# A Stand-Class Growth and Yield Model for Mexico’s Northern Temperate, Mixed and Multiaged Forests

## Abstract

**:**

^{2}each, nine Weibull distribution techniques of parameter estimation were fitted to the diameter structures of pines and oaks. Statistical equations using stand attributes and the first three moments of the diameter distribution predicted and recovered the Weibull parameters. Using nearly 1200 and 100 harvested trees for pines and oaks, respectively, I developed the total height versus diameter at breast height relationship by fitting three non-linear functions. The Newnham model predicted stem taper and numerical integration was done to estimate merchantable timber volume for all trees in the stand for each diameter class. The independence of the diameter structures of pines and oaks was tested by regressing the Weibull parameters and projecting diameter structures. The model predicts diameter distributions transition from exponential (J inverse), logarithmic to well-balanced distributions with increasing mean stand diameter at breast height. Pine diameter distributions transition faster and the model predicts independent growth rates between pines and oaks. The stand-class growth and yield model must be completed with the diameter-age relationship for oaks in order to carry a full optimization procedure to find stand density and genera composition to maximize forest growth.

## 1. Introduction

## 2. Experimental Section

#### 2.1. Methodology

#### 2.1.1. Fitting, Predicting and Recovering the Weibull Distribution Parameters

^{2}each were distributed throughout the forest. At least three sampling sites were randomly placed in each of the 837 forest stands. In each sample plot, the following characteristics of all trees that meet the inventory (DBH ≥ 7.5 cm) scheme were measured: diameter at breast height (DBH), top height (H), canopy cover (Cc), species (S), and sociological position (SP). Age was measured in 3–5 trees of each sample plot. At the stand scale, the ecological interactions between selected hardwood and pine species were observed by fitting the Weibull density function to the diameter distributions, predicting parameters from stand attributes, relating statistically parameters between oaks and pines, projecting the diameter structures with stand attributes, and developing the diameter-age of pines and oaks. These features form the core of the stand-class growth and yield model.

#### 2.1.2. The Weibull Density Function

_{B}[17,18,19,20,21,22,23]. The Weibull density function has gained extensive popularity because of its flexibility and closed form [16,24]. The Weibull density function (pdf) is given by Equation (1) and, as a cumulative density function (cdf), by Equation (2) [24];

_{x}(X) = probability of the random variable, DBH = diameter at breast height; α, β and ε are shape, scale and location parameters, respectively.

_{i}= random variable (diameter at breast height), n = number of observations; α and β = shape and scale parameters.

#### 2.1.3. Hypothesis Testing and Goodness-of-Fit

^{2}and Kolmogorov-Smirnoff (K-S) statistics—Equations (6) and (7), respectively, were used to test the null hypothesis of equal diameter distributions between observed and estimated frequencies:

_{i}= absolute observed diameter frequency; e

_{i}= absolute expected diameter frequency; P

_{x}(X) = cumulative observed density function, and S

_{n}(X) = cumulative expected density function of X.

#### 2.1.4. Predicting and Recovering Distribution Parameters

**Table 1.**Tree dimensional features for 587 stands for constructing and 250 stands for validating the diameter-class model.

Model | Stands | Group of Species | Density (No ha^{−1}) | DBH (cm) | S.D (cm) | H (m) | S.D (m) |
---|---|---|---|---|---|---|---|

Construction | 587 | Pinus spp. | 631 | 23.4 | 8.7 | 12.6 | 4.5 |

587 | Quercus spp. | 212 | 12.8 | 12.8 | 9.3 | 3.2 | |

Validation | 250 | Pinus spp. | 602 | 23.9 | 8.0 | 11.4 | 4.1 |

250 | Quercus spp. | 231 | 12.3 | 11.3 | 9.5 | 3.8 |

#### 2.2. Testing the Independence of the Diameter Distributions of Pines and Oaks

^{−1}were discarded from further data analysis. The average tree density for selected forest stands was 310 and 125 trees ha

^{−1}for pines and oaks, respectively.

#### 2.3. The Stand-Class Growth and Yield Model

^{3}·ha

^{−1}) classified as: (i) sawnwood (DBH ≥ 20 cm); (ii) plywood (DBH ≥ 40 cm); and (iii) secondary forest products (DBH ≤ 20 cm).

