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.