Introduction
Understanding the formation of forest size structure (i.e. the
size-frequency distribution of trees) is crucial for forest protection
and management. Although a great deal of statistics and theoretical
investigations have been made, I would say that an essential defect in
this research area results from the disadvantages in mathematical
analysis, rather than lack of revealing new biological or ecological
processes.
Two general governing equations, the McKendrick-von Foerster equation
(Eq. 1) and the Kolmogorov forward equation (Eq. 2), have long been
proposed for the size structure dynamics in plant communities(Von
Foerster 1959; Hara 1984; Kohyama 1991; Condit & Sukumar 1998; Coomeset al. 2003; Muller-Landau et al. 2006b), in which the
former is commonly used, it can be considered as a simplification of the
latter, by ignoring the variance of tree growth rate in the same
size-class, i.e. letting V (D ) in Eq. 2 equals zero.
(1)
(2)
where N represents individual number with size D (measured
by trunk diameter) and at time t , G (D ),V (D ), and M (D ) are mean growth rate, growth
rate variance, and mortality in each size-class, respectively.
Although growth rate variance is inevitable in natural world, the reason
for rarely using Kolmogorov forward equation is largely due to
difficulties in solving the second-order partial differential equation
analytically, and ecologists believe that discarding the variance term
has little impact in analyzing forest size structure under demographic
equilibrium state, so most studies are based on the equilibrium solution
of
McKendrick-von
Foerster equationN (D )
=N 0exp[−∫M (D )G (D )-1dD ]G (D )-1(where N 0 is a constant representing the ideal
number of trees in the smallest size-class, i.e. when D closes to
zero.)(Hara 1984; Kohyama 1991; Condit & Sukumar 1998; Coomes et
al. 2003; Muller-Landau et al. 2006b; Moore et al. 2020).
However, the deviation of observations to the mathematical prediction
has
been causing debates, some of which cannot be simply attributed to
disturbances or the non-ideality of field data(Coomes et al.2003; Muller-Landau et al. 2006a; Muller-Landau et al.2006b; Stegen & White 2008; Farrior et al. 2016; Zhou & Lin
2018; Moore et al. 2020). One puzzling problem rises from the
non-self-consistency of metabolic scaling theory of forest
(MSTF)(Enquist et al. 2009; West et al. 2009). In this
theory, power-law functions of N (D ), G (D )
and M (D ) are proposed and well fitted with convincing data
cases, which are N (D ) =N 0D -2,G (D ) = kD 1/3, andM (D ) = 2kD -2/3,
respectively(Enquist et al. 2009; West et al. 2009).
However, putting the functional form of G (D ) andM (D ) into the equilibrium solution leads toN (D ) =N 0D -7/3 rather thanN (D ) =N 0D -2. Lin and I (2018)
have interpreted that the non-self-consistency results from the
neglection of growth rate difference among size-classes in math
derivations of MSTF(Zhou & Lin 2018), which actually led to an equation
of N (D ) =N 0exp[−∫M (D )G (D )-1dD ]
in self-thinning. Therefore, the data were actually well fitted with a
flawed theory, there is still a missing link in explaining the formation
of forest size structure under demographic equilibrium state.
Here I will show how the stochasticity in tree growth rates in the same
size class, which has been ignored in previous studies, affects forest
size structure significantly, and solves the paradox in MSTF.