## 3. Results and Discussion

#### 3.1. Parameter Estimators

**Table 2.**Statistics of parameters calculated by nine techniques for 587 mixed forest stands of Durango, Mexico.

Method | Parameters of the Weibull distribution | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

α * | β ^{†} | ε ^{‡} | ||||||||||

Pine | Oak | Pine | Oak | Pine | Oak | |||||||

A ^{§} | SE ^{||} | A | SE | A | SE | A | SE | A | SE | A | SE | |

MNP | 1.6 | 0.0206 | 1.4 | 0.0165 | 26.3 | 0.1403 | 27 | 0.1445 | 12.1 | 0.1114 | 11.1 | 0.1156 |

MPP | 1.1 | 0.0083 | 0.8 | 0.0124 | 10.8 | 0.1527 | 9.4 | 0.1692 | 14.4 | 0.0495 | 14.5 | 0.0454 |

MCM | 1.7 | 0.0413 | 1.5 | 0.0454 | 26 | 0.1445 | 25.3 | 0.1527 | 11.7 | 0.2724 | 11.8 | 0.2559 |

MV2 | 2.0 | 0.0165 | 2.0 | 0.0248 | 28 | 0.1238 | 28.8 | 0.1445 | 13.5 | 0.1238 | 13.8 | 0.1032 |

MRZ | 1.4 | 0.0165 | 1.1 | 0.0165 | 26.6 | 0.1568 | 27 | 0.2394 | 13.8 | 0.1073 | 14.3 | 0.0949 |

MDS | 1.8 | 0.0248 | 1.5 | 0.0289 | 15.7 | 0.2311 | 16.9 | 0.3096 | 10.9 | 0.1238 | 10.4 | 0.1445 |

MRM | 1.6 | 0.0206 | 1.5 | 0.0165 | 13.7 | 0.227 | 15 | 0.227 | 12.3 | 0.1156 | 11.4 | 0.1156 |

MZM | 1.0 | 0.0083 | 0.8 | 0.0124 | 26.3 | 0.1445 | 26.1 | 0.161 | 14.6 | 0.033 | 14.6 | 0.0413 |

MV3 | 1.2 | 0.0289 | 0.8 | 0.0289 | 12 | 0.26 | 10.4 | 0.26 | 13.3 | 0.1362 | 14.3 | 0.1568 |

*****Shape parameter,

^{†}Scale parameter,

^{‡}Location parameter,

^{§}average,

^{||}Standard error.

**Figure 1.**An example of the maximum likelihood two-parameter Weibull density function fitted to diameter structures of pines and oaks.

#### 3.2. Goodness of Fit Tests

^{2}and K-S tests. MV2 and MV3 consistently accepted the highest percentage of null hypotheses (76.5%), in contrast to MZM and MDS (34.9% and 44.6%, respectively). MV2 and MV3 procedures also accepted the largest percentage of null hypotheses, according to the K-S goodness of fit test (95.1%). The MDM and MPP techniques, however, recorded the least goodness of fit with 33.0% and 58.1% of null hypotheses accepted, respectively (Figure 2).

^{2}model, the approaches MV2 and MV3 had the largest percentage of accepted null hypotheses (60.1) for oak trees. The MCM and MDS techniques recorded the worst goodness of fit with only 25.5% and 29.4% of accepted null hypotheses. Using K-S, the MCM and MV2 methods recorded the largest percentages of accepted null hypotheses (92.9% and 90.4%, respectively). On the other hand, MDS and MRZ had the worst goodness of fit with only 38.8% and 43.8% of null hypotheses accepted, respectively (Figure 2).

**Figure 2.**Goodness of fit tests χ

^{2}and K-S conducted on nine different techniques of parameter estimation of the Weibull density function for 587 forest stands of fitting (

**a**,

**b**) and 250 forest stands of validation (

**c**,

**d**) parameters.

#### 3.3. Parameter Variance and Bias

**Table 3.**Efficiency and consistency of parameter estimators for α, β and ε calculated by nine methods for pines and oaks in mixed forest stands of Durango, Mexico.

