{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times " 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 } {PSTYLE "List Item" -1 14 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 14 5 }{PSTYLE "List Subitem" 14 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 3 12 0 0 0 0 273 0 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "List Subitem" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 3 12 1 0 2 2 14 5 }} {SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 41 "ORDINARY DIFFERENTIAL EQUATIONS POWERTOOL" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 54 "Unit 3 -- Applicat ion: Exponential and Logistic Growth" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {URLLINK 17 "Prof. Douglas B. Meade" 4 "http://www .math.sc.edu/~meade/" "" }}{PARA 257 "" 0 "" {URLLINK 17 "Industrial M athematics Institute" 4 "http://www.math.sc.edu/~IMI/" "" }}{PARA 257 "" 0 "" {URLLINK 17 "Department of Mathematics" 4 "http://www.math.sc. edu/" "" }}{PARA 257 "" 0 "" {URLLINK 17 "University of South Carolina " 4 "http://www.sc.edu/" "" }}{PARA 257 "" 0 "" {TEXT -1 19 "Columbia, SC 29208\n" }}{PARA 257 "" 0 "" {TEXT -1 7 "URL: " }{URLLINK 17 "ht tp://www.math.sc.edu/~meade/" 4 "http://www.math.sc.edu/~meade/" "" }} {PARA 257 "" 0 "" {TEXT -1 25 "E-mail: meade@math.sc.edu" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 38 "Copyright \251 \+ 2001 by Douglas B. Meade" }}{PARA 257 "" 0 "" {TEXT -1 19 "All rights reserved" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 67 "--------------------------------------------------------- ----------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 18 "Outline fo r Unit 3" }}{EXCHG {PARA 14 "" 0 "" {HYPERLNK 17 "3.A" 1 "" "3.A" } {TEXT -1 29 " Exponential Growth and Decay" }}{PARA 258 "" 0 "" {HYPERLNK 17 "3.A-1" 1 "" "3.A-1" }{TEXT -1 17 " General Solution" }} {PARA 258 "" 0 "" {HYPERLNK 17 "3.A-2" 1 "" "3.A-2" }{TEXT -1 14 " Dou bling Time" }}{PARA 258 "" 0 "" {HYPERLNK 17 "3.A-3" 1 "" "3.A-3" } {TEXT -1 10 " Half-Life" }}{PARA 14 "" 0 "" {HYPERLNK 17 "3.B" 1 "" "3 .B" }{TEXT -1 18 " Logistic Equation" }}{PARA 258 "" 0 "" {HYPERLNK 17 "3.B-1" 1 "" "3.B-1" }{TEXT -1 18 " General Solution" }}{PARA 258 "" 0 "" {HYPERLNK 17 "3.B-2" 1 "" "3.B-2" }{TEXT -1 18 " Carrying Capa city" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 14 "Initialization" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with( D Etools ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with( plots ):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with( linalg ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "3.A" {TEXT -1 32 "3.A Exponential Growth and Decay" }}{SECT 0 {PARA 4 "" 0 "3.A-1" {TEXT -1 22 "3.A-1 General Solution" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "The general model for exponential growth and decay with \"growt h\" constant " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 4 ", is" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "ode := diff( x(t), t ) = k * x(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "and initial c ondition" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "ic := x(0) = A;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "This equation is seen to be separable, and Maple agrees with this classification" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "odeadvisor( ode, [separable] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "To find the g eneral solution from first principles," }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "impl_soln := subs( _t=t , map( int, ode/x(t), t=0.._t ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "impl_part_soln := subs( ic, impl_soln );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "expl_part_soln := op(solve( impl_pa rt_soln, \{x(t)\} ));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Of course, the same result could be obtained from" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 23 "infolevel[dsolve] := 3:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 48 "soln2 := dsolve( \{ode, ic\}, x(t), [separable] ); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "infolevel[dsolve] := 0:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "simplify( soln2 );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "3 .A-2" {TEXT -1 19 "3.A-2 Doubling Time" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 142 "The doubling time for a process growing exponentially is the time needed for the quantity to double from its original size. Th at is, the time " }{XPPEDIT 18 0 "t[d]" "6#&%\"tG6#%\"dG" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "x(t[d]) = 2*x(0)" "6#/-%\"xG6#&%\"tG6 #%\"dG*&\"\"#\"\"\"-F%6#\"\"!F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "double_eqn \+ := subs( t=t[d], x(t[d])=2*x(0), ic, expl_part_soln );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "double_time := solve( double_eqn, \+ \{t[d]\} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 213 "One of the important characteristics of \+ the doubling time is that not only is it the time needed for the initi al size to double, it is the time needed for the size to double at any point in an exponential process." