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	Logistic function - Wikipedia, the free encyclopedia
		Equation (2) is the continuous version of the logistic map.
		The Verhulst equation, (1), was first published by Pierre F. Verhulst in 1838 after he had read Thomas Malthus' An Essay on the Principle of Population.
		Verhulst derived his équation logistique (logistic equation) to describe the self-limiting growth of a biological population.
	en.wikipedia.org /wiki/Logistic_function (953 words)

	Interesting Facts about Population Growth Mathematical Models
		Verhulst was born in 1804 in Brussels, Belgium.
		Comparing it with equation (1) it is nonlinear in the sense that one can’t simply multiply the previous population by a factor.
		Equation (4) simply says that the population for the next time period is a function (or depends) on the present population.
	arcytech.org /java/population/facts_math.html (3912 words)

	[No title] (Site not responding. Last check: 2007-10-14)
		In 1835, Verhulst was appointed professor of mathematics at the University of Brussels where he offered courses on geometry, trigonometry, celestial mechanics, astronomy, differential and integral calculus, and the theory of probability.
		In 1841, Verhulst was elected to the Belgium Academy.
		Verhulst's research on the law of population growth showed that forces, which tend to obstruct population growth, increase in proportion to the ratio of the excess population to the overall population.
	www.mhhe.com /math/calc/smithminton2e/cd/tools/timeline/verhulst.html (230 words)

	Verhulst (print-only) (Site not responding. Last check: 2007-10-14)
		Verhulst showed in 1846 that forces which tend to prevent a population growth grow in proportion to the ratio of the excess population to the total population.
		The non-linear differential equation describing the growth of a biological population which he deduced and studied is now named after him.
		In 1841 Verhulst was elected to the Belgium Academy and in 1848 he became its president.
	www-groups.dcs.st-and.ac.uk /~history/Printonly/Verhulst.html (408 words)

	Verhulst, Pierre-Francois (1804-1849) -- from Eric Weisstein's World of Scientific Biography
		His interest in probability theory had been triggered by a new lottery game, but he soon applied it to political economy and later to demographical studies, a field that was rapidly developing due to Malthus' theory and the increasing use of statistics in human sciences.
		Verhulst, however, opposed any attempt to apply mathematical models to ethical judgement.
		There he was elected president in 1848, shortly before his fragile health, which had troubled him for many years, tore him from life at the young age of 45.
	scienceworld.wolfram.com /biography/Verhulst.html (233 words)

	Search Results for equation*
		He solved his operator equation in the particular cases which arise in the study of the physical problem in his thesis (and in the paper which appeared in 1900 based on that thesis) while the general case was solved by Fredholm somewhat later and not published until 1908.
		In 1799 Ruffini published a book on the theory of equations with his claim that quintics could not be solved by radicals as the title shows: General theory of equations in which it is shown that the algebraic solution of the general equation of degree greater than four is impossible.
		Although an algebraic equation of the fifth degree cannot be solved in radicals, a result which was proved by Ruffini and Abel, Hermite showed in 1858 that an algebraic equation of the fifth degree could be solved using elliptic functions.
	www-groups.dcs.st-and.ac.uk /~history/Search/historysearch.cgi?SUGGESTION=equation*&CONTEXT=1 (17342 words)

	X-next (Verhulst) Logistic Equation
		Logistic equations are ones that are iterated (calculated over and over), recursive (output of the last calculation is input for the next), and normalized (population size ranges from zero - extinction - to one - maximum conceivable population).
		These equations have been used in biology to study population dynamics, for example how the population of an organism this year (X) will change in the next and subsequent years (X-next).
		Logistic equations are extremely useful, and if you go to a search engine such as Google and search the key words "logistic equation" you get many hits.
	www.jmu.edu /geology/evolutionarysystems/programs/xnext.shtml (323 words)

