In Defence of the -3/2 Boundary Rule: a Re-evaluation of Self-thinning Concepts and Status

The -3/2 power rule, or -3/2 self-thinning rule, was accepted 10 years ago as an important generalization, but has recently been questioned by a number of authors. This paper assesses what remains of the rule. While it has been empirically established that size-density trajectories followed by self-thinning plant populations do not necessarily follow a -3/2 slope, a more general power rule describing a density-dependent upper limit to mean shoot biomass per plant (the '-3/2 boundary rule') remains largely intact. Principal component analysis (PCA) overestimates the steepness of the thinning slope if y:x variance ratio is greater than 1:1. Lonsdale's (Ecology71: 1373-1388) overall mean PCA slope of -0·6 for biomass-density suggests a true mean slope close to the theoretical value of -0·5. Reduced major axis (RMA) regression appears a reasonable approximation for the -3/2 but not the -1/2 formulation of the rule. Fitting of a linear functional relationship (LFR) is a more appropriate slope estimation procedure, not previously used for data on thinning. None of these procedures estimates a boundary line that is not transgressed by any data point except through errors of measurement. Mortality due to overcrowding ensues when a small, suppressed plant no longer holds its leaves high enough in the canopy to maintain a positive carbon balance. It follows that LAI should remain constant during thinning, and that self-thinning theory should be developed in terms of maximum leaf area index and the biomass required to support it. A derivation is presented and some of its consequences are examined.