Can an Exponential Function Be Applied to the Asymptotic Density–Size Relationship? Two New Stand-Density Indices in Mixed-Species Forests

Abstract

This study presents two stand-density indices (SDIs) based on exponential density decline as a function of quadratic mean diameter for all species combined in mixed-species forests with 22 species mix grouped in four species groups. The exponential-based density–diameter relationship, as well the density index corresponding to the slope or instantaneous mortality rate parameters, was compared with those based on power-law density–diameter relationship. A dataset of 202 fully stocked circular plots at maximum density was used for fitting the models, and a dataset of 122 circular plots was used for validation stand density index for all species combined of mixed-species stands. The dataset for validation was independent of dataset for model development. The first stand-density index showed a density management graphic (DMG) with a variable intercept and common instantaneous mortality rate, and the second index showed a DMG with common intercept and variable mortality rate. Additionally, the value of the initial density of the fitted line was more realistic than those generated by the potential model for all species combined. Moreover, the density management diagrams showed a curvilinear trend based on the maximum stand density index in graphical log–log scale. The DMGs could be interpreted as forest scenarios based on variable initial density and common management objectives or the same density and different management objectives for forest-rotation periods involving all species combined in mixed-species stands. The fitting of exponential and potential equations for species or species groups showed that the density–size relationships in mixed-species forests should be modeled for all species combined because the disaggregation of mixture species represented a weak tendency for each species or species group and the resultant fitted equations were unrealistic.

Relación densidad-tamaño máxima para bosques mixtos de Nuevo San Juan Parangaricutiro, Michoacán, México

“…The use of the potential and exponential models in mixed-species forests has been analyzed and compared in few studies (Quiñonez-Barraza & Ramírez-Maldonado, 2019). Due to differences in the mathematical structures of both models, the value of the slope varies significantly, fueron más pronunciadas en RC que en RFE.…”

“…According to some studies (Ducey & Knapp, 2010;Torres-Rojo & Velázquez-Martínez, 2000), the original Reineke (1933) relationship, and its SDI, have been reported to be inadequate for stands with species mixture and complex structures (Tang et al, 2016). For this reason, alternatives have been developed to model MSDR (Rivoire & Moguedec, 2012;Sterba & Monserud, 1993;Torres-Rojo & Velázquez-Martínez, 2000); furthermore, alternative SDI have been proposed for uneven-aged stands (Long & Daniel, 1990) and to incorporate species composition (Ducey & Knapp, 2010;Torres-Rojo & Velázquez-Martínez, 2000), so they can be used to construct density guides (Quiñonez-Barraza & Ramírez-Maldonado, 2019). However, forestry researchers face great challenges for including composition within a mathematical structure that would explain its influence on the trajectory of the MSDR (Quiñonez-Barraza & Ramírez-Maldonado, 2019;Rivoire & Moguedec, 2012;Torres-Rojo & Velázquez-Martínez, 2000).…”

“…De acuerdo con algunos estudios (Ducey & Knapp, 2010;Torres-Rojo & Velázquez-Martínez, 2000), se ha encontrado que la relación original de Reineke (1933), así como su IDR, resulta inadecuada para rodales con mezcla de especies y estructuras complejas (Tang et al, 2016). Por ello, se han desarrollado alternativas para modelar la RDTM (Rivoire & Moguedec, 2012;Sterba & Monserud, 1993;Torres-Rojo & Velázquez-Martínez, 2000); asimismo, se han propuesto alternativas de IDR para rodales incoetáneos (Long & Daniel, 1990) y para incorporar la composición de especies (Ducey & Knapp, 2010;Torres-Rojo & Velázquez-Martínez, 2000), de tal forma que puedan usarse para construir guías de densidad (Quiñonez-Barraza & Ramírez-Maldonado, 2019). No obstante, los investigadores forestales se enfrentan a grandes retos para incluir la composición dentro de una estructura matemática que explique su influencia en la trayectoria de la RDTM (Quiñonez-Barraza & Ramírez-Maldonado, 2019;Rivoire & Moguedec, 2012;Torres-Rojo & Velázquez-Martínez, 2000).…”

“…Por ello, se han desarrollado alternativas para modelar la RDTM (Rivoire & Moguedec, 2012;Sterba & Monserud, 1993;Torres-Rojo & Velázquez-Martínez, 2000); asimismo, se han propuesto alternativas de IDR para rodales incoetáneos (Long & Daniel, 1990) y para incorporar la composición de especies (Ducey & Knapp, 2010;Torres-Rojo & Velázquez-Martínez, 2000), de tal forma que puedan usarse para construir guías de densidad (Quiñonez-Barraza & Ramírez-Maldonado, 2019). No obstante, los investigadores forestales se enfrentan a grandes retos para incluir la composición dentro de una estructura matemática que explique su influencia en la trayectoria de la RDTM (Quiñonez-Barraza & Ramírez-Maldonado, 2019;Rivoire & Moguedec, 2012;Torres-Rojo & Velázquez-Martínez, 2000). Además de la composición de especies, la RDTM con mezcla de especies también se ha analizado bajo otros enfoques (del Río et al, 2016;Weiskittel et al, 2009) con base en la integración de valores de gravedad específica de la madera (Ducey & Knapp, 2010;Woodall et al, 2005), variables climáticas (Andrews et al, 2018;Condés et al, 2017;de Prado et al, 2020), calidad del sitio (Reyes-Hernández et al, 2013;Weiskittel et al, 2009) o la combinación de la gravedad específica y tolerancia a la sombra (Ducey et al, 2017).…”

Modeling Dynamics of Structural Components of Forest Stands Based on Trivariate Stochastic Differential Equation

“…Stands of small quadratic mean diameter have a large number of trees, while stands of large quadratic mean diameter have relatively few [20]. To develop the number of trees per hectare trajectory for all stands, it is necessary to determine a relationship describing development of the number of trees per hectare development versus the quadratic mean diameter [16,35,36]. The relationship between the quadratic mean diameter (increasing over age) and the number of live trees per hectare (decreasing over age) are commonly described by means of a "limiting relationship" [13].…”