	Athens Intercollegiate Connection - Logistic Equation Theory & Applications (via CobWeb/3.1 planetlab2.cs.virginia.edu) (Site not responding. Last check: 2007-10-14)
		The factor (1-N/K) is present in the equation to attenuate unrestricted growth, but the details of its origin are often presented without explanation.
		Since the resources of a population are generally limited, it is reasonable to assume that as its density increases and it approaches the carrying capacity of its environment, the birth rate will decline and the death rate increase.
		Having used the logistic population model as earlier in the course as one of our fundamental models of a nonlinear differential equation, it is natural to augment this model to take into account the effect of harvesting on the population.
	www.aic.gr.cob-web.org:8888 /articles/logistic4.htm (764 words)

	Population assignment
		Verhulst, and the American demographer, Raymond Pearl, in 1838 and 1920 respectively, independently derived a gradual leveling of increase as numbers approached a defined asymptote.
		This equation which corresponds to an S-shaped growth curve is known as the Verhulst-Pearl or logistic equation.
		This equation has been widely applied to populations of animals whose density seems to be limited by some environmental carrying capacity, such as space, food resources, or nesting sites.
	www.rw.ttu.edu /2301_wallace/population_assignment.htm (450 words)

	Logistic Equation -- from Wolfram MathWorld
		The logistic equation (sometimes called the Verhulst model or logistic growth curve) is a model of population growth first published by
		The discrete version of the logistic equation (3) is known as the logistic map.
		Similarly, a normalized form of equation (3) is commonly used as a statistical distribution known as the
	mathworld.wolfram.com /LogisticEquation.html (229 words)

	Verhulst biography
		Pierre Verhulst was educated in Brussels, then in 1822 he entered the University of Ghent.
		On 28 September 1835 Verhulst was appointed professor of mathematics at the Université Libre of Brussels.
		Verhulst's research on the law of population growth is important.
	www-groups.dcs.st-and.ac.uk /~history/Biographies/Verhulst.html (391 words)

	The Logistic Equation: Introduction (Site not responding. Last check: 2007-10-14)
		This equation thus describes how a population grows if its per-capita birth rate is constant, and mortality and migration is neglected.
		He suggested that when the population gets high, there is a tendency for individuals to spend more time competing (and even going to war) over scarce resources which would increase the death rate.
		Verhulst's equation has come to be known as the Logistic Growth Equation.
	www.math.jhu.edu /~js/Math107/coursenotes2/node20.html (578 words)

	Advances in the Philosophy of Technology
		In the nonlinear differential equation of Verhulst dynamics (e.g., the growth of a population), the initial exponential growth is damped by a quadratic feedback term.
		In 1845, Verhulst already analyzed a nonlinear difference equation as a recursive iteration scheme for modeling the growth of a population xn during discrete time points, n= 1, 2, 3,.
		Verhulst dynamics can easily be visualized in a phase space with different attractors of a fixed point, periodic oscillation between two points, and a complete irregularity without any periodicity.
	scholar.lib.vt.edu /ejournals/SPT/v4_n1html/MAINZER.html (4682 words)

	2.7 Logistic Systems \| GIACS (via CobWeb/3.1 planetlab2.cs.virginia.edu) (Site not responding. Last check: 2007-10-14)
		Logistics equations were studied in the context of complex systems initially for the wrong reasons: their deterministic yet unpredictable (chaotic) solutions for more then 3 nonlinear coupled equations and their fractal behavior in the discrete time-step version seemed for a while as a preferred (if not royal) road to complexity.
		The application of the Logistic Equation has been used to describe social change diffusion: the rate of adoption is proportional [12] to the number of people that have adopted the change times the number of the agents that still haven’ t.
		The extension to spatially extended logistic systems in terms of partial differential equations was first formulated in [25] in the context of the propagation of a mutant superior gene in a population.
	www.giacs.org.cob-web.org:8888 /expertreport2p7 (1570 words)