Method | Weibull Distribution Parameters | ||||||||
---|---|---|---|---|---|---|---|---|---|

α * | β ^{†} | ε ^{‡} | |||||||

BP ^{§} | A ^{||} | S² ^{¶} | BP | A | S² | BP | A | S² | |

Pines | |||||||||

MNP | 0.027 | 1.24 | 0.009 | 0.085 | 25.52 | 0.067 | −0.023 | 13.89 | 0.048 |

MPP | −0.085 | 1.23 | 0.004 | −0.022 | 11.48 | 0.086 | 0.015 | 14.01 | 0.013 |

MCM | 0.234 | 1.76 | 0.696 | −1.641 | 25.77 | 14.97 | −4.216 | 11.17 | 40.220 |

MV2 | 0.035 | 2.86 | 0.001 | −0.053 | 28.23 | 0.084 | ** | 12.50 | ** |

MRZ | 0.227 | 1.91 | 0.010 | 0.300 | 25.80 | 0.233 | −0.160 | 10.59 | 0.127 |

MDS | 0.050 | 2.34 | 0.009 | 0.358 | 22.90 | 0.378 | −0.400 | 4.16 | 0.476 |

MRM | 0.024 | 1.27 | 0.008 | 0.065 | 11.62 | 0.237 | −0.017 | 13.93 | 0.059 |

MZM | −0.059 | 0.98 | 0.018 | −0.085 | 25.80 | 0.098 | 0.025 | 14.53 | 0.016 |

MV3 | −0.057 | 1.18 | 0.058 | −0.882 | 12.04 | 13.19 | 0.322 | 13.33 | 1.559 |

Oaks | |||||||||

MNP | 0.013 | 0.97 | 0.008 | 0.168 | 25.84 | 0.328 | 0.075 | 13.82 | 0.221 |

MPP | 0.027 | 0.92 | 0.007 | 0.401 | 11.37 | 0.665 | −0.128 | 14.24 | 0.071 |

MCM | 1.525 | 2.39 | 5.966 | −1.479 | 26.64 | 18.71 | −9.453 | 5.25 | 27.969 |

MV2 | −0.006 | 2.17 | 0.007 | 0.088 | 29.50 | 3.497 | ** | 13.00 | ** |

MRZ | 0.217 | 1.55 | 0.030 | 0.250 | 26.25 | 0.736 | −0.173 | 9.77 | 0.650 |

MDS | 0.091 | 1.86 | 0.007 | 1.060 | 25.39 | 0.909 | −0.770 | 3.57 | 0.610 |

MRM | 0.016 | 0.97 | 0.008 | 0.107 | 12.01 | 0.975 | 0.037 | 13.81 | 0.241 |

MZM | 0.041 | 0.75 | 0.010 | 0.432 | 26.38 | 0.475 | −0.009 | 14.56 | 0.285 |

MV3 | −0.091 | 0.86 | 0.50 | −0.231 | 10.47 | 31.94 | 0.321 | 14.31 | 10.74 |

*****Shape parameter,

**Scale parameter,**

^{†}**Location parameter,**

^{‡}**Average bias,**

^{§}**Mean,**

^{||}**variance.**

^{¶}#### 3.4. Parameter Prediction and Recovery

**Table 4.**Weibull parameter prediction and recovery equations for pine and oaks growing in Mexico’s northern mixed multiaged temperate forests.

Group of Species | Parameter | Empirical Equation | n | r^{2} | Sx |
---|---|---|---|---|---|

Pinus spp. | α | 0.91Dm^{13.86}Dq^{−12.31}N^{−0.49}BA^{0.41} | 587 | 0.52 | 0.20 |

β | 2.9 + 2.2Dm − 0.01N − 1.2Dq + 0.13BA + 0.005Cc + 0.033H | 587 | 0.93 | 0.89 | |

Xp | 98.5N^{−0.46}BA^{0.46} | 587 | 0.96 | 0.55 | |

Std | 1.36 + 0.29Xp − 0.008N + 0.13BA | 587 | 0.68 | 1.10 | |

Sk | 0.00000021Std^{2.62}N^{3.14}BA^{−2.93} | 587 | 0.42 | 0.51 | |

Quercus spp. | α | 0.30β^{0.99}Dm^{4.18}Dq^{−4.48}IDR^{−0.068} | 587 | 0.51 | 0.21 |

β | 12.76 + 1.77Dm − 0.035N − 1.13Dq + 0.56BA | 587 | 0.84 | 1.39 | |

Xp | 92.8N^{−0.43}BA^{0.43}Cc^{−0.031} | 587 | 0.94 | 0.76 | |

Std | 799902177Xp^{−3.5}N^{−2.5}BA^{2.5}Cc^{0.061} | 587 | 0.88 | 1.61 | |

Sk | 0.00004Std^{1.9}N^{1.9}BA^{−1.8}Cc^{0.15} | 587 | 0.50 | 0.57 |

^{2}and K-S tests, respectively. When using the moment recovery approach, 49.8% and 78.4% of the population fit the two-parameter Weibull distribution when using the χ

