Modeling the Forest Stand Growth Dynamics Based on the Thermodynamic Approach

Abstract

The forest ecosystem is a common example of the functioning of open thermodynamic systems. The work analyzes the change in the entropy of an open thermodynamic system where the following processes can be realized: absorption of short-wave solar radiation – differentiation process; total biomass growth process associated with the consumption of resources for respiration and competition. As a result of these processes, the negative entropy flow enters the system, and the positive entropy is produced in the system. As the stand grows, its biomass reaches a maximum, which corresponds to the steady state in the ecosystem. It is shown that, in accordance with the Prigogine’s theorem, the specific entropy production in an open system takes on a minimum positive value. With a further increase in the age of the stand, the steady state of the open thermodynamic system evolves to an equilibrium state, at which a decrease in the plant biomass is observed, and the entropy tends to a maximum value in accordance with the 2nd law of thermodynamics (ecosystem decay). The analysis of the behavior of an open thermodynamic system forms the basis of a new ecological and physiological model of the stand growth dynamics. The model proposed uses the following parameters: the biomass of an individual tree and the number of trees per hectare. In order to model the biomass growth dynamics of an individual tree, the von Bertalanffy equation is used. It contains a dynamic equation describing growth of an individual due to resource uptake and limitation of growth due to resource consumption. The equation that characterizes the dynamics of stand size derives from the condition of reaching the maximum biomass of the stand during the stand’s growth. In general, the stand’s dynamics model has only three independent parameters. They are the onset time of the steady state, the resource consumption rate and the factor linking the area and biomass of the organism. The model verification is presented by calculating the biomass dynamics for full (normal) pine plantations of the first five quality classes (Ib, Ia, I, II, III). The model’s quality is assessed by the dimensionless Nash-Sutcliffe model efficiency coefficient, the value of which is usually greater than 0.95. This corresponds to a description of the data that is close to ideal.

Entropy Production Using Ecological and Physiological Models of Stand Growth Dynamics as an Example

“…Note that the importance of thermodynamic laws for substantiation of ecological and physiological models of the forest stands growth dynamics was first considered in the works of G. A. Aleksandrov and G. S. Golitsyn [19]. Below, we present the main ideas and provisions of ecological and physiological models, a detailed description of which was published in our papers [9,10]. We based our model [9] on the fact well-known in forestry that when the biomass of a stand reaches a certain age, it reaches its maximum, after which the biomass begins to gradually decrease (this dependence is typical of many stands (see, for example, [13,20])).…”

“…where α is a function of time, b is a constant whose value is determined in the ecological and physiological model [9,10], and F is the surface area of the stand.…”

“…Below, we present the main ideas and provisions of ecological and physiological models, a detailed description of which was published in our papers [9,10]. We based our model [9] on the fact well-known in forestry that when the biomass of a stand reaches a certain age, it reaches its maximum, after which the biomass begins to gradually decrease (this dependence is typical of many stands (see, for example, [13,20])).…”

“…As a result of these processes, a negative entropy flux enters the system, while positive entropy is produced in the system itself. During the growth of the stand over time, the biomass reaches the maximum value, which corresponds to the stationary state in the ecosystem [9,10]. As the age of the stand increases further, the stationary state of an open thermodynamic system transitions to a state in which the biomass decreases and the entropy of the system increases (ecosystem decay occurs).…”

“…The aim of this work is to show that, using the ecological-physiological models of the forest stand growth dynamics used in modern forestry, it is possible to obtain an analytical expression for the specific entropy production as a function of time, which has a minimum at the point in time at which the biomass of a stand reaches its maximum value. In contrast to a number of works in which empirical data on heat fluxes, radiation energy, and energy of chemical reactions were used to calculate entropy production, in the present work the analytical expression for specific entropy production was obtained based on phenomenological models in which the parameters of ecological and physiological models developed earlier by the authors of the article [9,10] were used.…”