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "q1 := x(T+t[d])/x(T)" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " = subs( t=T+t[d]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 " rhs(expl_part_soln) ) / subs( t=T , rhs(expl_part_soln) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 " q1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "`` = simplify(eval( rhs(q1), double_time ));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 0 {PARA 4 "" 0 "3.A-3" {TEXT -1 15 "3.A-3 Half-Life" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 145 "The half-life for a quantity that is dec aying according to an exponential model is the time after which exactl y half the original amount remains." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "half_eqn := subs( t=t[h], \+ x(t[h])=1/2*x(0), ic, expl_part_soln );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "half_time := solve( half_eqn, \{t[h]\} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 91 "<0 for a decaying process. Thus, the half-life for an exponential model with \+ \"growth\" rate " }{XPPEDIT 18 0 "-k" "6#,$%\"kG!\"\"" }{TEXT -1 76 " \+ is the same as the doubling time for an exponential model with growth \+ rate " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}} {SECT 0 {PARA 3 "" 0 "3.B" {TEXT -1 21 "3.B Logistic Equation" }} {SECT 0 {PARA 4 "" 0 "3.B-1" {TEXT -1 22 "3.B-1 General Solution" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 120 "The general logistic equation is \+ a modification of the exponential model in which the growth is tempere d by the factor (" }{XPPEDIT 18 0 "K-x(t)" "6#,&%\"KG\"\"\"-%\"xG6#%\" tG!\"\"" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "logistic_ode := diff( x(t), t ) = A * x(t) * ( K - x(t) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "with initial condition" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "logistic_ic := x(0)=X[0];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "This ODE is easily sep arated," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "sep_log_ode := logistic_ode / (x(t)*(K-x(t)));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "integrated from the initial time, 0, to any other time t, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "part_log_soln := subs( _t=t, logistic_ic, map(int,sep _log_ode,t=0.._t) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "and solved for " }{XPPEDIT 18 0 " x(t)" "6#-%\"xG6#%\"tG" }{TEXT -1 32 " to obtain the explicit solution " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "part_expl_soln := collect(op( solve( part_log_soln, \+ \{x(t)\} ) ),exp);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "A quick che ck that this is, in fact, a solution to the original ODE shows" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "odetest( part_expl_soln, logistic_ode );" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "and, f or the initial condition," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "eval( part_expl_soln, t=0 );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "3 .B-2" {TEXT -1 23 "3.B-2 Carrying Capacity" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 141 "Before discussing the generic properties of the logistic model, it is instructive to use graphical methods to examine a specif ic example. Let" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 26 "param := \{ A = 1/2, K=3 \};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "l_ode := subs(param,logistic_ode);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Then, the direction field for this model is" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "DEpl ot( l_ode, x(t), t=0..10, x=0..10, arrows=SMALL );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "The equilibrium solution at " }{XPPEDIT 18 0 "x=3" "6#/%\"xG\"\"$" } {TEXT -1 64 " is obvious in the graph (and the equation); the equilibr ium at " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 97 " is obviou s from the equation. Initial conditions that will produce the equilibr ium solutions are" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 35 "equil_ic := [ [x(0)=0], [x(0)=3] ];" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "equil_plot := DEplot( l_ode, x(t), t=0..5, equil_ic, x=0..10," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 " arrows=SMALL, linecolor=CYAN ):" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 11 "equil_plot;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "A sample of s olutions with initial conditions between the two equilibria" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "i c1 := [ [x(0)=i/2] $ i=1..5 ]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "s oln_plot1 := DEplot( l_ode, x(t), t=0..5, ic1, x=0..10, arrows=SMALL, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 " linecolor =GREEN ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "soln_plot1;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Note that all of these solutions are increasing and appea r to approach " }{XPPEDIT 18 0 "x=3" "6#/%\"xG\"\"$" }{TEXT -1 4 " as \+ " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 23 " continues to increase." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "Next, a sample of solutions with initial conditions above the positive equili brium" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "ic2 := [ [x(0)=2*i] $ i=2..5 ]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "soln_plot2 := DEplot( l_ode, x(t), t=0..5, ic2, x=0.. 10, arrows=SMALL," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 " \+ linecolor=BLUE, stepsize=0.1 ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "soln_plot2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "These solutions are all decreasing and also appear to approach the " }{XPPEDIT 18 0 "x=3" "6#/%\"xG\"\"$ " }{TEXT -1 4 " as " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 11 " incre ases." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 " The composite plot is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "display( [equil_plot, soln_plot1, s oln_plot2] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "To investigate some of the genera l properties of solutions to the logistic equation, note that the equi librium solutions are" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "equil_sol := solve( rhs(logistic_od e)=0, \{x(t)\} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "A quick inspection of the ODE show s that " }{XPPEDIT 18 0 "diff(x(t),t)" "6#-%%diffG6$-%\"xG6#%\"tGF)" } {TEXT -1 14 " > 0 when 0 < " }{XPPEDIT 18 0 "x(t)" "6#-%\"xG6#%\"tG" } {TEXT -1 3 " < " }{XPPEDIT 18 0 "K" "6#%\"KG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "diff(x(t),t)" "6#-%%diffG6$-%\"xG6#%\"tGF)" }{TEXT -1 10 " < 0 when " }{XPPEDIT 18 0 "x(t)" "6#-%\"xG6#%\"tG" }{TEXT -1 3 " \+ > " }{XPPEDIT 18 0 "K" "6#%\"KG" }{TEXT -1 97 ". The long-term behavio r of the solutions can be determined from the explicit solution to the IVP" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "limit_size := Limit( part_expl_soln, t=infinity );" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "value( limit_size );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "Maple is unable to evaluate this limit because it is unab le to decide if " }{XPPEDIT 18 0 "exp(A*K*t)" "6#-%$expG6#*(%\"AG\"\" \"%\"KGF(%\"tGF(" }{TEXT -1 10 " tends to " }{XPPEDIT 18 0 "0" "6#\"\" !" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "infinity" "6#%)infinityG" } {TEXT -1 62 " (or does not exist). However, under the physical assumpt ions," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "assume( A>=0, K>0 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "the limiting \+ size is found to be" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "value( limit_size );" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Beca use all solutions with " }{XPPEDIT 18 0 "x(0)" "6#-%\"xG6#\"\"!" } {TEXT -1 3 " > " }{XPPEDIT 18 0 "K" "6#%\"KG" }{TEXT -1 13 " decrease \+ to " }{XPPEDIT 18 0 "x=K" "6#/%\"xG%\"KG" }{TEXT -1 28 " and all solut ions with 0 < " }{XPPEDIT 18 0 "x(0)" "6#-%\"xG6#\"\"!" }{TEXT -1 3 " \+ < " }{XPPEDIT 18 0 "K" "6#%\"KG" }{TEXT -1 13 " increase to " } {XPPEDIT 18 0 "x=K" "6#/%\"xG%\"KG" }{TEXT -1 16 ", the parameter " } {XPPEDIT 18 0 "K" "6#%\"KG" }{TEXT -1 60 " is known as the carrying ca pacity of the logistic equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "unassign('A','K');" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 9 "[Back to " }{HYPERLNK 17 "ODE Powertool Table of Content s" 1 "unit00.mws" "" }{TEXT -1 1 "]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 1 0" 14 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }