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Traditional unevenaged forest management seeks a balance between equilibrium stand structure and economic profitability, which often leads to harvesting strategies concentrated in the larger diameter classes. The sustainability (
The goal of traditional unevenaged forest management is a balance between equilibrium stand structure and economic profitability. In pursuing this balance, many harvesting strategies concentrateon the larger diameter classes. The sustainability (
Matrix models commonly serve to analyze the evolution, management and harvesting of tree populations [
This concept of stability relates closely to the concept of perturbation: A system is stable if it always returns to an equilibrium position following small perturbations (otherwise, the system is unstable). The stable distributions are closely dependent on recruitment, removal and stem migration throughout the diameter at breast height (dbh)classes over time [
Therefore, in the present study, we combine that matrix model to describe the population dynamics with an economic model (objective function) that summarizes the net present value (
Optimal control theory enables the solution of a wide variety of dynamic problems, for which the evolution of the dynamic system (discrete or continuous) can be partially controlled by the agent’s decision. In every moment,
Many authors have considered optimal control theory to model forest stands (e.g., continuous case, [
Thus, the main purpose of this study is to determine economically optimal harvesting strategies of unevenaged
In the following sections, we describe the model and the estimates of the corresponding parameters. The general calculations, including regressions, were run using Maple version 16 [
The model for recruitment, harvesting and stem migration throughout the dbh classes over time is based on the model proposed by López
The starting assumptions were as follows: (a) the forest is in a steady state; (b) the average diameter growth curves for Qualities I
Harvesting operations in this study area generally took place every 10 years, so we adopted this period as the range of the projection intervals in the model. Considering this time period and the diameter growth functions corresponding to Qualities I
Since the main objective for the stands in the study area is the production of timber, we use, as the objective function of the optimization problem, the net present value of all the management operations over a time horizon of
As commented on in the Introduction, the initial conditions must represent the stable diameter distributions of the stand or
We also introduce the following constraints:
By applying the “sustainable/stable” harvesting strategy to each scenario and setting
We calculated the transition probabilities by applying the method introduced by López
Using these diameter growth models, the transition probabilities from class
Diameter growth models: Quality I,
The above mentioned transition probability formula is a direct application of the one introduced by López
Transition probabilities between diameter classes for each quality.
Quality I  Quality II  Quality III  

(0,6) → (6,12)  
(6,12) → (12,18)  
(12,18) → (18,24)  
(18,24) → (24,30)  
(24,30) → (30,36)  
(30,36) → (36,42)  
(36,42) → (42,48)    
(42,48) → (48,→)     
According to the model assumptions and the methodology used, we had to consider that the same transition probability,
Although
We estimated natural mortalities using previous studies. Specifically, Misir
We thus assumed the following 10year individual tree constant mortality rates for each diameter class:
As shown in the Introduction, the “sustainable/stable” harvesting strategy is aimed at reaching in each harvest the proportions of stems/ha in each class corresponding to the stable diameter distribution of the stand, yielding the following “sustainable/stable” harvest rate:
As commented on in
The results obtained for
Numerical values for
Q I 

1.305091  1.477671  1.594240  

0.233770  0.323259  0.372742  

16.857068  14.888296  13.799677  
Est. Dis.  [186.1, 122.9, 96.4, 77.2, 61.4, 47.5, 34.6, 22.1, 7.2]  [416.9, 239.8, 162.6, 110.9, 73.4, 45.7, 25.5, 11.4, 2.4]  [615.9, 325.9, 202.3, 125.4, 74.6, 41.1, 19.8, 7.4, 1.2]  

5,347.36  6,118.03  6,383.55  

1.292499  1.458959  1.571264  

0.226305  0.314580  0.363570  

18.568685  16.450081  15.274322  
Est. Dis.  [188.3, 125.7, 99.8, 81.0, 65.4, 51.4, 38.3, 25.3, 8.6]  [423.2, 246.9, 169.9, 117.8, 79.4, 50.5, 28.9, 13.4, 2.9]  [626.4, 336.8, 212.6, 134.2, 81.5, 45.9, 22.7, 8.8, 1.5]  