	Jimmy Jiang and Graham Thorsteinson Project 2 (Site not responding. Last check: 2007-10-14)
		Problem 2 - We rewrote Verhulst's equation in Maple language and used the given values of a, b, and p0.
		We took the limit of the population equation as t approaches infinity to determine the population limit.
		Then we used the population limit and substituted back into the Verhulst equation as p0 and found the answer of the population limit to be the same.
	coweb.math.gatech.edu:8888 /calculus/1253 (271 words)

	growth.html
		The Belgian mathematician Pierre Verhulst (1804-1849) published a mathematical model of population growth in 1845, and used the census data taken in the United States from 1790 to 1840 inclusive to extrapolate into the future.
		Verhultst solved this differential equation and fitted the solution to US population data to estimate the constants in the model.
		All you need to know is that Maple gave you a bunch of equations which look like "P(t)=blabla (t, P0)", where t is time and P0 is the population at time 0, and that these equations list those functions which satisfy the differential equation.
	cerebro.xu.edu /math/math171/99s14/growth9.html (508 words)

	LOGISTIC
		When the population P is small compared to M, P << M, the growth is nearly exponential, but as the size of the population increases, as P approaches M the growth rate goes to 0.
		Verhulst used the 1790 through 1840 population data from the the U. census, under the assumption that U.S. population growth would continue to satisfy his logistic equation.
		The rate of growth of the population is proportional to the size of the population.
	isolatium.uhh.hawaii.edu /m206L/lab8/Logistic/Logistic.htm (676 words)

	Fractal Science Gallery - multi-dimensional iterations in complex space
		Verhulst's equation is one of the best-known bifurcating equations that lead to deterministic states of bounded chaos.
		Verhulst's equation shows the boundary between chaos and determinate structure.
		The donor set of Verhulst's equation as graphed in the complex plane.
	www.referencesystemk.com /fractals-science-gallery.htm (443 words)

	qualitative differntial equations
		Finding analytical solutions to differential equations can be very difficult or impossible, yet often the behavior of the equations can be examined from a qualitative perspective to determine types of behavior, which can lead to insight into modeling problems.
		There is a solution to the logistic growth differential equation, which can be found in a hyperlink to this section (Solution to the Logistic Growth Model).
		Find all equilibria for the differential equation, which are simply all points where the derivative of the unknown function is zero.
	www-rohan.sdsu.edu /~jmahaffy/courses/f00/math122/lectures/qual_de/qualde.html (1665 words)

	The Verhulst Equation
		The complicated behaviour exhibited by the logistic map is typical for a whole class of families of one-dimensional maps of a finite interval that have a single smooth maximum, known as unimodal maps.
		For the differential equations the dimension of the problem plays an important role in determining the dynamics.
		For the difference equations the situation is different: for maps it is possible even in one dimension to obtain chaotic (or seemingly random) orbits.
	www.gris.uni-tuebingen.de /projects/dynsys/latex/dissp/node15.html (624 words)

	CSC4.html (Site not responding. Last check: 2007-10-14)
		One especially well-known model was proposed by Verhulst in 1838 to describe the growth of human populations.
		The same model was independently used by Pearl and Reed (1920) to describe the growth of the U.S. population.
		in the Verhulst equation that models the growth of the population from 1790 to 1990, and then project the model over the next several centuries.
	www.dartmouth.edu /~math3f98/csc_archive/CSC4part2/CSC411.html (228 words)

	Prime numbers
		One of Cauchy's contribution to differential equations is given by Cauchy's Functional equation.
		The basic mathematical model based in population studies is that the population size for one generation is proportional to the size of the previous generation.
		This inversion method was found in a similar way as was done in [2] in the case of generalizations of the Charlier polynomials.
	hypatia.math.uri.edu /~kulenm/diffeqaturi/m381f00fp/michael/michaelmp.html (1566 words)

	Population Dynamics
		The Verhulst Model, named after Pierre-Francois Verhulst (1804-1849) of Belgium, improved upon the exponential growth model of Malthus by incorporating a limiting population value that the environment can support.
		The Malthusian equation assumes that population growth is proportional to the current population.
		Since the Verhulst Model does not take into account time-dependent changes in demographic age distribution, the true rate of population change may be slightly higher or lower than calculated by Verhulst Equation.
	brneurosci.org /population.html (821 words)

