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

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. • Three diameter-based maximum size-density relationships were fitted well and showed different behaviors. • Relative density applied adequately for the lower boundary of self-thinning zone. • Maximum size-density relationships differed by initial planting density and stand development stage. • Models and observations implied a different slope rather than a fixed slope on the relationship of ln DQ -ln N . • Combined maximum size-density relationship with relative density was suggested according to diameter range for practicability.