5,743.33  6,611.13  6,918.95  

1.281305  1.442342  1.550881  

0.219546  0.306683  0.355205  

20.291808  18.026241  16.764668  
Est. Dis.  [190.3, 128.3, 102.9, 84.5, 69.1, 55.3, 42.0, 28.5, 10.1]  [429.0, 253.5, 176.8, 124.3, 85.3, 55.3, 32.4, 15.5, 3.5]  [636.1, 347.0, 222.4, 142.6, 88.2, 50.8, 25.7, 10.2, 1.9]  

6,130.45  7,096.29  7,447.43  
Q II 

1.262093  1.412527  1.514472  

0.207665  0.292049  0.339704  

17.431369  15.574919  14.526515  
Est. Dis.  [233.3, 147.0, 113.7, 90.3, 71.1, 53.9, 37.2, 16.6]  [516.1, 280.5, 185.4, 123.4, 79.1, 46.6, 23.0, 6.5]  [757.0, 376.4, 226.4, 135.8, 77.3, 39.6, 16.4, 3.7]  

4,239.22  4,986.61  5,284.13  

1.251140  1.396188  1.494358  

0.200729  0.283764  0.330816  

19.182509  17.189656  16.060414  
Est. Dis.  [236.3, 150.7, 118.0, 95.0, 76.1, 58.9, 41.9, 19.5]  [524.6, 289.4, 194.4, 131.7, 86.1, 52.0, 26.5, 7.8]  [771.0, 389.9, 238.8, 146.1, 85.0, 44.7, 19.1, 4.5]  

4,542.73  5,373.95  5,710.91  

1.241406  1.381684  1.476521  

0.194462  0.276245  0.322732  

20.943989  18.817621  17.608961  
Est. Dis.  [239.1, 154.0, 122.0, 99.5, 80.9, 63.8, 46.6, 22.6]  [532.4, 297.7, 202.8, 139.6, 93.0, 57.4, 30.1, 9.2]  [783.9, 402.5, 250.5, 156.0, 92.7, 49.8, 22.0, 5.4]  

4,838.92  5,754.14  6,131.06  
Q III 

1.220604  1.350816  1.439536  

0.180733  0.259707  0.305332  

18.023866  16.286454  15.282699  
Est. Dis.  [295.4, 179.0, 138.1, 110.0, 86.7, 64.6, 32.8]  [644.2, 332.8, 216.0, 140.9, 87.0, 46.5, 14.8]  [937.6, 440.2, 257.8, 149.7, 80.5, 36.1, 9.2]  

3,202.56  3,906.48  4,224.82  

1.211164  1.336630  1.422004  

0.174348  0.251850  0.296767  

19.815651  17.955611  16.877591  
Est. Dis.  [299.6, 183.8, 143.8, 116.4, 93.7, 71.8, 38.0]  [655.8, 344.3, 227.4, 151.3, 95.6, 52.8, 17.6]  [956.3, 457.2, 273.0, 162.1, 89.4, 41.4, 11.0]  

3,421.69  4,194.95  4,548.76  

1.202780  1.324044  1.406468  

0.168593  0.244738  0.288999  

21.616583  19.636810  18.486024  
Est. Dis.  [303.4, 188.3, 149.1, 122.5, 100.4, 79.0, 43.6]  [666.3, 355.0, 238.2, 161.4, 104.2, 59.2, 20.4]  [973.6, 473.1, 287.6, 174.1, 98.3, 46.9, 12.9]  

3635.05  4477.23  4866.59 
The
Since the optimal harvesting strategies produced by the solutions of the discrete optimal control problem described in
The maximum value of Δ is one, and the minimum of zero occurs when the diameter distributions,
The stumpage value model applied in this study is a generalization of the model proposed for Quality I in [
Stumpage prices (€/m^{3}) of
Products  Diameter classes (cm)  

