{VERSION 3 0 "SUN SPARC SOLARIS" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 8 255 0 0 1 2 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "helvetica" 1 8 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE " Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 }{PSTYLE "New Page" -1 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 1 0 -1 0 }{PSTYLE "" 18 257 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 1 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 18 258 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 1 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 18 259 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 18 260 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 1 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 5 " " 0 "" {TEXT 258 10 "BDH1-1.mws" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 36 "Modelling Via Differential Equations" }}{PARA 257 "" 0 "" {TEXT 259 38 "Section 1.1 - Blanchard, Devaney, Hall" }}{PARA 258 "" 0 "" {TEXT 260 15 "11 January 2000" }{TEXT -1 0 "" }}{PARA 19 "" 0 "" {TEXT -1 36 "updated to Release 5.1 by Doug Meade" }}{PARA 260 "" 0 " " {TEXT 261 15 "18 January 1998" }}{PARA 19 "" 0 "" {TEXT -1 28 "Doug \+ Meade (modified by M^2)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Introduction" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 336 "The purpose of this worksheet is to intr oduce you to some of the ways Maple can be used to assist with the ana lysis of differential equations. You are not expected to currently und erstand everything that is presented in this worksheet. As the course \+ progresses you will be expected to become proficient at these types of \"computations\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 381 "This worksheet is intended to be use while reading secti on 1.1 of the text. The definitions are not repeated in this worksheet . There are a few places where this worksheet differs from the text. T hese are clearly identified. There are a number of questions for you t o answer as you work through this worksheet. Please take the time to d o what is asked. Ask questions as they arise." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "Getting Started" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 140 "Every Maple worksheet should begin by re-initializing the Maple \+ \"kernel\" and loading the additional packages that we are most likely to use." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with( plots ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with( DEtools ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 27 "Unlimited Population Growth" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "O ne model for unlimited population growth is represented by the followi ng first-order ordinary differential equation." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 36 "MODEL := diff( P(t), t ) = k * P(t);" }}} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Question" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 87 "Identify all independent variables, dependent variables , and parameters in the problem." }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Equilibrium Solution" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "(Se e the text for a definition of equilibrium solution.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "EQUILeqn := rhs( MODEL ) = 0;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "EQUILsol := solve( EQUILeqn, \{ P(t) \} );" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Qualitative A nalysis" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "InitCond := P(0) \+ = p[0];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "The text provides a m ethod by which we can determine that solutions to this model are incre asing if " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 11 " > 0 and " } {XPPEDIT 18 0 "p[0]" "6#&%\"pG6#\"\"!" }{TEXT -1 25 " > 0 and decreas ing if " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 11 " < 0 and " } {XPPEDIT 18 0 "p[0]" "6#&%\"pG6#\"\"!" }{TEXT -1 89 " > 0. This type o f analysis can be done for other equations but is often quite difficul t." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 17 "Analytic Solution" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 167 "We will learn specific techniques for solving special cl asses of differential equations. The Maple command for finding explici t solutions to differential equations is " }{TEXT 19 6 "dsolve" } {TEXT -1 167 ". The arguments are the initial value problem (a set con taining the differential equation and the initial condition) and the f unction for which we are trying to solve." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 44 "SOLN := dsolve( \{ MODEL, InitCond \}, P(t) );" }}} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 7 "Warning" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 390 "While Maple is good at solving many differential equatio ns, there are times when the results produced by dsolve are quite diff erent from what we obtain by hand. If you encounter this, you should n ot automatically assume Maple is correct. Instead, I recommend that yo u use Maple to test which (if either) of the solutions actually satisf ies the differential equation (and initial condition)." }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "The U.S. Population" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "The \+ computations used to determine values for the " }{TEXT 257 10 "paramet ers" }{TEXT -1 10 " k and " }{XPPEDIT 18 0 "p[0]" "6#&%\"pG6#\"\"! " }{TEXT -1 263 " from the census data are easily reproduced in Maple . The idea is that two data points can be used to construct a system o f two equations for the two unknown constants. While it is possible t o exactly duplicate the steps shown in the text to determine k and \+ " }{XPPEDIT 18 0 "p[0]" "6#&%\"pG6#\"\"!" }{TEXT -1 50 ", it is simpl er to ask Maple to solve the system." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "EQ1790 \+ := subs( P(t)= 3.9, t= 0, SOLN );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "EQ1800 := subs( P(t)= 5.3, t=10, SOLN );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "PARAMETERS := solve( \{ EQ1790, EQ1 800 \}, \{ k, p[0] \} );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Thus, this model predicts the U.S. population is given by the following fun ction." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "solP := eval( rhs (SOLN), PARAMETERS );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Here is the actual data from the text." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 328 "data:= [1790,3.9 ], [1800,5.3], [1810,7.2], [1820,9.6], [1830,12], [1840,17], [1850,23] , [1860,31], [1870,38], [1880, 50], [1890,62], [1900,75], [1910,91], [ 1920,105], [1930,122], [1940,131], [1950,151], [1960,179], [1970,203], [1980,226], [1990,249], [2000,`NA`], [2010,'NA'], [2020,'NA'], [2030, 'NA'], [2040,'NA'], [2050,'NA'];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "shifted_data:= seq( [ 10*(k-1), data[k,2] ], k = 1 .. nops([data]) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 441 "To evaluate the reasonableness of this prediction we will often plot the function. However, since the t ext provides a table with census values from 1790 through 2050, let's \+ produce predicted populations for each decade (and a few decades into \+ the next millenium). Notice that t in the formula is not \"date\" -- t he year 1790 corresponds to the value t = 0. We produce a second list, called shifted_data, that uses t = 0 as the starting time. " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "soln_points:= NULL:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "for t from 0 to 260 by 10 do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 " printf( `%4d %10.4f %10.4a \\n`, t+1790, solP, shifted_data[t/10+1,2] );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 " soln_points:= soln_points, [t, solP]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "t:= 't':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "plot( [ [sol n_points], [shifted_data] ], color=[red, blue], style = POINT );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Question" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 170 "How well \+ does the plotted solution (or at least the points we've sampled at 10 \+ year intervals) fit the real word data?. Can you explain why they are \+ not exactly the same?" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Questio n" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Determine the values of " } {XPPEDIT 18 0 "p[0]" "6#&%\"pG6#\"\"!" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 261 " when the populations in 1 790 and 1890 are used as data for the system of equations. Repeat the \+ calculations using the populations in 1790 and 1990 as data for the sy stem of equations. How do the different choices of data change the qua lity of the prediction? " }}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 26 "Logistic Population Grow th" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 116 "The logistic population mod el is motivated and derived in the text (pp. 8-9). The resulting diffe rential equation is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "MODE L := diff( P(t), t ) = k * ( 1 - P(t)/N ) * P(t);" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 26 "with the initial condition" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 24 "InitCond := P(0) = p[0];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Equi librium Solution" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 196 "The equilibri um solutions are found in the same manner as in the first example: det ermine all constant populations (i.e., for which the derivative, dP/dt , and hence the RHS of the model, is zero)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "EQUILeqn := rhs( MODEL ) = 0;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 40 "EQUILsol := solve( EQUILeqn, \{ P(t) \} );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Note that there are now two equ ilibria. In addition to the zero solution, the " }{TEXT 256 17 "carryi ng capacity" }{TEXT -1 20 " of the population, " }{XPPEDIT 18 0 "N" "6 #%\"NG" }{TEXT -1 29 ", is an equilibrium solution." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Qualitative Analysis" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "The text \+ gives a good description of the qualitative analysis of the logistic m odel. Assume " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 256 " > 0. The basic idea is that (for equations of this type - we'll be more specif ic shortly) the solutions are either always increasing or always decre asing in regions between the equilibria. In this case, for 0 < P < N, the RHS of the model is positive " }{XPPEDIT 18 0 "k*(1-P/N)*P" "6# *(%\"kG\"\"\",&\"\"\"F%*&%\"PGF%%\"NG!