Discussion
Therefore, under particular circumstances, the inevitable variance in growth rate surely has a probability in pulling back N (D ) from D -7/3, which determined byG (D ) and M (D ) in MSTF, to approximatelyD -2 in statistics, but not precisely (Fig. 2 ). The so-called energetic equivalence among different size classes in forests(Perkins et al. 2019), is not a result of biological mechanism (i.e. considering a forest as a tree(West et al. 2009)), but a statistical coincidence. Meanwhile, sinceV (D ) is significantly affected by the accounting time scale of G (D ), which is reflected as the bin width of size classes in size structure estimation, it is no wonder that the estimated forest size structures deviate from the power-law distribution with the changes of growth rate, bin width selection, or estimation methods(Whiteet al. 2008).
Although the evidences of growth rate variance, mean growth rate, mortality and forest size structure were well linked in the special case of MSTF, more generalized interpretation and quantification of their relationships are still needed. Basically, an explicit equilibrium solution to the Kolmogorov forward equation, which in the form ofN (D ) = f (G (D ), V (D ),M (D )), if exist, would be expected, so that foresters can get an intuitive understanding to the respective roles ofG (D ), V (D ), and M (D ) on forest size structure formation, and predict the large scale forest size structure from small scale inventories. Further than the static analysis on demographic equilibrium state, time dynamic analysis of forest size structure would be more challenging, especially in considering of the stochasticity in growth rate. Although ecologists believe that forests would internally tend to equilibrium in the absence of disturbance, how a forest started with any arbitrary size-density distribution finally converge to the approximately power-law distribution has never been strictly proved, mathematical approaches in stability theory may be adopted for the analysis.
However, as a posteriori function, Kolmogorov forward equation does not provide any biological inferences to forest dynamics, it simply reveals the physical truth in how growth and mortality affect individual numbers in different size classes, biological insights on specific processes in growth and mortality, e.g. size dependent growth, age dependent death, or the effects of competition, etc., require additional experimental and theoretical investigations. But without a clear understanding to the certain mathematical results, ecological phenomena may be mis-interpreted with inappropriate anticipation to biological mysteries.