

**b**) stand density index (SDI); (

**c**) relative spacing (RS); and (

**d**) crown length ratio (CR).

**Figure 3.**Relationships between stand density measures of loblolly pine plantations planted at sites of two different SIs (18 m and 24 m) and at an initial density of $\ge $1500 trees per hectare (D) or $<$1500 trees per hectare (N): (

**a**) between stand density index (SDI) and basal area per ha (BA); (

**b**) relative spacing (RS) and BA; (

**c**) crown ratio (CR) and BA; and (

**d**) CR and RS.

Variable * | Mean | Std Dev | Maximum | Minimum |
---|---|---|---|---|

Last cycle | 7.94 | 2.36 | 12 | 3 |

Age of the last cycle (years) | 25.98 | 7.58 | 45 | 12 |

SI (m) | 20.03 | 2.75 | 27.45 | 8.96 |

IPT (trees ha^{−1}) | 1749 | 373.26 | 3363 | 897 |

DBH (cm) | 16.21 | 6.68 | 34.50 | 0.37 |

HT (m) | 13.51 | 6.12 | 30.52 | 1.51 |

**Table 2.**Selected models, model parameter estimates (standard errors), effects of high-quality sites or densely planted sites on model parameters, and the model performance.

Relationship | Selected Model | Estimates for Reference (N + SI = 0) Sites | SI * | D Sites * | R^{2} | RMSE (RMSE%) | ||
---|---|---|---|---|---|---|---|---|

b_{00} | b_{10} | b_{20} | ||||||

${\mathrm{B}\mathrm{A}}_{ha}$–Age | BA = ${b}_{0}\times {(1-{e}^{-{b}_{1}\times age})}^{{b}_{2}}$ | 16.702 (3.98) | 0.001 (0.00) | 4.884 (0.25) | b, _{01}b_{11} | b_{12} | 0.93 | 5.00 (21.70%) |

SDI–Age | SDI = ${b}_{0}\times {(1-{e}^{-{b}_{1}\times age})}^{{b}_{2}}$ | 329.469 (83.42) | 0.005 (0.00) | 5.152 (0.33) | b, _{01}b_{11} | b, _{02}b_{12} | 0.92 | 105.00 (19.93%) |

RS–Age | RS = ${b}_{0}\times {age}^{{b}_{1}}$ | 5.422 (0.18) | −0.436 (0.04) | b_{11} | b, b_{02}_{12} | 0.92 | 0.12 (28.50%) | |

CR–Age | CR = ${b}_{0}/(1-{e}^{1+{b}_{1}\times age})$ | −2.006 (0.03) | 0.017 (0.00) | b_{11} | 0.91 | 0.08 (16.32%) | ||

${\mathrm{P}\mathrm{A}\mathrm{I}}_{\mathrm{D}}\u2013{\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$ | ${\mathrm{P}\mathrm{A}\mathrm{I}}_{\mathrm{D}}=1/({b}_{0}+{b}_{1}\times {BA}^{{b}_{2}}$) | 1.584 (0.14) | 0.005 (0.00) | 1.838 (0.09) | b_{01}, b_{11} | 0.89 | 0.23 (28.05%) | |

${\mathrm{P}\mathrm{A}\mathrm{I}}_{\mathrm{D}}\u2013$SDI ** | ${\mathrm{P}\mathrm{A}\mathrm{I}}_{\mathrm{D}}=1/({b}_{0}+{b}_{1}\times {SDI}^{{b}_{2}}$) | 1.616 (0.17) | 0.033 (0.01) | 2.574 (0.18) | b_{01}, b_{21} | b_{22} | 0.89 | 0.27 (32.92%) |

${\mathrm{P}\mathrm{A}\mathrm{I}}_{\mathrm{D}}\u2013$RS | ${\mathrm{P}\mathrm{A}\mathrm{I}}_{\mathrm{D}}={b}_{0}+{b}_{1}\times RS$ | 0.052 (0.11) | −1.999 (0.34) | b_{01}, b_{11} | b_{02} | 0.66 | 0.32 (36.58%) | |

${\mathrm{P}\mathrm{A}\mathrm{I}}_{\mathrm{D}}\u2013$CR | ${\mathrm{P}\mathrm{A}\mathrm{I}}_{\mathrm{D}}={b}_{0}/{b}_{1}\times \mathrm{l}\mathrm{n}(1+{e}^{{b}_{1}\times \left(CR-{b}_{2}\right)})$ | 0.931 (0.59) | 3.589 (0.25) | 0.823 (0.15) | b_{01} | 0.94 | 0.23 (28.05%) | |

SDI–${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$ | SDI = ${b}_{0}+{b}_{1}\times BA$ | 0.708 (1.78) | 24.45 (0.76) | 0.95 | 27.31 (4.88%) | |||

RS–${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$ | RS = ${b}_{0}\times {BA}^{{b}_{1}}$ | 0.430 (0.22) | −0.435 (0.04) | b, b_{01}_{11} | b_{02} | 0.94 | 0.05 (18.52%) | |

CR–${\mathrm{B}\mathrm{A}}_{\mathrm{h}\mathrm{a}}$ | CR = $1/(1+{e}^{{-b}_{0}+{b}_{1}\times BA})$ | 0.924 (0.26) | 0.094 (0.01) | b, b_{01}_{11} | 0.93 | 0.09 (18.00%) | ||

CR–RS | CR = ${RS}^{{b}_{1}}/({b}_{0}+{RS}^{{b}_{1}})$ | 0.125 (0.01) | 2.020 (0.04) | b_{01} | b_{02} | 0.93 | 0.08 (16.00%) |

_{0}, b

_{1}, and b

_{2}) mean that these estimates increased with increasing SI or were larger in sites planted with a dense density (D sites) than a normal density, while the italic parameters represent reversal results. ** SDI was divided by 100 to facilitate model convergence.

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## Share and Cite

**MDPI and ACS Style**

Weng, Y.; Coble, D.; Grogan, J.; Ding, C.; Lou, X.
Evaluating Stand Density Measures for Regulating Mid-Rotation Loblolly Pine Plantation Density in the Western Gulf, USA. *Sustainability* **2024**, *16*, 9452.
https://doi.org/10.3390/su16219452

**AMA Style**

Weng Y, Coble D, Grogan J, Ding C, Lou X.
Evaluating Stand Density Measures for Regulating Mid-Rotation Loblolly Pine Plantation Density in the Western Gulf, USA. *Sustainability*. 2024; 16(21):9452.
https://doi.org/10.3390/su16219452

**Chicago/Turabian Style**

Weng, Yuhui, Dean Coble, Jason Grogan, Chen Ding, and Xiongwei Lou.
2024. "Evaluating Stand Density Measures for Regulating Mid-Rotation Loblolly Pine Plantation Density in the Western Gulf, USA" *Sustainability* 16, no. 21: 9452.
https://doi.org/10.3390/su16219452