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Linear or nonlinear system of sustainable forest management?

Abstract: A forest is a chaotic non-linear dynamic system. The rules of forest growth can be expressed with complex equations of growth and development. The solutions are complex numbers, while their mapped forms are dendrograms in which vertical directions show amplitudes or multidimensional vectors and horizontal directions show space and time. Complex numbers are sets that represent possible physical states and form an abstract complex vector space of growth and increment. Integration of complex numbers yields increment and further integration yields the growth of diameter and height structure. Complex numbers are the topological dimension of a forest, while sets of complex numbers are the fractal dimension of a forest. Complex equations are qualitative models for quantitative, numerical predictions of forest growth and development.

Key words: Complex equations, complex numbers, dendrogram, vector space


In forests, period is the time that must elapse before the crown flushes again. Growth repeated from year to year is periodic movement. Such movement may be mapped with points in the phase space.

To obtain a clear system that can be modeled and investigated as well as draw valid conclusions, scientists remove unlinearity from a system model. Complex systems are reduced to linear ones so as to make quantitative assessments of future development with statistical methods. In the Republic of Croatia forest management follows the Forest Management Code (Mestrovic & Fabijanic 1995). The purpose of applying linear models in forestry is to evaluate future ten-year increment on the basis of past ten-year increment. A large number of samples are taken for statistical processing. Mechanical classification of dendrometric measurement data provides risky and coarse approximation.


Tools for numerical predictions of forest growth and development, multidimensional modeling, numerical site classification, and detection of dissipative forest structure are complex equations of growth of diameter structure:

[[PSI].sub.d] = [Ae.sup.-kt] sin ([[omega].sub.upd] t - [phi]) (1)

and growth of height structure:

[[PSI].sub.h] = [Ae.sup.-kt] sin ([[omega]] t - [phi])-A sing(g t) (2)

The symbols in the equations are: [[PSI].sub.d]--complex numbers of diameter growth; [[PSI].sub.h]--complex numbers of height growth; A--wave amplitudes; e--basis of natural logarithm; k--coefficient of growth resistance; t--time; [[omega].sub.upd]--coefficient of diameter growth pulsation; [[mega].sub.D]--coefficient of crown expansion pulsation; [[omega]],--coefficients of height structure pulsation; g -gravitational constant of height structure; [phi]--phase space of growth.

Equations that contain rates of change are called differential equations. The rate of change of a function is determined by the difference between the values of this function at two close moments. This difference is marked with the Greek letter [PSI]. The pulsation coefficient [[omega].sub.p] is the multiplication product of period increment [lambda] and constant [alpha] = 1/137 = 0.0072993. The pulsation coefficients of diameter growth of pedunculate oak (Quercus robur L.) are: [[omega].sub.pd] = 0.0729927, of crown expansion uD = 0.1824817, and of height growth [[omega] = 0.1459854.

Numerical site classification, the construction of tariffs and the calculation of stand volume are performed using a complex equation of height growth (2). The spatio-temporal development of height curves of pedunculate oak per age class is unambiguously determined by bifurcations for all times. The stem length--crown length ratio is strictly mathematical: 0.533: 0.467. It is determined by the stem length amplitude [A.sub.hd] = 4.669 (eigenvalue) and the crown length amplitude [] = 4.090. The sum of stem length amplitude and crown length amplitude gives the amplitude of height growth [A.sub.h] = 8.759. The rate of forced height growth, after the second culmination of height increment, is determined with the gravitational constant of height structure:

g = 10 [(1/137).sup.2] = 0.000532793 [god.sup.-1] (3)

The phase space of diameter growth for pedunculate oak [[phi].sub.d] = 0.001, crown expansion [[phi].sub.D] = 0.027, and the phase space of height structure [[phi].sub.h] = 0.876.

The coefficient of growth resistance k is the only nonlinear parameter that coordinates the rate of model growth with the growth rate of every tree or stand.


The solutions of complex equations (1 and 2) are complex numbers, i.e. the topological dimension, whereas sets of complex numbers are the fractal dimension of a forest. These are dendrograms in which vertical directions show amplitudes or multidimensional vectors, while horizontal directions show space and time. In the context of complex growth dynamics, the pulsation coefficients [[omega].sub.upd] : [[omega].sub.uph] : [[omega].sub.D] = 1 : 2 : 4 indicate critical points of the phenomenon of resonance. Period three leads forests into chaos. A dendrogram (Fig. 1.) of diameter growth [[PSI].sub.d] and crown expansion [[PSI].sub.D] is damped motion, whereas that of height growth [[PSI].sub.h] and stem length [[PSI].sub.ld] is forced motion.


