{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 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{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 "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 "Normal" -1 256 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" -1 257 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 2 270 5 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 41 "ORDINARY DIFFERENTIAL EQUATIONS POWERTOOL" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 41 "Unit 25 -- Applica tion: Population Models" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {URLLINK 17 "Prof. Douglas B. Meade" 4 "http://www.math.sc.edu/ ~meade/" "" }}{PARA 256 "" 0 "" {URLLINK 17 "Industrial Mathematics In stitute" 4 "http://www.math.sc.edu/~IMI/" "" }}{PARA 256 "" 0 "" {URLLINK 17 "Department of Mathematics" 4 "http://www.math.sc.edu/" " " }}{PARA 256 "" 0 "" {URLLINK 17 "University of South Carolina" 4 "ht tp://www.sc.edu/" "" }}{PARA 256 "" 0 "" {TEXT -1 19 "Columbia, SC 292 08\n" }}{PARA 256 "" 0 "" {TEXT -1 7 "URL: " }{URLLINK 17 "http://ww w.math.sc.edu/~meade/" 4 "http://www.math.sc.edu/~meade/" "" }}{PARA 256 "" 0 "" {TEXT -1 25 "E-mail: meade@math.sc.edu" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 38 "Copyright \251 2001 b y Douglas B. Meade" }}{PARA 256 "" 0 "" {TEXT -1 19 "All rights reserv ed" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 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 of Unit 25" } }{EXCHG {PARA 14 "" 0 "" {HYPERLNK 17 "25.A" 1 "" "25.A" }{TEXT -1 31 " Malthusian (Exponential) Model" }}{PARA 14 "" 0 "" {HYPERLNK 17 "25. B" 1 "" "25.B" }{TEXT -1 15 " Logistic Model" }}{PARA 14 "" 0 "" {HYPERLNK 17 "25.C" 1 "" "25.C" }{TEXT -1 15 " Gompertz Model" }}} {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( DEtools ):" }}{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 "25.A" {TEXT -1 35 "25.A Ma lthusian (Exponential) Model" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "A \+ population with constant birth and death rates can be modeled as" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "ode1 := diff( p(t), t ) = k * p(t);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 18 "ic := p(0) = P[0];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "p(t)" "6#-%\"pG6#%\"tG" }{TEXT -1 44 " denotes the size of the population at time " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "k=k[birth]-k[death]" "6#/%\"kG,&&F$6#%&birthG\"\" \"&F$6#%&deathG!\"\"" }{TEXT -1 29 " is the net growth rate, and " } {XPPEDIT 18 0 "P[0]" "6#&%\"PG6#\"\"!" }{TEXT -1 39 " is the initial s ize of the population." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "The solution to any Malthusian model is an exponenti al function" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "sol1 := dsolve( \{ ode1, ic \}, p(t) );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "If the net growth rate is positive, " }{TEXT 256 4 "i.e. " }{TEXT -1 2 ", " }{XPPEDIT 18 0 "k[birth]" "6#&%\"kG6#%&birthG" } {TEXT -1 3 " > " }{XPPEDIT 18 0 "k[death]" "6#&%\"kG6#%&deathG" } {TEXT -1 228 ", then the population grows (exponentially) without boun d. If the net growth rate is negative, then the population decays expo nentially to zero. And, if the net growth rate is zero, the population remains constant for all time: " }{XPPEDIT 18 0 "p(t)=P[0]" "6#/-%\"p G6#%\"tG&%\"PG6#\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 144 "While an exponential model can be us eful for short-term projections, the unbounded growth is a major limit ation to making long-term projections." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 " " 0 "25.B" {TEXT -1 19 "25.B Logistic Model" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 190 "The logistic model addresses the unbounded growth of the Malthusian model by assuming that the growth rate starts at the net g rowth rate but decreases (linearly) as the population increases." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "ode2 := diff( p(t), t ) = k * ( 1 - p(t)/A ) * p(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "ic;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "This autonomo us ODE is separable; the solution is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "q1 := dsolve( \{ ode2, ic \}, p(t) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "A more typical form for this solution is " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "sol2 := subsop(2=map( collect, rhs(normal(q1)), exp ) , q1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "There are two equilibria for the logistic model : " }{XPPEDIT 18 0 "p=0" "6#/%\"pG\"\"!" