Dynamic growth model for oak stands in Moscow, Russia

. The most objective information about the state of Moscow’s forests is provided by long-term observation data on permanent trial plots. Such data makes it possible to identify changes in forest stands under the influence of recreationists, environmental pollution, and climate change. Long-term observational data are particularly valuable in modeling forest stand growth and productivity. The goal of the study is to develop a dynamic model of the growth of oak stands in Moscow based on long-term observation data. The modelling data were obtained from 7 plots of the permanent sample plot network established by the Forest Experimental


Introduction
Oak (Quercus robur L.) is one of the main forest-forming species in the Moscow region [1][2][3].Oak stands make up 10% of the forested area of Moscow.All Moscow forests perform environment-forming and environmental protection functions.At the same time, they are exposed to the influence of a large number of recreationists, emissions from motor vehicles, industrial enterprises, etc [4][5][6].The most objective information about the state of Moscow's forests is provided by long-term observation data on permanent trial plots.Such data makes it possible to identify changes in forest stands under the influence of recreationists, environmental pollution, and climate change [7].In addition, long-term observational data are particularly valuable in modeling forest stand growth and productivity.
In the practice of forestry in Russia, tables of the growth of forest stands continue to be used.Abroad, such tables are being replaced by numerous models of growth and productivity of forest stands [8][9][10].Numerous studies [11][12] have shown that climate change is leading to changes in forest stands and trees growth trajectories.Therefore, in Russia new models of forest stand dynamics that reflect actual growth are needed to replace growth progress tables.In Moscow, forest stand dynamics models can be a reliable tool for predicting growth, carbon accumulation in woody biomass, and other indicators.Therefore, the goal of the study is to develop a dynamic model of the growth of oak stands in Moscow based on long-term observation data.

Data
The modelling data were obtained from 7 plots of the permanent sample plot network established by the Forest Experimental Station of the Russian State Agrarian University -Moscow Timiryazev Agricultural Academy.The ecological conditions of the Forest Experimental Station are typical for Moscow.There are a total of 42 inventories and the number of inventories per plot range from 3 to 9. Inventories were carried out between 1927 and 2009.The age of forest stands during inventories ranged from 44 to 223.Summary statistics including the mean, minimum (min), maximum (max), and standard deviation (std) of the main stand variables for total plots are shown in Table 1.During inventories on permanent sample plot, the diameters of all growing trees were measured with an accuracy of 0.1 cm, with further grouping by thickness classes and calculation of the quadratic mean diameter.Heights were measured for 15-20 trees on a permanent sample plot with an accuracy of 0.5 m.The average height was determined graphically.The stand basal area per hectare was calculated by multiplying the quadratic mean diameter by the number of trees per hectare.The stand volume on the permanent sample plot was calculated using regional tables of trunk volumes.

Model structure
At the first stage of developing the model structure, candidate models were considered for compliance with our data set.Candidate models were selected from literature sources [14][15][16].To model the dynamics of mean height and quadratic mean diameter, the best result was shown by the GADA-equation: where y -mean height or quadratic mean diameter at age t, y0 -mean height or quadratic mean diameter at age t0, a and b -equation parameters.
This equation ( 1) was obtained by GADA from the Richard-Chapman growth function by replacing the third parameter responsible for the shape of the curve.Since the forest stands of permanent sample plots are in similar soil and climatic conditions, the entire set of growth curves has a common asymptote.
To model mortality in forest stands, the following equation was used: where y -number of trees per hectare at age t, y0 -number of trees per hectare at age t0, b -equation parameters.
The basal area of the stand was calculated using the known values of the quadratic mean diameter and the number of trees per hectare: where G -basal area of the stand per hectare, m 2 , N -number of trees per hectare, QMD -quadratic mean diameter, cm.
Stand volume was calculated using known values of mean height, stand basal area and average form factor: where M -stand volume per hectare, m 3 , G -stand basal area per hectare, m 2 , H -mean height, m, F -average form factor.
The form height of stands in equation ( 4) was determined using a linear dependence on the mean height: where HF -form height, H -mean height of stand, m, b -equation parameters.
The set of equations (1-5) is a dynamic growth model that allows to predict the main attributes of stands: mean height, quadratic mean diameter, number of trees per hectare, stand basal area and stand volume.Prediction by the model is based on the values of the mean height, quadratic mean diameter and number of trees known at the initial age.Other forest stand indicators can be calculated using equations (4-6).

