## GROWTH-RATE PARAMETER
^{2)}

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The factor which controls growth rate.

Since no growth process is independent of its environmental conditions and/or intrinsic characteristics, none can be linear at long-term. This must be reflected into growth equations if real situations are to be satisfactorily modelled.

Thus a growth-rate parameter must be introduced into the equation.

According to J. GLEICK: "In the physical systems from which these equations were borrowed, that parameter corresponded to the amount of heating, or the amount of friction, or the amount of some other "messy" quantity. In short, the amount of nonlinearity" (1987, p.63).

R. JENSEN uses as an example a very simple difference equation:

X_{n+1} = ax_{n} (1-x_{n}) or X_{n+1 }= aX_{n} – ax_{n}

He writes: "The time-evolution of x_{n} generated by this single algebraic equation exhibits an extraordinary transformation from order to chaos as the parameter a, which measures the strength of nonlinearity is increased" (1987, p.170).

For x included between 0 and 1 and for different values of a, JENSEN distinguishes the following cases:

- for a = 1 – Growth followed by monotonic damping and extinction;

- for a = 2,9 Growth toward an asymptotic stability with a punctual attractor (variable according to the value of a);

- for a = 3,2 Growth followed by fluctuations around a stable cycle limit;

- for a = 4,0 Growth followed by apparently random fluctuations (chaos).

For a clearer understanding, see reference.

### Categories

- 1) General information
- 2) Methodology or model
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### Publisher

Bertalanffy Center for the Study of Systems Science(2020).

To cite this page, please use the following information:

* Bertalanffy Center for the Study of Systems Science (2020).* Title of the entry. In Charles François (Ed.), International Encyclopedia of Systems and Cybernetics (2). Retrieved from www.systemspedia.org/[full/url]

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