The self-thinning surface

Abstract

This article introduces a generalized expression of the self-thinning rule, B = KSαNβ, where B is stand biomass per unit area, N is stand density, S is relative site index, and K, α and β are parameters. On log scales, this equation becomes a self-thinning surface that defines a density-dependent upper frontier of stand biomass over a gradient of site productivity for a given species. This equation is formulated for parameter estimation as a stochastic frontier function with two error components that have different distributional properties. As an example, maximum likelihood estimates of the self-thinning surface and its confidence envelope were shown for Pinus radiata (D. Don). Furthermore, site occupancy was estimated through one of the error components of the stochastic frontier function. The conditional response of mortality at any given site occupancy was revealed by using regression quantiles. Light mortality was associated with increases in site occupancy, while heavy mortality caused a reduction in site occupancy. Changes in the estimated site occupancy had a linear relationship with changes in log stand density. The dynamic interplay between site occupancy and mortality, together with the random external effects on the self-thinning frontier, was suggested to drive the growth trajectories of individual stands during stand growth and self-thinning. Consequently, individual stands seldom travel along their self-thinning frontiers but are more likely to converge toward them during the self-thinning phase of stand development.