Complex equations are a set of wave functions that represent possible physical states. A set has properties of an abstract mathematical object called abstract vector space. Sets of all physically logical solutions of linear and wave differential equations always form vector spaces, and their vectors are definable complex functions of space and time. Numerical assessment of diameter growth rate using a complex equation of diameter growth (1) is simple. To coordinate the rate of the model with the rate of tree diameter growth with iterations of growth resistance coefficient k we only need to know tree diameter d and tree age. The linear relation:

[i.sub.d] = [[PSI].sub.d] + b d (4)

provides a practical formula for numerical assessments of stand diameter increment [i.sub.d] without boring trees with the Pressler's drill.


Growth rate and dissipative structure are obtained with iterations of the growth resistance coefficient k in the intervals: 001[right arrow]0.027[right arrow]0.050[right arrow]0.073[right arrow]0.999.

Complex equations are universal tools for numerical site classification of a stand. The values of height resistance coefficient for dominant trees k < 0.050 indicate the first site class, those between 0.050 < 0.055 indicate the second site class, while stands with k ranging from 0.055 < 0.060 indicate the third site class.


Figure 3 shows a spatio-temporal shift in stand heights of pedunculate oak (Quercus robur L.) in the 1st site class. The curvature of the fifth dimension s is clearly manifested.


A forest is a complex and chaotic nonlinear dynamic system. A dynamic system is a rule that describes changed conditions in a given space in dependence on time. The rule that describes a changed condition of the system through time is deterministic. The rule of forest growth and development are complex equations.

The choice of the models Logarithmic Spiral and Law of Damped Sinusoidal Oscillations (Savelov, 1960) was a personal reductionist approach to the study of forest growth and increment. Integrating the subtle relations of universal constants of nature, wave amplitudes, coefficients or numbers with forest growth constitutes a holistic approach.

The synthesis of globally recognized theories (Capra, 1968), Newton's classical theory, the Theory of Natural Philosophy (Boskovic, 1763), the Quantum Theory (Ponomarev, 1988), the Theory of Dissipative Structures, the Theory of Deterministic Chaos (Stewart, 1996), personal insights and experimental measurements have resulted in a formula that imitates forest growth and development.

Every forest has its own rate of growth, which may be simply and quickly determined with complex equations. The vitality of a stand can also be detected at the same time. By classifying the condition of stands according to the scheme we obtain a dissipative structure, qualitative and quantitative forest production that may be transferred to maps. Every tree species has its own rate of growth. The knowledge of periodicity, pulsation coefficients and wave amplitudes of other tree species allows for multidimensional modeling of stable stands of the highest quality. Complex equations are universal tools for national inventories and assessments of sustainable forest management.

Universal laws of nature provide a model for making a passage from rigid linear management into nonlinear multidimensional modeling for the purpose of achieving sustainable management and preventive forest protection.


Using a reductionist and holistic approach, the author has gained an insight into a universal law of forest growth and development. Nature obeys relatively few basic laws. Laws of forest growth and development are complex equations. They are universal tools for modeling multidimensional forest dynamics. Complex equations of forest growth and development are qualitative models for quantitative, numerical predictions of forest growth and development.

In conditions of global warming, the transition from rigid linear management into nonlinear and dynamic management is of decisive importance for sustainable development.

The study of complexity and growth dynamics of all tree species is of prime importance for sustainable management and protection of forest ecosystems.


Boskovic, J. R. (1763). Theoria philosophiae naturalis, Translated by Stipisic, J. (1974). Theory of Natural Philosoph, Sveucilisna naklada Liber, Zagreb

Capra, F. (1997): The Web of Life, Translated by Zafirovic, L. (1998). Liberata, ISBN 953-6673-05-3, Zagreb

Mestrovic, S. & Fabijanic, G. (1995). The Forest Management Code, Ministry of Agriculture and Forestry of Croatia : Croatian Forests, ISBN 953-6253-04-6, Zagreb

Ponomarev, L. I. (1988). The Quantum dice, Translated by Trbojevic, D. (1995). Skolska knjiga, ISBN 953-0-609043, Zagreb

Savelov, A. A. (1960). [??]OCKHE KPHBbIE, Translated by Kucinic, B. & Hozjan, S., (1979). Areal Curves, Skolska knjiga, B-649/1-60940, Zagreb

Stewart, I., (1996): Does God play dice?, Translated by Lopac, V. (2003). Jesenjski-Turk, ISBN 953-222-117-4, Zagreb
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Author:Bezak, Karlo
Publication:Annals of DAAAM & Proceedings
Article Type:Report
Geographic Code:4EUAU
Date:Jan 1, 2007
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