^{2}and K-S tests, respectively. For oaks, the moment prediction approach and the recovering of parameters by MNP accepted 37.5% and 73.7% of the population, as tested by the χ

^{2}and K-S statistics, respectively. The parameter prediction approach was accepted by only 43.7% and 66.7% of the population, as tested by the χ

^{2}and K-S goodness-of-fit tests, respectively. When averaging the results of χ

^{2}and K-S tests, the parameter recovery approach fitted diameter distributions better than the parameter prediction approach, which is consistent with the findings of Hyink [34] and Gove and Patil [53].

#### 3.5. Sensitivity Analysis

^{2}and K-S goodness-of-fit tests. Changes in the acceptance of null hypotheses were hardly noticed when the standard error was added to the equation to predict β. The percentage of accepted null hypotheses notoriously shifted to 76.6% when the standard errors of each parameter were added (Table 5). The sensitivity analysis reflects the need to predict and recover α with the greatest precision. Návar et al. [38] demonstrated that growth models based on predicting the Weibull distribution parameters were most sensitive to changes in α values as well. Návar-Cháidez [37] developed a simple regression equation to predict α with the skew coefficient, Sk, consistent with the mathematical theory.

**Table 5.**Sensitivity analysis of equations for predicting and recovering the Weibull distribution parameters, and the goodness-of-fit test for pine and oak stands of Durango, Mexico.

Parameter | Ho Accepted (χ^{2}) | Ho Accepted (K-S) | ||
---|---|---|---|---|

Prediction Approach | Pine | Oak | Pine | Oak |

No change MV2 | 51.3 | 43.7 | 74.7 | 66.7 |

α ± EES | 32.5 | 36.7 | 71.9 | 65.2 |

β ± EES | 49.8 | 41.7 | 73.2 | 65.6 |

α ± EES y β ± EES | 36.2 | 29.1 | 60.2 | 59.2 |

Predicting Moments Approach | ||||

No change MNP | 49.8 | 37.5 | 78.4 | 73.7 |

Sk ± EES | 34.8 | 28.1 | 73.1 | 61.3 |

Std ± EES | 47.4 | 38.0 | 78.6 | 71.3 |

Xp ± EES | 48.4 | 38.8 | 81.9 | 70.5 |

Sk, Std, Xp ± EES | 30.3 | 25.9 | 59.2 | 61.6 |

#### 3.6. Regressing Distributional Parameters of Oaks and Pines

^{−1}when there are no pines in the stand, but when pine density increases to 500 trees ha

^{−1}, oak density diminishes to 110 trees ha

^{−1}. When pine density increases to 1000 trees ha

^{−1}, oak density diminishes only to 81 trees ha

^{−1}. Oak density diminishes, on average, only by 6% when pine density increases 100%. That is, even though there is a statistical relationship, the slope is so small that it can be attributed to other causes of oak and pine distribution, such as subtle changes in altitude above sea level, which may modify the ratio of pine/oak diversity [9].

**Figure 3.**The statistical relationships between stand parameters of oaks and pines (r

^{2}= coefficient of determination, Sx = Standard error, P = probability).

**Figure 4.**The statistical relationships between the Weibull distribution parameters of oaks and pines (r

^{2}= coefficient of determination, Sx = Standard error, P = probability).

#### 3.7. The Stand-Class Growth and Yield Model

**Figure 5.**Graphical representation of the stand class growth and yield model for pines and oaks of Mexico’s northern mixed temperate forests. Capital letters indicate the sequence in mean quadratic diameter.

**Table 6.**Empirical prediction equations that form the core of the stand-class growth and yield model for pines and oaks.

Attributes/Species | Pinus spp. | Quercus spp. |
---|---|---|

Stand Density | 1591.5dmp^{−0.3392} (r^{2} = 0.28) | 293.15Dmq^{−0.3051} (r^{2} = 0.36) |

IDR | 30.61Np^{0.4307} (r^{2} = 0.65) | 296.9exp^{0.0007Nq} (r^{2} = 0.43) |

Dq | 1.1108Dmp^{−1.3924} (r^{2} = 0.98) | 0.69Dmq^{1.1397} (r^{2} = 0.99) |

H | 1.5158Dmp^{0.7217} (r^{2} = 0.47) | 0.9054Dmq^{0.8499} (r^{2} = 0.55) |

Cc | 69.403Dmp^{−0.1429} (r^{2} = 0.87) | 2.29Dmq^{0.8626} (r^{2} = 0.56) |

**Figure 6.**Three equations fitted to the total height—diameter at breast height relationship for pines and oaks of northern temperate forests of Mexico.

