Is De Liocourt's Distribution Stable?

Abstract

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.

Optimizing the management of uneven-aged Pinus nigra stands between two stable positions

“…Following López et al (2007López et al ( , 2008López et al ( , 2012López et al ( , 2013, we calculated the transition probabilities considering that pk is an average global property shared by all of the individuals in the same class k, and that, given the basal area constraints and the low basal area scenarios of this study, stand density has a small influence on diameter growth. Moreover, in the case of constrained optimization matrix models, in which the search for optimal paths throughout the phase space is mainly governed by the objective function, optimal decisions are essentially conditioned by the stumpage prices, i.e., by selective harvesting, and thus stand density has a minor influence on the objective function.…”

“…The population dynamics model that integrates recruitment, harvesting, and stem migration throughout the diameter classes over time is based on the model proposed by López et al (2007López et al ( , 2008 for Fagus sylvatica L., and by López et al (2012López et al ( , 2013 for Pinus nigra Arn. stands.…”

“…We calculated the transition probabilities by applying the method introduced in López et al (2012) considering that, given the basal area constraints and the low basal area scenarios of this study, stand density has a small influence on diameter growth. Thus, the diameter growths were defined by the Bertalanffy-Richards function (Von Bertalanffy 1938, Richards 1959: adjusted using regression analysis to data points [t, D(t)] obtained from tables by Gómez Loranca (1998).…”

“…Stand management practices aimed at conserving this equilibrium are important because they represent the stand's ability to persist in the face of external disturbances. Managing a stand by applying the "sustainable/stable" harvesting strategy (López et al 2007(López et al , 2008(López et al , 2012(López et al , 2013) is a good way to reach and maintain that equilibrium with sustainability. In fact, the "sustainable/stable" harvesting strategy is aimed at reaching the stable diameter distribution of the stand for each harvest, yielding for all the diameter classes and time steps the following "sustainable/stable" harvest rate (eqn.…”