Growth in System Perspective

Equations of this type are found in a variety of fields, and we can use system (3.1) to illustrate the formal identity of system laws in various realms, in other words, to demonstrate the existence of a general system theory.

This may be shown for the simplest case—i.e., the system con­sisting of elements of only one kind. Then the system of equations is reduced to the single equation:

which may be developed into a Taylor series:

This series does not contain an absolute term in the case in which there is no “spontaneous generation” of elements. Then dQ/dt must disappear for Q = 0, which is possible only if the absolute term is equal to zero.

The simplest possibility is realized when we retain only the first term of the series:

This signifies that the growth of the system is directly propor­tional to the number of elements present. Depending on whether the constant a1 is positive or negative, the growth of the system is positive or negative, and the system increases or decreases. The solution is:

Q0 signifying the number of elements at t = 0. This is the exponential law (Fig. 3.3) found in many fields.

In mathematics, the exponential law is called the “law of natural growth,’’ and with (a1 > 0) is valid for the growth of capital by compound interest. Biologically, it applies to the in­dividual growth of certain bacteria and animals. Sociologically, it is valid for the unrestricted growth of plant or animal popula­tions, in the simplest case for the increase of bacteria when each individual divides into two, these into four, etc. In social science, it is called the law of Malthus and signifies the unlimited growth of a population, whose birth rate is higher than its death rate. It also describes the growth of human knowledge as measured by the number of textbook pages devoted to scientific discoveries, or the number of publications on drosophila (Hersh, 1942). With negative constant (a1 < 0), the exponential law applies to radio­active decay, to the decomposition of a chemical compound in monomolecular reaction, to the killing of bacteria by rays or poison, the loss of body substance by hunger in a multicellular organism, the rate of extinction of a population in which the death rate is higher than the birth rate, etc.

Going back to equation (3.12) and retaining two terms, we have:

A solution of this equation is:

Keeping the second term has an important consequence. The simple exponential (3.14) shows an infinite increase; taking into account the second term, we obtain a curve which is sigmoid and attains a limiting value. This curve is the so-called logistic curve (Fig. 3.4), and is also of wide application.

In chemistry, this is the curve of an autocatalytical reaction, i.e., a reaction, in which the reaction product obtained accelerates its own production. In sociology, it is the law of Verhulst (1838) describing the growth of human populations with limited re­sources.

Mathematically trivial as these examples are, they illustrate a point of interest for the present consideration, namely the fact that certain laws of nature can be arrived at not only on the basis of experience, but also in a purely formal way. The equations discussed signify no more than that the rather general system of equation (3.1), its development into a Taylor series and suitable conditions have been applied. In this sense such laws are “a priori,” independent from their physical, chemical, biological, sociological, etc., interpretation. In other words, this shows the existence of a general system theory which deals with formal characteristics of systems, concrete facts appearing as their special applications by defining variables and parameters. In still other terms, such examples show a formal uniformity of nature.

Source: Bertalanffy Ludwig Von (1969), General System Theory: Foundations, Development, Applications, George Braziller Inc.; Revised edition.

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