Figure 1. Equations and graphs illustrating the effect of tree size and competition on demographic rates.

The equations relate diameter growth (top panels: A,B), longevity (middle panels: C,D), and ingrowth (bottom panel: E) to tree diameter (D; left panels: A,C) and crown area index (CAI_h and CAI₀; right panels: B, D). The effect of diameter on longevity (L_S(D), middle left) is specified as the product of an increasing power function and a decreasing logistic function (grey dashed lines). To aid with model fitting, we reparameterized the slope (λ) of the logistic function so that it takes a value of 0.01 at diameter D_0.01, and the inflection point is expressed as a proportion (θ) of D_0.01, using the relationship λ = ln(99)/(D_0.01·(1−θ)). Ingrowth is capped at the maximum rate when crown area index is lower than CAI_0.05, the lowest 5^th percentile of observed CAI₀ values in plots where the species is present. Including this 5^th-percentile cap avoids overestimating recruitment in open stands, for which there were few observations. All functional forms were plotted using parameter values for sugar maple (Table 1).