# Modeling Production Processes in Forest Stands: An Adaptation of the Solow Growth Model
## 1 Introduction  
The growth of individuals and populations can be presented as the totality of all births (the birth of a new individual or a change in the population density, biomass of an individual, or biomass of the entire population) and dissipation (the death of an individual or a population). Growth processes are usually described by using logistic models, including the Verhulst, Gompertz, Mitscherlich, Chapman-Richards functions, etc. [1; 2; 3; 4; 5; 6; 7; 8]. Such models describe the growth process with one independent variable \(x\), which characterizes the state of the system (the number of individuals in the population, the mass of one individual or the entire population) and a set of free parameters such as birth rates or biomass synthesis rates. However, descriptions of processes observed in forest stands by growth models are not in complete agreement with the real processes occurring in the forest. For instance, the death of the forest stand is as natural as its growth, and mortality processes should be described in the model too [9]. According to logistic models, consumption at an arbitrary time of any amount of phytomass that is smaller than its current value never causes the death of a forest stand. However, a fir tree (_Abies sibrica_ [PERSON]) dies if larvae of the Siberian silk moth (_Dendrolimus sibricus_ [PERSON]) have consumed all its needles, although the phytomass of the needles constitutes about 3% of the total phytomass of a 100-year-old tree [10; 11]. In a valid model of forest stand growth, there must be a steady state with \(x=0\), which describes a rather common situation when the forest is dead or cut down, but no reforestation occurs. In thelogistic models, however, there is only one steady state, with \(x=A\), while the state with \(x=0\) is not steady. Logistic models only evaluate phytomass growth rate, but GPP (gross primary production) and NPP (net primary production) cannot be determined by using those models. Phytomass loss in the forest stand, a very significant ecological parameter, cannot be estimated either. Finally, those growth models do not take into account the effects of climate factors on forest stand growth. Today, the most common approach to estimating the NPP of forest stands is the use of balance equations, which contain the data on the increment, fall-off, and loss of forest stand phytomass, standing timber volume, and allometries. However, despite the apparent conceptual simplicity of balance equations, it is very difficult to exactly evaluate the NPP by using the forest inventory data. Exact evaluations are only possible regarding the increment in the standing timber volume or phytomass between ages \(T\) and \(T+\Delta T\). The increments in branch and leaf phytomass can be evaluated with lower accuracy [12; 13]. The main challenge associated with NPP evaluation is the difficulty or even impossibility of measuring the phytomass and phytomass increment of the tree roots. Another challenge is the evaluation of the phytomass loss, as there are no exact data on the rates of dying of roots, branches, and bark [14; 15; 16]. In many cases, the data on the phytomass of tree parts and changes in the phytomass are lacking, and the only available data are inventories of the standing timber or measurements of the height and diameter of the trees in the stand. Then, calculations of the phytomass of each tree part and subsequent NPP evaluations are performed using regression equations that characterize relationships between the phytomasses of tree parts and the standing timber stock or tree diameters and heights [17; 18; 19; 20; 10]. For instance, evaluations of the NPP of Russian forests based on these approaches vary widely [21]. In the present study, growth processes of tree stands are estimated using a model based on [PERSON]'s model of economic growth, which is more complex than logistic models and which includes two independent variables describing processes of product generation and accumulation in the model system.