<20  20–40  >40  
Poles  0  11.85  0 
Sawlog  0  9.8  11.76 
High quality sawlog  0  0  9.96 
Total average price  6  22.85  22.32 
Stumpage price models: Quality I,
The last row in
By substituting the transition probabilities in
We thus obtain four main results. First,
The tables and figures mentioned in the above paragraph are included below.
The optimal harvesting strategy, designed to maximize
Population dynamics,
Optimal harvesting strategies,
Optimal strategy
Q I  6205.07  16.04  7188.49  17.50  7468.18  17.00  
6636.65  15.55  7767.29  17.49  8095.43  17.00  
7056.80  15.11  8316.44  17.19  8718.00  17.06  
Q II  4788.21  12.95  5754.83  15.40  6139.05  16.18  
5124.07  12.80  6203.83  15.44  6623.80  15.99  
5453.96  12.71  6647.77  15.53  7098.14  15.77  
Q III  3,502.94  9.38  4,389.27  12.36  4,820.65  14.10  
3,724.85  8.86  4,701.48  12.07  5,185.97  14.00  
3,941.58  8.43  5,007.24  11.84  5,529.35  13.62 
Keyfitz’s Δ values, corresponding to the distance between the diameter distribution associated with the optimal harvesting strategy (for
R = 200 stem/ha  R = 520 stem/ha  R = 840 stem/ha  

Δ 

Δ  λ_{T}  Δ 


Q I  G = 22  0.450415  0.056033  0.397311  0.002602  0.394624  0.000226 
G = 24  0.434304  0.178754  0.392813  0.002866  0.390207  0.000284  
G = 26  0.425244  0.210573  0.437235  0.002866  0.387282  0.000559  
QII  G = 22  0.467745  0.310556  0.490225  0.006628  0.361989  0.004161 
G = 24  0.467935  0.333705  0.487651  0.011089  0.351967  0.004161  
G = 26  0.468382  0.354182  0.478031  0.014921  0.346997  0.004161  
Q III  G = 22  0.348297  0.604092  0.392534  0.275817  0.335904  0.365103 
G = 24  0.422880  0.551102  0.385367  0.303478  0.323309  0.336591  
G = 26  0.417756  0.562400  0.378934  0.337522  0.324559  0.312709 
Applying the optimal harvesting strategy, for Qualities I, II and III, the diameter distribution of the stand evolved through time, leaving the initial equilibrium distribution and tending toward a diameter distribution more characteristic of evenaged stands. For high recruitments (
Evolution of the diameter distribution with the economically optimal harvesting strategy throughout a harvest cycle of 70 years for Quality I,
The influence of small variations in the discount rate or the natural mortality rates on the optimal harvesting strategies was minimal, because they did not change the general pattern of the solutions.
This study proposes a model that combines an objective function comprising the
Following López
López
The basal area indirectly resulted in optimization problems, such as a “bangbang control” variable, which oscillated between
By applying optimal harvesting strategies, the schedules aim for the highest
For high recruitments, the intense harvest applies to the upper diameter class at the beginning (first period) and the end (seventh period) of the planning horizon. In addition, for medium and high recruitments, the intense harvest gets performed in every period of the planning horizon for the diameter classes corresponding to poles, while low recruitments follow a different pattern. In this case, the intense harvest is performed in the upper diameter classes, since low recruitments are associated with a lower number of stems per hectare with a higher average diameter.
For the three qualities, this harvest pattern caused a decrease in the number of stems of the first diameter class, such that at the end of the time horizon, the diameter range of the distribution narrowed and shifted from a reversed Jshaped curve to a bellshaped curve, centered on the medium diameter classes (18–36 cm), as is typical of an evenaged stand distribution (see
There is not a significant change in the harvest pattern for the optimal strategy when higher discount rates are considered, except for a slight trend to advance harvest in the upper diameter classes. This result, leading to an earlier harvest, is consistent with an increase in the discount rate.
The
By applying the optimal harvesting strategies, at the end of
Regarding the sustainability of the optimal harvesting strategies, all the scenarios were unsustainable (see
In conclusion, the proposed discrete optimal control model was used to analyze if the introduction of an
The authors declare no conflict of interest.