**Figure 7.**Graphical representation of the standing timber volume classified in merchantable forest products (sawnwood, plywood and secondary forest products) for pine and oak species of northern temperate forests of Mexico. Note capital letters indicate the sequence in mean quadratic diameter.

**Figure 8.**Diameter growth curve for pines (source: 41) (Note: r

^{2}= coefficient of determination, Sx = Standard error, P = probability).

^{2}≤ 0.50. The empirical equation that predicts the shape parameter of the two-parameter maximum likelihood Weibull density function presents the smallest coefficient of determination. A different approach would be to use the empirical equation developed by Návar-Cháidez [37] that uses only the skewness coefficient to predict α. The empirical equation to predict Sk has also one of the smallest coefficients of determination for both pines and oaks and the improvement of this equation is a matter of further research. Other stand variables must better predict either α or Sk, such as the quartiles of the diameter distribution function, basal area, site index, and altitude [22,39]. A parameter that explains the deviance from the normal distribution could also improve future α assessments. Regardless of these shortcomings the model predicts robust tendencies and therefore it deserves further interpretation.

^{−1}) are left in the forest after long-term harvesting operations or natural forest disturbances or establish first because of their reproductive advantages leaving large forest openings. Pines, with a few exceptions, are shade intolerant and regenerate sexually via seed dispersal and establish well in these openings. Oaks may improve microhabitat conditions, making it possible for pines to establish successfully in open spaces in between oaks, although I had seen pines growing well beneath the canopy of large isolated oaks probably because the canopy does not interfere with sunlight entering the forest floor or perhaps because pine seedlings grow at the periphery of the canopy. Lafon et al. [57] recorded changes in the fertilization status of soils, given by the C:N ratio, which facilitated the establishment of pines under the canopy of oaks in east Tennessee. In some stands, pioneer pine species do not appear to establish well under the shade of oaks [58]. Oaks, on the other side establish well under the canopy of pines as secondary species of succession or in openings as pioneer species of succession [5].

^{−1}) present in the stand; and (ii) pines grow quickly in height searching for full sunlight to reach a dominant position in the forest. Over time, pines outgrow oaks in DBH as well, and several oak trees of most species remain dominated during the life cycle across the altitude gradient in the eastern ridges of the Sierra Madre Occidental mountain range. However, in late successional stages, several oak species attain a dominant sociological position and share this place with dominant pine trees. Therefore, the differential displacement rate of pine and oak diameter distributions may be explained by their differential growth rates and symbiotic mechanisms rather than by inter-specific competition. Domínguez and Návar [59] supported this observation by demonstrated that by reducing 50% of stand basal area by harvesting oak trees did not improve the diameter growth of the remaining pine trees, even though pine trees were approximately 50 years-old. Therefore, resource partitioning may be playing an important role in these forests. For example, oaks and pines do not appear to compete for sunlight and it is likely they do not compete for nutrients and soil water either. That is, inter-specific competition is not as strong as it is intra-specific competition in these mixed and multiaged forests [60]. Differential timing in the usage of resources and, most likely, the exploitation of different soil compartments, could explain the potential lack of inter-specific competition. However, further research is required on the physiological or metabolic processes of both oaks and pines to better understand as well as to put into prospective the findings of this research on the mechanisms of tree coexistence in Mexico’s northern natural forests.

## 4. Conclusions

## Conflicts of Interest

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**MDPI and ACS Style**

Návar, J. A Stand-Class Growth and Yield Model for Mexico’s Northern Temperate, Mixed and Multiaged Forests. *Forests* **2014**, *5*, 3048-3069.
https://doi.org/10.3390/f5123048

**AMA Style**

Návar J. A Stand-Class Growth and Yield Model for Mexico’s Northern Temperate, Mixed and Multiaged Forests. *Forests*. 2014; 5(12):3048-3069.
https://doi.org/10.3390/f5123048

**Chicago/Turabian Style**

Návar, José. 2014. "A Stand-Class Growth and Yield Model for Mexico’s Northern Temperate, Mixed and Multiaged Forests" *Forests* 5, no. 12: 3048-3069.
https://doi.org/10.3390/f5123048