	Population dynamics and iteration of functions
		What we have not yet accounted for is the ``death rate.'' This might be assumed to be proportional to the population again, which would again lead to a difference equation of exponential growth if the birth rate exceeds the death rate, or of exponential decay if the opposite occurs.
		This equation is known as the Verhulst equation or the logistic equation.
		Solving this differential equation is a great pleasure of learning separation of variables in a first course in differential equations.
	www.math.okstate.edu /mathdept/dynamics/lecnotes/node51.html (479 words)

	Ratio Study (via CobWeb/3.1 planetlab2.cs.virginia.edu) (Site not responding. Last check: 2007-10-14)
		We use the fact that the exact solution of the Dahlquist test is similar to that of the Verhulst equation: we weigh the ratio using the result of the previous analysis, that is fixing
		The same guide procedure was used to prove the stability of a critical point of a non-linear equation, by the proposition 1.23, i.e.
		We obtain a differential equation composed of a linear part and of a residual, which vanishes when it is valued at zero.
	www.gris.uni-tuebingen.de.cob-web.org:8888 /projects/dynsys/latex/dissp/node28.html (1623 words)

	Modeling With Differential Equations (Site not responding. Last check: 2007-10-14)
		Certainly, there are many mathematicians who study differential equations from a more theoretical point of view and who solve certain kinds of equations with no known applications to the "real world," but the vast progress made in the last several hundred years of studying differential equations has come from attempts to understand the physical world.
		Modeling is the process of creating an equation or system of equations that describes or predicts--more or less accurately--some physical situation.
		As you encounter differential equations in whatever field of study you pursue, you should always assess what assumptions have to be made for the model in question to be valid.
	math.ucsd.edu /~math20d/Spring/Lab4S/Lab4S.html (2535 words)

	Handprint : Mandelbrot
		This equation involves a more complicated set of repeated calculations, because both real and imaginary numbers are involved.
		Each of the complex numbers in this equation, z and c, can be represented as a point on the (real, imaginary) dimensions of the complex plane: z by the point (x,y) representing the numbers x and iy, and c by the point (a,b) representing the numbers a and ib.
		The Mandelbrot set is thus an image of the effects on the Verhulst process behavior of variations in the value of the constant c.
	www.handprint.com /PS/KAO/mbrot.html (1320 words)

	Striking Regularities In The World Population Curve
		The one equation that stands out is the Malthus-Verhulst equation dp/dt = k p (u-p)/u which describes a population curve of the form p(t) = u/(1 + exp(k(t0-t))) with a minimum of 0, a maximum of u and an inflection time at t0 at which p(t0) = u/2.
		In general, what one is finding is that the world population is a birth-death process that satisfies a differential equation of the form p'(t) = f(p(t)) where f(p) is piecewise quadratic or linear; each piece associated with a separate phase.
		The phase boundaries occur roughly in the vicinity of 1 billion (corresponding to the time of the dawn of the Industrial Revolution) and 3.5 billion with the latter boundary being associated with a transition region between 2.5 - 4.5 billion that corresponds to the Second World War and the consequent rise of the post-Industrial era.
	w4.lns.cornell.edu /spr/2005-10/msg0071733.html (1408 words)

	Limited and Unlimited Growth
		He suggested that when the population gets high, there is a tendency for individuals to spend more time interfereing with each other - fighting for food, or for scarce resources, killing each other over wars for land claims, or somehow increasing the rate of death.
		It was noted that the same equation can be written in a variety of forms which are essentially the same but carry slightly different interpretations.
		The Logistic equation is used as a fundamental yardstick with which to understand population behaviour, and biologists talk about "r-selection" and "K-selection" to refer to various strategies that species adopt to outwit their competitors.
	www.ugrad.math.ubc.ca /coursedoc/math100/notes/mordifeqs/logistic.html (1050 words)

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