# Projection Matrix Models: A Suitable Approach for Predicting Sustainable Growth in Uneven-Aged and Mixed Hyrcanian Forests

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## Abstract

**:**

## 1. Introduction

- Can individual-tree increment models and matrix models be combined to reliably estimate growth rate and allowable cut rate in uneven-aged mixed forests managed by single-selection silvicultural techniques in Hyrcanian forests?
- Are the total operated allowable cut (OAC) and estimated allowable cut (EAC) volumes for Hyrcanian forests consistent?
- How does the estimated volume from the developed individual tree increment/matrix model compare to the predicted volume according to the Hyrcanian forest plan?

## 2. Materials and Methods

#### 2.1. Study Area

#### 2.2. Data Collection

- (A)
- Operated allowable cut (OAC): is the by volume of trees harvested in three compartments of 305, 306 and 309 according to forestry plan.
- (B)
- Predicted allowable cut (PAC): is the amount of tree harvesting predicted when formulating a forestry plan for the whole district.
- (C)
- Estimated allowable cut (EAC): is the amount of tree harvesting that has been obtained in the present study.

#### 2.3. Projection Matrix Model and Determination of Harvest Rate

_{t+1}is the stem density (ha

^{−1}) in class-j at the final time of projection, I is identity matrix, Nt is the stem density (ha

^{−1}) in class-j at the initial time of projection, and H is diagonal matrix with the actual harvest rate.

_{j}is recruitment coefficients (the number of offspring’s living at time t + 1 of projection that were produced in the interval (t, t + 1) by an average tree in class j at time t). P

_{j}is transition probabilities between two consecutive pair of diameter classes, which is separately calculated for each group from the following equation:

_{j}represents the diameter of the tree at the end of the period for a tree with diameter of d

_{j}. In Equation (4), instead of the numerator of fraction, the diameter increment of each tree from every species group can be computed. For this purpose, by using the results of Salehnasab, et al. [46], the diameter increment of each species group (Equations (5)–(8)) is placed in the numerator of Equation (4).

_{cy}is nine-year-old tree c from sample plot y (cm), d is diameter at breast height (cm), BAL is basal area of the largest trees (m

^{2}/ha), g

_{m}is mean of the basal area at sample plot (m

^{2}), and H

_{d}is size diversity index, and Hs species diversity index (Table 1).

_{0}> 1, the long-term sustainable harvest rates can be determined as the proportion of trees removed in each class so that the dominant eigenvalue of matrix G(I − H) is λ = 1 [44]. There are, of course, different harvesting strategies to meet these conditions. Some strategies focus on the largest diameter class. Thus, with increasing diameter, the probability of a tree being harvested increases [44]. The strategy used in this model was to obtain the three main conditions proposed in the study of Torres, Belda, Pérez, and Fernández [45], which result in a stable diameter distribution. In this strategy, in order to perform a harvesting operation, the dominant eigenvalue λ of matrix G must always be greater than one (λ

_{0}> 1), and the harvest from the forest stand will continue as long as the dominant eigenvalue of matrix G(I − H) is equal to one. Also, through solving GW0 = λW0 and obtaining to the right eigenvector W0 corresponding to the dominant eigenvalue λ

_{0}of the matrix G, the stable diameter distribution is defined and then the long-term dynamic of harvested trees is also achieved. Therefore, by solving the linear system of GHW0 = (λ

_{0}− 1) W0, the above conditions are rewritten to maintain a sustainable harvest rate. Finally, using Equation (11), the harvest rate is estimated.

## 3. Results

_{t+}

_{1}) was estimated. Afterwards, Chi-square test was used to evaluate the fitted models. The results showed that at 0.01 level, there was no significant difference between operated and estimated N

_{t+}

_{1}by models (in all species groups). Therefore, the resulting model was accepted and used to calculate the eigenvector.

_{t}), the harvesting proportion was obtained per diameter class (Figure 2, Figure 3 and Figure 4) Here, the harvesting proportion refers to the harvesting of wood from all diameter classes, not just larger diameter classes or trees. The results indicated that in the long-term the diameter distribution of the forest stands change.

^{3}and 2415.41 m

^{3}, respectively. The results of the study depict that the sum of operated allowable cut of all groups was higher than estimated allowable cut (Table 5). The dominant eigenvalue of the transition matrix of all groups, with regard to logging, was estimated to be less than one (0.964). Finally, for three compartments the annual estimated allowable cut is 3.03 m

^{3}/ha.

