hyperbolic growth

19/03/2014 14:42Hyperbolic growth - Wikipedia, the free encyclopedia

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The reciprocal function, exhibitinghyperbolic growth.

Hyperbolic growthFrom Wikipedia, the free encyclopedia

When a quantity grows towards a singularity under a finite variation(a "finite-time singularity") it is said to undergo hyperbolicgrowth.[1] More precisely, the reciprocal function has ahyperbola as a graph, and has a singularity at 0, meaning that thelimit as is infinite: any similar graph is said to exhibithyperbolic growth.

Contents1 Description

1.1 Comparisons with other growth2 Applications

2.1 Population2.2 Queuing theory2.3 Enzyme kinetics

3 Mathematical example4 See also

4.1 Mathematics4.2 Growth

5 References6 References

DescriptionIf the output of a function is inversely proportional to its input, or inversely proportional to the differencefrom a given value , the function will exhibit hyperbolic growth, with a singularity at .

In the real world hyperbolic growth is created by certain non-linear positive feedback mechanisms.[2]

Comparisons with other growthLike exponential growth and logistic growth, hyperbolic growth is highly nonlinear, but differs in importantrespects. These functions can be confused, as exponential growth, hyperbolic growth, and the first half oflogistic growth are convex functions; however their asymptotic behavior (behavior as input gets large)differs dramatically:

logistic growth is constrained (has a finite limit, even as time goes to infinity),exponential growth grows to infinity as time goes to infinity (but is always finite for finite time),hyperbolic growth has a singularity in finite time (grows to infinity at a finite time).

Applications



PopulationCertain mathematical models suggest that until the early 1970s the world population underwent hyperbolicgrowth (see, e.g., Introduction to Social Macrodynamics (http://urss.ru/cgi-bin/db.pl?cp=&page=Book&id=37484&lang=en&blang=en&list=14) by Andrey Korotayev et al.). It was also shownthat until the 1970s the hyperbolic growth of the world population was accompanied by quadratic-hyperbolicgrowth of the world GDP, and developed a number of mathematical models describing both thisphenomenon, and the World System withdrawal from the blow-up regime observed in the recent decades.The hyperbolic growth of the world population and quadratic-hyperbolic growth of the world GDP observedtill the 1970s have been correlated by Andrey Korotayev and his colleagues to a non-linear second orderpositive feedback between the demographic growth and technological development, described by a chain ofcausation: technological growth leads to more carrying capacity of land for people, which leads to morepeople, which leads to more inventors, which in turn leads to yet more technological growth, and on andon.[3] Other models suggest exponential growth, logistic growth, or other functions.

Queuing theoryAnother example of hyperbolic growth can be found in queueing theory: the average waiting time ofrandomly arriving customers grows hyperbolically as a function of the average load ratio of the server. Thesingularity in this case occurs when the average amount of work arriving to the server equals the server'sprocessing capacity. If the processing needs exceed the server's capacity, then there is no well-definedaverage waiting time, as the queue can grow without bound. A practical implication of this particularexample is that for highly loaded queuing systems the average waiting time can be extremely sensitive to theprocessing capacity.

Enzyme kineticsA further practical example of hyperbolic growth can be found in enzyme kinetics. When the rate of reaction(termed velocity) between an enzyme and substrate is plotted against various concentrations of the substrate,a hyperbolic plot is obtained for many simpler systems. When this happens, the enzyme is said to followMichaelis-Menten kinetics.

Mathematical exampleThe function

exhibits hyperbolic growth with a singularity at time : in the limit as , the function goes to infinity.

More generally, the function

exhibits hyperbolic growth, where is a scale factor.



Note that this algebraic function can be regarded as analytical solution for the function's differential:[4]

This means that with hyperbolic growth the absolute growth rate of the variable x in the moment t isproportional to the square of the value of x in the moment t.

Respectively, the quadratic-hyperbolic function looks as follows:

See alsoHeinz von FoersterTechnological singularityParadigm shiftList of paradigm shifts in scienceScientific mythologySocial effect of evolutionary theoryDeep ecology

MathematicsMathematical singularity

GrowthExponential growthLogistic growth

References(http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B83WC-4N0HJMK-2&_user=1300184&_coverDate=12%2F31%2F2007&_rdoc=6&_fmt=summary&_orig=browse&_srch=doc-info(%23toc%2333783%232007%23999839995%23671853%23FLA%23display%23Volume)&_cdi=33783&_sort=d&_docanchor=&_ct=9&_acct=C000052237&_version=1&_urlVersion=0&_userid=1300184&md5=d9c2663e7fbd6a77385d61334953d75d)Alexander V. Markov, and Andrey V. Korotayev(2007). "Phanerozoic marine biodiversity follows a hyperbolic trend". Palaeoworld. Volume 16. Issue4. Pages 311-318.Kremer, Michael. 1993. "Population Growth and Technological Change: One Million B.C. to 1990,"The Quarterly Journal of Economics 108(3): 681-716.Korotayev A., Malkov A., Khaltourina D. 2006. Introduction to Social Macrodynamics: CompactMacromodels of the World System Growth. (http://cliodynamics.ru/index.php?option=com_content&task=view&id=124&Itemid=70) Moscow: URSS. ISBN 5-484-00414-4 .



Rein Taagepera (1979) People, skills, and resources: An interaction model for world populationgrowth. Technological Forecasting and Social Change 13, 13-30.

References1. ^ See, e.g., Korotayev A., Malkov A., Khaltourina D. Introduction to Social Macrodynamics: Compact

Macromodels of the World System Growth (http://cliodynamics.ru/index.php?option=com_content&task=view&id=124&Itemid=70). Moscow: URSS Publishers, 2006. P. 19-20.

2. ^ See, e.g., Alexander V. Markov, and Andrey V. Korotayev (2007). "Phanerozoic marine biodiversity follows ahyperbolic trend". Palaeoworld. Volume 16. Issue 4. Pages 311-318 (http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B83WC-4N0HJMK-2&_user=1300184&_coverDate=12%2F31%2F2007&_rdoc=6&_fmt=summary&_orig=browse&_srch=doc-info(%23toc%2333783%232007%23999839995%23671853%23FLA%23display%23Volume)&_cdi=33783&_sort=d&_docanchor=&_ct=9&_acct=C000052237&_version=1&_urlVersion=0&_userid=1300184&md5=d9c2663e7fbd6a77385d61334953d75d).

3. ^ See, e.g., Korotayev A., Malkov A., Khaltourina D. Introduction to Social Macrodynamics: CompactMacromodels of the World System Growth (http://cliodynamics.ru/index.php?option=com_content&task=view&id=124&Itemid=70). Moscow: URSS Publishers, 2006; Korotayev A. V. ACompact Macromodel of World System Evolution // Journal of World-Systems Research 11/1 (2005): 7993.(http://jwsr.ucr.edu/archive/vol11/number1/pdf/jwsr-v11n1-korotayev.pdf); for a detailed mathematical analysis ofthis issue see A Compact Mathematical Model of the World System Economic and Demographic Growth, 1 CE -1973 CE (http://arxiv.org/abs/1206.0496).

4. ^ See, e.g., Korotayev A., Malkov A., Khaltourina D. Introduction to Social Macrodynamics: CompactMacromodels of the World System Growth (http://cliodynamics.ru/index.php?option=com_content&task=view&id=124&Itemid=70). Moscow: URSS Publishers, 2006. P. 118-123.

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