Comparison and analysis of self-thinning models based on diameter-based maximum size-density relationships

Abstract

Identifying self-thinning phase is one of the key issues to deal with stand density management and simulate the growth and yield in a forest stand. This study was conducted to figure out the carrying capacity of silver birch plantations and provide the self-thinning zone for practicability. The analyzed data were the stand-wise observations from the experimental plots of silver birch plantations in southern and central Finland, which were established and measured between 1977 and 2020. Models for the diameter-based maximum size-density relationship (MSDR) were fitted only to the censored data after classifying the stand observations undergoing self-thinning phase. The applied diameter-based MSDR in this study were Reineke’s self-thinning rule (STR), competition-density (C-D) rule, and Nilson’s stand sparsity index (SSI). Model fitting was executed using linear quantile mixed-effect model for Reineke’s STR and Nilson’s SSI and nonlinear mixed-effect model for the C-D rule. For practical purpose, a lower boundary of the self-thinning zone based on the developed MSDR was analyzed using the concept of relative density (RD) according to the ratio of stem number (N) to maximum stem number (N_max) at quadratic mean diameter (DQ). Linear quantile mixed models were fitted well with the 0.99 level for Reineke’s STR and with the 0.01 level for Nilson’s SSI to find the MSDR between DQ and N. Among the fitting methods for the C-D rule, the three-parameter method performed better than the four-parameter method or the method with Reineke’s slope of −1.605. The fitted slope of Reineke’s STR in this study was −1.5848, which was close to the original slope from Reineke’s. Our results from the developed models and the observations undergoing self-thinning phase implied that the slope of MSDR is not always invariant. Moreover, a different slope for MSDR was suggested by initial planting density and the stand development stage; a self-thinning phase occurred earlier with lower initial planting density, which meant a steeper slope. When the lower boundary of the self-thinning zone was analyzed applying RD to the MSDR models, the results suggested that RD 0.7 for Reineke’s STR at DQ ≤ 18.65 cm and RD 0.8 for Nilson’s SSI at DQ ≥ 18.65 cm provided adequate level for self-thinning phase. It is considered in this study that the diameter-based MSDR measures were analyzed and examined adequately, and the practical self-thinning zone was provided using RD for silver birch plantations.