^{3}, respectively. Predicted volumes are higher than the estimated volumes, for these compartments (unlogged), the annual estimated allowable cut is 2.72 m

^{3}/ha.

## 4. Discussion

^{3}/ha) and 0.076 (2.72 m

^{3}/ha), respectively. Due to differences of species proportions, estimated harvest rates were different for two compartment groups. The highest harvest rate belonged to chestnut-leaved oak group (0.1 and 0.08 M

^{3}/ha per year). However, due to the volume inventory information at the beginning of the period, the highest value of harvest rate was for the beech group (1.26 m

^{3}/ha per year).

^{3}, respectively, which has significant differences with the results of the present study. Also, for the mixed type (which can be compared with the total volume of harvest), the annual harvest per hectare was estimated to be 3.75 m

^{3}/ha, which is more than the findings of the present study (2.72 m

^{3}/ha per year). Therefore, the findings of our study suggest that, compared to other methods, the projection matrix model estimates allowable cut lower and closer to the net increment, and thus can prevent the possible consequences of harvest over increment of the single-selection method. Forest cutting is used as a tool for forest cultivation (biological production) and wood harvesting (mechanical production) and leads the forest toward desirable quantitative and qualitative production [50]. To use this tool properly, it is necessary to accurately estimate the allowable cut [51]. Given such importance of the allowable cut, for each forest type based on their species and increment rate and other influential characteristics, it is necessary to perform comprehensive studies to determine an appropriate method for estimation of allowable cut in any forest area.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Iran’s position in the world, Location of the study area in northern Iran, and the network of permanent sample plots in the form of red dots, respectively (

**a**). Photographs from Hyrcanian forests (

**b**).

**Figure 2.**Trend of the proportion of operated and estimated harvest in the diameter classes of the beech group.

**Figure 3.**Trend of the proportion of operated and estimated harvest in the diameter classes of the hornbeam group.

**Figure 4.**Trend of the proportion of operated and estimated harvest in the diameter classes of the other species group.

Variable | Minimum | Maximum | Standard Deviation |
---|---|---|---|

Diameter (cm) | 7 | 188 | 24.7 |

The mean of basal area at sample plot (m^{2}) | 0.02 | 0.633 | 0.1 |

The basal area of the largest tree (m^{2}/ha) | 0 | 52 | 8.4 |

Size diversity index of the sample plot | 0 | 2.468 | 0.314 |

Shannon-Wiener index | 0 | 1.8 | 0.663 |

Compartments Number | Number of Marked Trees in Compartments | Predicted Allowable Cut in Forestry Plan (m^{3}) | |||||
---|---|---|---|---|---|---|---|

Beech | Hornbeam | Chestnut-Leaved Oak | Other Species | Total | Without Coefficient | With Coefficient 0.9 | |

305 | 73 | 97 | 0 | 7 | 177 | 600 | 540 |

306 | 16 | 216 | 0 | 10 | 246 | 520 | 468 |

309 | 81 | 237 | 0 | 9 | 327 | 1420 | 1278 |

Total | 170 | 550 | 0 | 26 | 750 | 2536 | 2282.4 |

**Table 3.**The volume of operated allowable cut (m

^{3}) in each species group and compartment in Gorazbon district.

Species Group | Compartment 305 | Compartment 306 | Compartment 309 | Total | ||||
---|---|---|---|---|---|---|---|---|

Volume | Volume with Coefficient 0.9 | Volume | Volume with Coefficient 0.9 | Volume | Volume with Coefficient 0.9 | Volume | Volume with Coefficient 0.9 | |

Beech | 480.82 | 432.74 | 74.79 | 67.31 | 632.33 | 569.1 | 1187.94 | 1069.15 |

Hornbeam | 296.75 | 267.7 | 586.35 | 527.72 | 865.34 | 778.8 | 1748.44 | 1573.6 |

Chestnut-leaved oak | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Other species | 14.41 | 12.97 | 12.1 | 10.89 | 46.94 | 42.25 | 73.45 | 66.105 |

Total | 791.98 | 712.78 | 673.24 | 605.916 | 1544.761 | 1390.15 | 3009.83 | 2708.847 |

