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The J-shaped exponential growth (left, blue) and the S-shaped logistic growth (right, red). Main article: Logistic curve In reality, initial exponential growth is often not sustained forever.
When a new technology requires significant investment in complementary intangible assets, then this can create a J-curve in productivity growth, at least as it is conventionally measured. Investments in intangibles require tangible costs, but they are not well-measured as part of output in the official national accounts .
Biological exponential growth is the unrestricted growth of a population of organisms, occurring when resources in its habitat are unlimited. [1] Most commonly apparent in species that reproduce quickly and asexually , like bacteria , exponential growth is intuitive from the fact that each organism can divide and produce two copies of itself.
The standard logistic function is the logistic function with parameters =, =, =, which yields = + = + = / / + /.In practice, due to the nature of the exponential function, it is often sufficient to compute the standard logistic function for over a small range of real numbers, such as a range contained in [−6, +6], as it quickly converges very close to its saturation values of 0 and 1.
Figure 1: A bi-phasic bacterial growth curve.. A growth curve is an empirical model of the evolution of a quantity over time. Growth curves are widely used in biology for quantities such as population size or biomass (in population ecology and demography, for population growth analysis), individual body height or biomass (in physiology, for growth analysis of individuals).
Exponential growth cannot continue indefinitely, however, because the medium is soon depleted of nutrients and enriched with wastes. The stationary phase is often due to a growth-limiting factor such as the depletion of an essential nutrient, and/or the formation of an inhibitory product such as an organic acid. Stationary phase results from a ...
By now, it is a widely accepted view to analogize Malthusian growth in Ecology to Newton's First Law of uniform motion in physics. [8] Malthus wrote that all life forms, including humans, have a propensity to exponential population growth when resources are abundant but that actual growth is limited by available resources:
A hockey stick graph or hockey stick curve is a graph, or curve shape, that resembles an ice hockey stick, in that it turns sharply from a nearly flat "blade" to a long "handle". In economics , [ 1 ] [ 2 ] marketing , [ 3 ] and dose–response relationships , [ 4 ] [ 5 ] a hockey stick graph is one in which the "blade" is near zero (hugging the ...