This study proposes a discrete optimal control model to obtain harvest strategies that maximize the net present value (NPV) of the timber harvested from uneven-aged Pinus nigra stands located in the Spanish Iberian System, between two stable positions. The model was constructed using an objective function that integrates financial data on the harvesting operations with a matrix model describing the population dynamics. The initial and final states are given by the stable diameter distribution of the stand, and the planning horizon is 70 years. The scenario analysis corresponding to the optimal solutions revealed that the stand diameter distribution does not deviate substantially from the equilibrium position over time and that the NPV for the optimal harvesting schedule was always greater than the NPV for the "sustainable/stable" harvesting strategy. The NPV increase for the different scenarios is between 5.36% and 14.43%, showing a greater increase in higher site index scenarios and higher recruitments.
This study analyzes the stability of De Liocourt's distribution, investigating the influence of factors such as site index, recruitment, and basal area. It is proved that De Liocourt's distribution is not stable, and some simple models providing better fit to the stable diameter distribution of the stand than De Liocourt's are introduced. The stable diameter distributions obtained were characterized by a decrease in stem density in relation to the corresponding De Liocourt's distributions for low-and high-diameter classes and an increase for intermediate-diameter classes. Despite their instability, De Liocourt's distributions have shown a high degree of fit to the corresponding stable diameter distributions. The goodness of fit between both distributions was better for high recruitment, high site quality, and low basal area. FOR. SCI. 58 (1) 2009). These models are defined by the finite difference linear system of equations N(t ϩ 1) ϭ AN(t), where N(t) and N(t ϩ 1) are column vectors that contain the number of stems/ha within each diameter class at time t and t ϩ 1, respectively, and A is a square primitive matrix that contains, for each time step, the transition probabilities between adjacent classes and individual recruitments. The population growth rate is the dominant eigenvalue 0 of matrix A. By asymptotic analysis (long-term behavior), we know that, independent of the initial conditions, when 0 Ͼ 1, the total number of stems/ha of the tree population increases exponentially over time (unless harvests are conducted), when 0 Ͻ 1, the population is decaying until extinction, and when 0 ϭ 1, a stable distribution proportional to the right eigenvector W 0 of A corresponding to 0 is obtained. Gotelli (2001) refers to this special case of stable distribution when 0 ϭ 1 as the "stationary distribution," and this is also the case that we are referring to here.In general, the concept of stability is closely associated with the concept of perturbation: a system is considered stable if it always returns to a reference position (equilibrium) after small perturbations (otherwise, the system is said to be unstable). In the case of these projection matrix models, this stability property is stronger, because the stable distribution (stationary distribution) is reached independently of the initial conditions. Applied to tree populations, the stable diameter distribution of a stand is reached when it neither increases in size nor changes in structure; that is, the number of stems/ha within each diameter class remains constant after each time step. These stable diameter distributions are closely dependent on recruitment, removal, and stem migration throughout the diameter classes over time (Schütz 2006). A method to obtain these distributions, closely related to the right eigenvector W 0 of the transition matrix A corresponding to the dominant eigenvalue 0 , the stand basal area G, and the global amount of recruitment R, was introduced in López et al.
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