Pj | Unlogged Compartments | Logged Compartments (305, 306 and 309) | ||||||
---|---|---|---|---|---|---|---|---|

Beech | Hornbeam | Chestnut-Leaved Oak | Other Species | Beech | Hornbeam | Chestnut-Leaved Oak | Other Species | |

P1 | 0.1564 | 0.1981 | 0.2278 | 0.2791 | 0.1433 | 0.1956 | 0.2152 | 0.2502 |

P2 | 0.2119 | 0.2317 | 0.3070 | 0.3459 | 0.1834 | 0.2313 | 0.2951 | 0.3005 |

P3 | 0.2418 | 0.2597 | 0.3597 | 0.3729 | 0.2456 | 0.2577 | 0.3587 | 0.3427 |

P4 | 0.2787 | 0.2842 | 0.4111 | 0.3919 | 0.2785 | 0.2828 | 0.4087 | 0.3724 |

P5 | 0.3174 | 0.3076 | 0.4602 | 0.4030 | 0.3352 | 0.3075 | 0.3895 | |

P6 | 0.3311 | 0.3268 | 0.4712 | 0.4098 | 0.3610 | 0.3286 | 0.4053 | |

P7 | 0.3343 | 0.3378 | 0.4678 | 0.4046 | 0.3846 | 0.3451 | 0.4088 | |

P8 | 0.3382 | 0.3601 | 0.5097 | 0.3978 | 0.3929 | 0.3598 | 0.4051 | |

P9 | 0.3556 | 0.3725 | 0.4985 | 0.3872 | 0.4000 | 0.3678 | 0.4075 | |

P10 | 0.3298 | 0.3912 | 0.3779 | 0.3674 | 0.3977 | 0.3949 | ||

P11 | 0.3293 | 0.3934 | 0.3427 | 0.2529 | 0.4128 | 0.3835 | ||

P12 | 0.3257 | 0.3711 | 0.3113 | 0.2462 | ||||

P13 | 0.3097 | 0.2949 | 0.2309 | |||||

P14 | 0.2880 | 0.2787 | 0.2053 | |||||

P15 | 0.2693 | 0.2525 |

The Group of Compartments | Unlogged Compartments | Logged Compartments | ||||||
---|---|---|---|---|---|---|---|---|

Species Group | Beech | Hornbeam | Chestnut-Leaved Oak | Other Species | Beech | Hornbeam | Chestnut-Leaved Oak | Other Species |

The largest right dominant eigenvalue | 1.062 | 1.079 | 1.113 | 1.074 | 1 | 0.91 | 1.121 | 1.083 |

The nine-year harvest rate | 0.06 | 0.073 | 0.101 | 0.067 | 0 | - | 0.108 | 0.077 |

Annual allowable cut (m^{3}/ha) | 1.26 | 0.55 | 0.08 | 0.83 | 0 | - | 0.1 | 0.75 |

Total of Nine-year allowable cut (m^{3}) | 8237.26 | 3564.46 | 537.68 | 5433.52 | 0 | −830.57 | 78.62 | 533.835 |

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**MDPI and ACS Style**

Salehnasab, A.; Burkhart, H.E.; Bayat, M.; Khaleghi, B.; Heidari, S.; Masood Awan, H.U.
Projection Matrix Models: A Suitable Approach for Predicting Sustainable Growth in Uneven-Aged and Mixed Hyrcanian Forests. *Sustainability* **2022**, *14*, 6777.
https://doi.org/10.3390/su14116777

**AMA Style**

Salehnasab A, Burkhart HE, Bayat M, Khaleghi B, Heidari S, Masood Awan HU.
Projection Matrix Models: A Suitable Approach for Predicting Sustainable Growth in Uneven-Aged and Mixed Hyrcanian Forests. *Sustainability*. 2022; 14(11):6777.
https://doi.org/10.3390/su14116777

**Chicago/Turabian Style**

Salehnasab, Abotaleb, Harold E. Burkhart, Mahmoud Bayat, Bagher Khaleghi, Sahar Heidari, and Hafiz Umair Masood Awan.
2022. "Projection Matrix Models: A Suitable Approach for Predicting Sustainable Growth in Uneven-Aged and Mixed Hyrcanian Forests" *Sustainability* 14, no. 11: 6777.
https://doi.org/10.3390/su14116777