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Unformatted text preview: HAS 222d2009 Lecture 16 week 9 Exponential growth P.Rhines Digression on exponential growth and global population: exponential growth occurs when the growth is proportional to the population..for example, if each person has the same number of children during their life; or, if lily pads grow on the surface of a pond by reproducing at a constant rate … the effect is very slow growth, almost invisible, followed by explosive growth until the pond is covered. What sort of physical processes lead to exponential growth? The key is that the slope of the curve e At is Ae At … slope is proportional to value. If t is time, this says that the rate of change of the curve is proportional to its value. If each human has more than 2 children, and life span is unchanging in time, then over time population will follow nearly exponential growth, with a growth rate A: the population increases by a factor e in time 1/A. Any process like lily pads that, for some time, increase in proportion to their numbers, will be exponential. Is global human population growing exponentially? exponential growth? linear and semilog plots of global population against year An exponential curve appears as a straight line on a semilog plot [log (population) plotted against time, where the vertical axis has equal intervals for each power of ten (or power of e)]. Apparently global popuation is b hyper ` exponential. • See the separate notes posted on the class website too. • notice that y = d e at and its inverse, t = (1/a) log(y/d) are complementary relations (the same curve turned round): the exponential curve eventually rises faster than any power of t, and the log curve rises slower than any power of y. (log here is log to the base e, e = 2.718 … ). If you plot an exponential curve on a distorted plot, with axes log(y) and t, it is a straight line with slope a and intercept log(d) (the value at t=0)....
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This note was uploaded on 03/08/2012 for the course H A&S 222b taught by Professor P.b.rhines during the Spring '09 term at University of Washington.
 Spring '09
 P.B.Rhines

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