# Modeling Production Processes in Forest Stands: An Adaptation of the Solow Growth Model
## Appendix A  
Calculating the root phytomass in a forest stand based on the data on the phytomasses of the stem with the bark, branches, and leaves (needles). The total phytomass of the non-photosynthesizing components of the forest stand, \(K(T)\), includes root phytomass (usually 15-29% of the total phytomass in the stand). However, the root phytomass of the trees in the stand usually cannot be measured in field studies. It was previously proposed to describe the distribution of the phytomass among tree or forest stand components by the Zipf-Pareto equation in the rank form [31; 32; 33]:  
\[M_{i}(T)=M_{1}(T)i^{-b(T)} \tag{10}\]  
Or  
\[\ln M(i,T)=\ln M_{1}(T)-b(T)\cdot\ln i \tag{11}\]  
where \(i\) is the rank of the phytomass component (the stem with the bark, roots, branches, leaves or needles) (starting with the component that has the greatest phytomass, \(M_{1}\)); \(M_{i}\) is the phytomassof the rank \(i\) component; and \(b(T)\) is the coefficient characterizing percentages of phytomasses of different ranks. If \(b(T)\to 0\), the phytomasses of all components are approximately equal to each other; when \(b(T)\) is increased, the phytomass of the rank \(1\) component (usually tree stems) is considerably greater than the phytomass of any other tree component in the stand. Regarding (A2) as a regression equation with the known values of the phytomass and ranks of phytomasses of aboveground components (stems, branches, leaves (needles)), one can find coefficient \(b\) and then calculate root phytomass:  
\[MR(T)=MS(T)\cdot 2^{-b(T)}\] (A3)  
From (IS) one can derive a formula for the total phytomass of the stand, \(M(T)\), at any age:  
\[M(T)=\sum_{i=1}^{4}M_{i}(T)=M_{1}\sum_{i=1}^{4}i^{-b(T)}=M_{1}(T)G(b(T))\] (A4)  
\[\tau a\text{e}\ G(b(T))=\sum_{i=1}^{4}i^{-b(T)}=\frac{1}{1^{b(T)}}+\frac{1}{2^ {b(T)}}+\frac{1}{3^{b(T)}}+\frac{1}{4^{b(T)}}.\]  
Function \(G(b(T))\) is not related to the current values of the stand phytomass; it characterizes the proportions of phytomass components at age \(T\). As the rank of the phytomass of leaves (needles) of forest stands aged above \(10\) years is always equal to \(4\)[33], by taking into account (A1) and (A4) we obtain:  
\[ML(T)=M_{4}(T)=M_{1}(T)\cdot 4^{-b(T)}=M(T)\frac{4^{-b(T)}}{G(b,T)}\] (A5)  
\[K(T)=M(T)-ML(T)=M(T)[1-\frac{4^{-b(T)}}{G(b,T)}]=M(T)\frac{1+2^{-b(T)}+3^{-b(T )}}{1^{-b(T)}+2^{-b(T)}+3^{-b(T)}+4^{-b(T)}}\] (A6)  
Thus, knowing function \(b(T)\), from the Zipf-Pareto equation for phytomass distribution, we can obtain the values of \"phyto-capital\" \(K(T)\) and the mass of the photosynthetic apparatus \(ML(T)\) for the forest stand of age \(T\). Figure 11 shows a typical function of the rank distribution of fir stand phytomass components at ages between \(40\) and \(300\) years. Figure 11: Functions of phytomass distribution among components of the Siberian fir _Abies sibirica_ Ldb. stand (the West Sayan Mountains, the upper reaches of the Kebezh River, \(52^{\circ}\)N, \(89^{\circ}\)E) at ages of \(40\) and \(300\) years. (**1**) aboveground components of tree phytomass in the \(300\)-year-old stand; (**2**) calculated root phytomass in the \(300\)-year-old stand; (**3**) aboveground components of tree phytomass in the \(40\)-year-old stand; (**4**) calculated root phytomass in the \(40\)-year-old stand [10]. Equation (A2) can be regarded as a regression equation with the known values of phytomasses and ranks of phytomasses, for which one should calculate its coefficients. Assuming that the root phytomass of the Siberian fir is a rank 2 component, the values of the phytomasses of aboveground tree components are described with high accuracy (R\({}^{2}\) > 0.99) by Equation (A2) (Figure A1). Knowing coefficients of regression Equation (A2), one can calculate root phytomass. The method described above was used to determine the ranks of roots and to calculate the values of root phytomass in forest stands. Is Equation (A2) sufficiently accurate?