Model evaluation
The models were assessed using to four statistical indicators: сoefficient of determination (R 2 ), adjusted coefficient of determination (R 2 -adj.),root mean square error (RMSE), and mean absolute percentage error (MAPE) [17][18].Model performance criteria selected for this study shows in Table 2.The parameters of equations (1-3) were estimated using the nonlinear least squares method.The parameters of equation ( 5) were estimated using the least squares method.The statistical significance of the equation parameter estimates was assessed using a t-test at a significance level of 0.01.All data analyzes in this study were conducted in R 3.6.3statistical software [19].
Table 2. Model performance criteria selected for this study.

Function name Equation
Coefficient of determination (R 2 ) Adjusted coefficient of determination (R 2 -adj.) .

Results and discussion
Fit parameters and statistics of models are shown in Table 3.All parameter estimates are statistically significant at p = 0.01.The asymptotic value of the mean height was 32.3 m, quadratic mean diameter was 56.1 cm.All growth curves of the mean height of oak stands on permanent sample plots have individual trajectories in a number of cases, deviating from monotonic growth functions.Therefore, a growth model of mean height has low R 2 = 0.688 and high MAPE = 9.597 %.The best fit metrics were obtained for the quadratic mean diameter growth model: R 2 = 0.806 and MAPE = 8.542 %.Quadratic mean diameter growth curves on permanent sample plots have better agreement than mean height growth curves.
For the number of trees per hectare model, the average absolute error is 16.781%.All obtained models meet the requirements for forest inventory in Russia in terms of error values of stand attributes.We used the resulting equations to visually examine the forest stand attributes considered.Figure 2 shows predictions for mean height (H40 = 8.0 m, H40 = 12.0 m, and H40 = 16.0 m), mean diameter (QMD40 = 6.0 cm, QMD40 = 9.0 cm, and QMD40 = 12.0 cm), number of trees (N40 = 1000 trees per hectare, N40 = 2200 trees per hectare, and N40 = 3400 trees per hectare), form height versus mean height.Each forest stand dynamics projection is illustrated for high, medium and low attributes.The resulting projections look reasonable.Mean heights and quadratic mean diameters correspond to general trends in the permanent sample plot data.Growth in height and thickness of stands occurs faster at the young age and then declines.The number of surviving trees depends on their initial number at the initial age and decreases with age.Finally, the error in predicting the basal area of oak stands and stand volume does not exceed 15 %.The modeling approach used in this work has been used to develop growth models for other tree species in different countries [10,18,20].We used three initial state variables for prediction, as in many studies.Taking into account the peculiarities of the inventory of forest stands in Russia, the mean height, quadratic mean diameter and number of trees per hectare were used as initial variables.The model in this study provides a simple and reliable system for predicting the growth and yield of Moscow oak stands.Using regression equations for tree stand biomass, tree stand attributes included in the model make it possible to calculate the biological productivity of Moscow oak stands and the amount of carbon sequestered in various biomass fractions.The model from this study can be included in a computer program to simulate the growth of oak stands in Moscow.

Conclusions
This study presents a dynamic model of the growth of oak stands in Moscow based on longterm observation data on 7 permanent sample plots in the Forest Experimental Station of the Russian State Agrarian University -Moscow Timiryazev Agricultural Academy.To predict the model, three attributes of the forest stand are used, taking into account the features of the inventory of forest stands in Russia.These include mean height, quadratic mean diameter and number of trees per hectare.The accuracy of the model's predictions meets the requirements for forest inventories in Russia.The model in this study is a simple and reliable tool for predicting the growth of oak stands in Moscow.The model from this study can be included in a computer program to simulate the growth of oak stands in Moscow.

Fig. 1 .
Fig.1.Projections of the mean height, quadratic mean diameter, number of trees per hectare and form height using models developed in this study (green line is data, and red line is prediction).

Table 1 .
Summary statistics of the main stand variables for total plots.

Table 3 .
Fit parameters and statistics of models.