# 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
To verify the EcoSolow model, let us consider the following balance equation for the phytomass of the forest stand of adjacent ages \(T\) and \(T+\Delta T\). For coniferous species, the youngest age classes are \(\Delta T\) = 20 years, and for hardwood species, \(\Delta T\) = 10 years:  
\[M(T+\Delta T)=M(T)+NPP(\Delta T)-D(\Delta T)-W(\Delta T) \tag{5}\]where \(NPP\) (\(T\), \(T+\Delta T\)) is primary production of the forest stand aged between \(T\) and \(T+\Delta T\) years; \(M(T)\) and \(M\) (\(T+\Delta T\)) are phytomasses of the forest stand aged \(T\) and \(T+\Delta T\); \(D(\Delta T)\) and \(W(\Delta T)\) are phytomass loss and fall-off for \(\Delta T\) years. Let us rewrite Equation (5) as follows:  
\[NPP(\Delta T)=M(T+\Delta T)-M(T)+D(\Delta T)+W(\Delta T) \tag{6}\]  
By inserting the NPP value from Equation (4) into the left-hand part of Equation (6), we obtain a recurrent equation relating the phytomasses of the forest stand of adjacent ages \(T\) and \(T+\Delta T\):  
\[A[K^{\alpha}(T)]\cdot[ML^{1-\alpha}(T)]-sM(T)=M(T+\Delta T)-M(T)+D(\Delta T)+W( \Delta T) \tag{7}\]  
To accurately evaluate the forest stand phytomass, one needs to know all phytomass values in Equation (7), including root phytomass, and determine the distribution of the phytomass synthesized at ages between \(T\) and \(T+\Delta T\) between the photosynthesizing and non-photosynthesizing components (Equation (7) does not enable this evaluation). Unfortunately, forest inventories usually either contain no data on root phytomass, as it is very difficult to measure it in the field, or present results of calculations performed by unreliable methods [26; 27; 28; 29; 30]. In this study, calculations of root phytomass were based on the data on the phytomass of aboveground components and the phytomass distribution between the photosynthesizing and non-photosynthesizing components. We calculated the root phytomass by using a model proposed previously, in which root phytomass of a tree or a forest stand was estimated from the data on the phytomasses of the stem (with the bark), branches and leaves (needles) [31; 32; 33]. The Appendix A section contains the description of the model used to calculate the root phytomass in the forest stand and the distribution of the phytomass at age \(T+\Delta T\) between the photosynthesizing and non-photosynthesizing components. The leaf (needle) fall-off is determined by using Equation \(W(\Delta T)=\frac{1}{\theta}\Delta T\cdot ML(T)\). ML(T) is needle phytomass, \(\Theta\) is the average lifespan of the foliage. For the birch, aspen, and larch, \(\Theta=1\) year. For conifers, \(\Theta\) varies between 2 and 6 years. Then the \(ML(T)/\theta\) ratio is the needle phytomass loss over 1 year, and \(\Delta T\cdot ML(T)/\theta\) is the mass of the leaf (needle) fall-off over the time \(\Delta T\), between two successive inventories. The loss of trees in the forest stand, in the case when the stand density \(\rho(T)\) and total phytomass are known for each age, is estimated as follows:  
\[D(\Delta T)=\frac{M(T)}{\rho(T)}\cdot[\rho(T+\Delta T)-\rho(T)] \tag{8}\]  
If the time series of the phytomass components in the forest stand, \(MS(T)\), \(MR(T)\), \(MB(T)\), and \(ML(T)\), and the time series of the fall-off \(D(\Delta T)\) and the loss \(W(\Delta T)\) of the forest stand phytomass are available, Equation (7) can be regarded as a nonlinear regression equation with unknown coefficients \(A\), \(\alpha\) and \(s\). Coefficients \(A\), \(\alpha\) and \(s\) in Equation (7) were calculated using the least squares method, with a nonlinear regression model. The coefficient of determination R\({}^{2}\) is used to evaluate the quality of data approximation by model Equation (7).