\"\"F+F%F)F%" }{TEXT -1 115 " > \+ 0, and so the solutions are increasing (to the carrying capacity). By \+ the same reasoning, for P > N or P < 0, " }{XPPEDIT 18 0 "k*(1-P/N)* P" "6#*(%\"kG\"\"\",&\"\"\"F%*&%\"PGF%%\"NG!\"\"F+F%F)F%" }{TEXT -1 61 " < 0 and solutions are decreasing (to the carrying capacity)." }}} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Questions" }}{EXCHG {PARA 15 "" 0 "" {TEXT -1 43 "Does the qualitative analysis suggest why " } {XPPEDIT 18 0 "N" "6#%\"NG" }{TEXT -1 63 " is called the carrying cap acity of the population? (Explain.)" }}{PARA 15 "" 0 "" {TEXT -1 46 "H ow would the qualitative analysis change if " }{XPPEDIT 18 0 "k" "6#% \"kG" }{TEXT -1 6 " < 0 ?" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 17 "Analytic Solution" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 167 "We will learn specific techniques for solving special classes of differential equations. The Maple comm and for finding explicit solutions to differential equations is " } {TEXT 19 6 "dsolve" }{TEXT -1 167 ". The arguments are the initial val ue problem (a set containing the differential equation and the initial condition) and the function for which we are trying to solve." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "SOLNlog := dsolve( \{ MODEL, InitCond \}, P(t) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "The U.S. Population" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 184 "To conclude this investigation of the mo dels, lets determine the values of the parameters that fit the U.S. po pulation in 1790, 1900, and 1990. (Why are three years used for this f it?)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "EQ1790 := eval( SOL Nlog, [P(t)=3.9, t=0] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "EQ1900 := eval( SOLNlog, [P(t)=75, t=110] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "EQ1990 := eval( SOLNlog, [P(t)=249, t=200] );" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 107 "The solution(s) to this system of three equations and \+ three unknowns can be found using the solve command. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "solve( \{ EQ1790, EQ1900, EQ1990 \} , \{ k, N, p[0] \} );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 383 "Note th at while there are many solutions, all but one of them involves comple x numbers. Since we know our answer should be real-valued (WHY?) we ca n assume that these are the parameter values that we should use for th is problem. There is a fancy way to have Maple select all real-valued \+ solutions but it's simpler to simply copy and paste the portion of the output that interests us." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "PARAMETERS := \{p[0] = 3.900000000, k = .2934545722e-1, N = 302.94 59184\};" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 107 "(optional) Fancy me thod of extracting real-valued solution from multiple solutions of a s ystem of equations" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "PARAME TERS := op( remove( has, \{%\}, I ) );" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "The logistic \+ model for the U.S. population based on the 1790, 1900, and 1990 popula tions is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "solPlog := eval ( rhs( SOLNlog ), PARAMETERS );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "soln_points:= NULL:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "A \+ table of predicted populations for 1790 - 2050 is easily created as be fore" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "for t from 0 to 260 by 10 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 90 " printf( `%4d %10.4f %10.4f %10.4a\\n`, t+1790, solP, solPlog, shifted_data[t/10+1,2] ); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 " soln_points:= soln_points, [t , solPlog]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "t := 't':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 452 "Note that th e logistic model does a much better job of following the actual U.S. p opulations (as reported in Table 1.1 on p. 7). A final comparison of \+ the two solutions can be performed by looking at the plot of the solut ions to each model. Here we want to overlay two different kinds of plo ts (not just two curves in one kind of plot). The way to do this is to give the plots names, end the commands with colons, and then ``displa y'' the two together" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "pl otA:= plot( [ solP, solPlog ], t=0..300, 0..400, color=[red, green], t itle=`Unlimited and Logistic Growth of the US Population (in millions) ` ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "plotB:= plot( [ [shifted_da ta] ], color = blue, style = POINT):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "display([plotA, plotB]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 156 "Note how one solution appears to level off around 300 (million) a nd the other appears to continue to grow forever. Which curve correspo nds with which model?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }