# Modeling Production Processes in Forest Stands: An Adaptation of the Solow Growth Model
## 2 Materials and Methods  
### The EcoSolow Model of Production Processes in the Forest Stand  
In this study, we have adapted [PERSON]'s model of economic growth to describe the growth of forest stands and estimate their NPP. In our adaptation of the [PERSON] model--the EcoSolow model (ESM), the process of phytomass growth in a forest stand is regarded as analogous to a manufacturing process, which is determined by two production factors: capital and labor. In the ESM, capital, \(K\), consists of the non-photosynthesizing components of the phytomass--tree stem, roots, and branches, and labor force, \(L\), is represented by leaves (or needles), which carry out photosynthesis. The GPP energy produced during photosynthesis is partly expended for the current \"consumption\" (respiration) of the plant, and the other part is \"invested\" and transformed into new phytomass, which is distributed between the non-photosynthesizing components of the trees and their photosynthetic apparatus. Tree mortality in the ESM is regarded as analogous to capital depreciation (wear and tear of the equipment etc.) in the manufacturing process. The non-photosynthesizing component, \(K(T)\), of the forest stand of age \(T\) comprises phytomasses of stems \(MS(T)\), roots \(MR(T)\), and branches \(MB(T)\):  
\[K(T)=MS(T)+MR(T)+MB(T) \tag{1}\]  
The photosynthesisizing component consists of needles (leaves) of the trees in the forest stand with phytomass \(ML(T)\). Energy production, \(GPP\), by trees during photosynthesis will be described by the Cobb-Douglas production function [24; 25]:  
\[Y(T)=GPP(T)=A[K(T)^{\alpha}]\cdot[ML(T)^{\beta}] \tag{2}\]  
where \(T\) is the forest stand age, \(A\), and \(\alpha\) and \(\beta\) are coefficients. Let us then consider the simplest case, when in Equation (2), \(\alpha+\beta=1\), and, hence, in Equation (2) there are only two free parameters: \(A\) and \(\alpha\). Part of the energy generated during photosynthesis is expended for maintaining the current functions of the trees (in fact, respiration). These expenditures, \(Z(T)\), will be assumed to be related to the total phytomass of the stand, \(M(T)=K(T)+ML(T)\):  
\[Z(T)=f(T)\cdot M(T) \tag{3}\]  
where \(f(T)\) is the function of current expenditures. In the simplest case, current expenditures can be assumed to be directly proportional to the stand phytomass, i.e., \(Z(T)=sM(T)\). Coefficient \(s\) can be regarded as expenditure for the functioning of the unit phytomass (we assume that these expenditures do not differ between tree components). The difference between production \(Y\) and current expenditures of energy \(Z(T)\) will characterize net primary production, NPP:  
\[NPP(T)=GPP(T)-Z(T)=A[K^{\alpha}(T)]\cdot[ML^{1-\alpha}(T)]-sM(T) \tag{4}\]  
In Equation (4), the values of coefficients of elasticity \(\alpha\) and \((1-\alpha)\) characterize an increase in the photosynthesis rate of the trees with an increase in the per unit values of \(K\) and \(ML\):  
\[\frac{\frac{\partial Y}{\partial K}}{\frac{Y}{K}}=\alpha;\frac{\frac{\partial Y }{\partial ML}}{\frac{Y}{\partial L}}=1-\alpha\]  
As can be inferred from Equation (2), production cannot occur if at least one production factor, \(K\) or \(L\), is equal to zero. Ecologically speaking, the death of the foliage due to some reasons (insect damage, fire, effects of pollutants, etc.) stops the \"production\" and leads to the death of the tree irrespective of the value of \(K\) of the non-photosynthesizing phytomass. The condition that follows from Equation (2) is ecologically valid, in contrast to the ecologically incorrect solution for the logistic models. This condition enables describing the processes of tree death caused by critical natural or human-induced events. If net primary production of the stand at age \(T\) is completely \"invested\" in the stand phytomass increment, values of \(K(T)\) and \(ML(T)\), and coefficients \(A\), \(\alpha\), \(s\) of Equation (4) must be known to be able to model the production process in the forest stand.