Deriving the mean mass-density trajectory by reconciling the competition-density effect law with the self-thinning law in even-aged pure stands

Abstract

The competition–density (C–D) effect law refers to the relationship between mean mass w and density ρ at a particular moment among a set of tree populations grown at a wide range of densities. The self-thinning law refers to the time trajectory of w and ρ in overcrowded stands. Because these two laws have not yet been theoretically harmonised, the aim of this paper is to achieve the unification of the two laws. Under the assumption that the reciprocal equation of the C–D effect in self-thinning stands and the self-thinning equation both hold, the slope of the reciprocal equation becomes the same as that of the self-thinning equation on logarithmic scales as the growth stage progresses. Finally, the reciprocal equation is converted to the w–ρ trajectory, eliminating the biological time from the reciprocal equation. The w–ρ trajectory of stands starting with any initial density has thus been explicitly formulated. Larger values of the relative mortality rate play an important role in relieving the C–D effect and cause the w–ρ trajectory to approach the self-thinning line at an earlier stage of growth. Stands exponentially decreasing in number obey the self-thinning law after a sufficient lapse of biological time, irrespective of their initial densities. Unknown functions, such as the survivorship curve and the ceiling biomass, have been explicitly represented as a function of biological time. The approximate expression for the w–ρ trajectory suitably mimics the time trajectory of mean stem and density in an eastern pine plantation.