Prediction and verification for the functional form of growth variance
In accepting N (D ) =N 0D -2,G (D ) = kD 1/3, andM (D ) = 2kD -2/3 in MSTF, the functional form of V (D ) can be expected by putting the three known functions into Kolmogorov forward equation (see Supporting information), under demographic equilibrium condition, this lead to: and:
In a prior perspective, it is basically acceptable that the variance of growth rate increases with tree size and mean growth rate, a power relationship between diameter and growth rate variance was also verified with data from Barro Colorado Island (BCI) forest(Condit et al.2012) (Fig. 1. B ).
Here I provide a statistical interpretation to the positive correlation between growth rate variance and tree size, that for G (D ) = kDλ , if growth rate variance derives from the intrinsic stochasticity in growth coefficient k and exponentλ , which mean values are μk andμλ , and variances are σ2k and σ2λ , then V (D ) can be expressed as (see Supporting information for full derivation(Limpert et al. 2001)): (5)
In acceptable range of the parameter values based on theoretical and empirical studies, where μλ = 1/3 and σ2λ varies around 0.01 ~ 0.1, functional curve of the statistical-basedV (D ) is quite close to the power-law function with the exponential value of 4/3 (Fig. 1. A ). Besides of the exponential value, the coefficient of V (D ), as expected to be 0.6 k when the coefficient of G (D ) is k , depends on the accounting time scale of G (D ). For instance, G (D ) accounted with 5-years’ time interval is 5 times more than that accounted with 1-year’s time interval, but variance of the former will be 25 times more than the latter, which is the square of the ratio in G (D )’s magnitude changes. Hence the relative magnitude between G (D ) and V (D ) will be different with the change of accounting time scale.
In estimating of forest size structure, if bin widths of size classes are at the magnitude of one to ten centimeters, as was once used in MSTF(Enquist & Nicklas 2001), that would be diameter increments in 5- to 10-years’ growth according to empirical evidences. In this premise, the estimated coefficient value of V (D ) based on the data from BCI forest is just close to that being expected to satisfy the −2 power-law distribution (Fig. 1. B ). However, if size classes are bound with smaller bin widths, as in later studies challenging MSTF(Muller-Landau et al. 2006b), the relative magnitude ofV (D ) will be quite small to lead significant deviation of size structure from that predicted with N (D ) =N 0exp[−∫M (D )G (D )-1dD ]G (D )-1.