Re-Expression of the Lorenz Asymmetry Coefficient on the Rotated and Right-Shifted Lorenz Curve of Leaf Area Distributions

The Gini coefficient, while widely used to quantify inequality in biological size distributions, lacks the capacity to resolve directional asymmetry inherent in Lorenz curves, a critical limitation for understanding skewed resource allocation strategies. To address this, we extend our prior geometric framework of the rotated and right-shifted Lorenz curve (RRLC) by introducing two original asymmetry metrics: the positional shift ratio (<i>P_L</i>, defined as xc/2, where xc is the <i>x</i>-coordinate of the RRLC's maximum value point) and the area ratio (<i>P_A</i>, defined as <i>A_L</i>/(<i>A_L</i> + <i>A_R</i>), where <i>A_L</i> and <i>A_R</i> denote the areas under the left and right segments of the RRLC). These indices uniquely dissect contributions of dominant versus small individuals to overall inequality, with <i>P_L</i> reflecting the peak position of the RRLC and <i>P_A</i> quantifying the area dominance of its left segment. Theoretically, <i>P_L</i> directly links to the classical Lorenz asymmetry coefficient <i>S</i> (defined as S=xc'+yc', where xc',yc' is the tangent point on the original Lorenz curve with a 45° slope) through <i>S</i> = 2 - 2<i>P_L</i>, bridging geometric transformation and parametric asymmetry analysis. Applied to 480 <i>Shibataea chinensis</i> Nakai shoots, our analysis revealed that over 99% exhibited pronounced left-skewed distributions, where abundant large leaves drove the majority of leaf area inequality, challenging assumptions of symmetry in plant canopy resource allocation. The framework's robustness was further validated by the strong correlation between <i>P_A</i> and <i>P_L</i>. By transforming abstract Lorenz curves into interpretable bell-shaped performance curves, this work provides a novel toolkit for analyzing asymmetric size distributions in ecology. The proposed metrics can be applied to refine light-use models, monitor phenotypic plasticity under environmental stress, and scale trait variations across biological hierarchies, thereby advancing both theoretical and applied research in plant ecology.