Math 19. Lecture 3Exponential GrowthT. JudsonFall 20041 Predicting the Population of the U.S.Given the following census data of the U.S. population, how can we predictthe population in 2010?Year t Actual P (t)Year t Actual P (t)1790 0 3.9 3.9 1910 120 91 471800 10 5.3 4.81920 130 105 581810 20 7.2 5.91930 140 122 721820 30 9.6 7.31940 150 131 881830 40 12 91950 160 151 1081840 50 17 111960 170 179 1331850 60 23 141970 180 203 1641860 70 31 171980 190 226 2021870 80 38 211990 200 249 2491880 90 50 252000 210 281 3061890 100 62 312010 220 — 3771900 110 75 382020 230 — 4642 The Exponential EquationConsider a population of P(t) at time t. During each unit of time, say∆t, a constant fraction of population will be having offspring. We will alsoassume that the population has a constant death rate. Thus, the change inthe population d uring the interval ∆t is∆P ≈ kbirthP (t)∆t − kdeathP (t)∆t1where kbirthis the fraction of the population having children during theinterval and kdeathis the fraction of the population that dies during theinterval. Therefore,∆P∆t≈ kP(t),where k = kbirth− kdeath. Since the derivative of P isdPdt= lim∆→∞∆P∆t,the rate of change of the population is proportional to the s ize of the popu-lation,dPdt= kPat time t.3 The Population of the U.S. RevisitedConsider the U.S. population model:dPdt= kPP (0) = 3.9P (200) = 249The general solution isP (t) = P (0)ekt.In the example,P (t) = 3.9ektk =1200ln2493.9≈ 0.0207824 Bufo Marinis—What Can Go WrongThe American marine toad (Bufo marinis) was introduced into Au s tr alia tocontrol sugar cane beetles. Unfortunately, the toads are nocturnal feedersand the beetles are abroad by day. The f ollowing table provides the landarea in Australia colonized by the toad from 1939–1974.Year Cumulative area occupied (km2)1939 32,8001944 55,8001949 73,6001954 138,0001959 202,0001964 257,0001969 301,0001974 584,0005 The Equationdqdt= aq + cThe equationdqdt= aq + chas solutionq(t) =q(0) +caeat−caWe can show this two ways: directly and by making the substitution p(t) =q(t) + c/a. The latter red uces th e equation to the exponential growth equa-tion.6 Sums of Exponential FunctionsBeware of sums of exponential functions of time such asf(t) = e−t+ e−4t,where t ≥ 0. This m ight be the level of the AIDS virus in a patient’s bloodpredicted for t dates after the beginning of a particular drug therapy. Theterm in the sum with the least negative or most positive exponential willdominate the sum for large t.37 Taylor’s TheoremThe second reason why dP/dt = aP occurs so often has to do with Taylor’stheorem. Any function f(x) can be approximated near a point x0by an nthdegree polynomialgn(x) = f(x0)+f′(x0)(x−x0)+12f′′(x0)(x−x0)2+· · ·+1n!f(n)(x0)(x−x0)n.Homework• Chapter 2. Exercises 1, 2, 4 (a, b, c, d), 5; pp. 43–44.Readings and References• C. Taubes. Modeling Differential Equations in Biology. Prentice Hall,Upper Saddle River, NJ , 2001. Chapter 2.• “HIV-1 Dynamics in Vivo: Virion Clearance Rate, Infected Cell Life-Span, and Viral Generation Time,” pp.
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