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.