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "p=A" "6#/%\"pG%\"AG" }{TEXT -1 253 ". The qualitative b ehavior can be determined from a sign chart, or from the direction fie ld. Unfortunately, Maple is unable to produce a direction field for ge neral parameters, but the essential features of the model are observab le with any reasonable, " }{TEXT 257 4 "i.e." }{TEXT -1 22 ", positive , values of " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "param2 := \{ k=1, A=7 \}:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "DEplot( eval( ode2, par am2 ), p(t), t=0..10, p=0..10 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "The direction field is suggests that, when " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 99 " > 0, any solution that begins with a positive initial populati on ultimately tends to the value of " }{XPPEDIT 18 0 "A" "6#%\"AG" } {TEXT -1 105 ". This shows how the logistic model overcomes the unboun ded growth for exponential models. The parameter " }{XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 121 " is frequently called the carrying capacity o f the population. This conjecture is supported with the symbolic compu tation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "assume( k > 0 ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "limit( sol2, t=infinity );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "u nassign( 'k' );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "To see what happens when k < 0, co nsider" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "param2 := \{ k=-1, A=7 \}:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "DEplot( eval( ode2, param2 ), p(t), t=0..10, p=0..10 \+ );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 11 "Thus, when " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 69 " < 0, the carrying capacity is a bifurcation value for solution s. If " }{XPPEDIT 18 0 "P[0]" "6#&%\"PG6#\"\"!" }{TEXT -1 3 " < " } {XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 33 ", solutions decay to zero an d if " }{XPPEDIT 18 0 "P[0]" "6#&%\"PG6#\"\"!" }{TEXT -1 3 " > " } {XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 99 ", solutions grow without bou nd. However, the symbolic computation of the limiting population yield s" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "assume( k < 0 ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "limit( sol2, t=infinity );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "u nassign( 'k' ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 164 "The problem is that when the net \+ growth rate is negative, the solution does not exist for all time. The solution exists only so long as the denominator is not zero." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "q2 := solve( denom(rhs(sol2))=0, t );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "This expre ssion is not easily simplified within Maple, but is easily seen to be \+ equivalent to" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Tblowup := 1/k * ln( 1 - A/P[0] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "P[0]" "6#&%\"PG6#\"\"!" }{TEXT -1 24 " are both positiv e, 0 < " }{XPPEDIT 18 0 "1 - A/P[0]" "6#,&\"\"\"F$*&%\"AGF$&%\"PG6#\" \"!!\"\"F+" }{TEXT -1 10 " < 1 when " }{XPPEDIT 18 0 "P[0]" "6#&%\"PG6 #\"\"!" }{TEXT -1 3 " > " }{XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 8 ". \+ Then, " }{XPPEDIT 18 0 "ln(1-A/P[0])" "6#-%#lnG6#,&\"\"\"F'*&%\"AGF'&% \"PG6#\"\"!!\"\"F." }{TEXT -1 56 " < 0 and the blow-up time is positiv e (and finite) when " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 73 " < 0. This finite time blowup is difficult to see in the direction field." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " } {XPPEDIT 18 0 "P[0]" "6#&%\"PG6#\"\"!" }{TEXT -1 3 " < " }{XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 69 ", the denominator is never zero and the \+ solution exists for all time." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Of course, if " } {XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 45 " = 0, then the population re mains unchanged, " }{XPPEDIT 18 0 "p=P[0]" "6#/%\"pG&%\"PG6#\"\"!" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "Another interesting feature of the logistic equation is the change of concavi ty for some solutions with 0 < " }{XPPEDIT 18 0 "P[0]" "6#&%\"PG6#\"\" !" }{TEXT -1 3 " < " }{XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 123 ". To \+ investigate the location of this inflection point, zeroes of the secon d derivative of the solution must be identified:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "DDsol2 := d iff( sol2, t$2 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "q3 := solve( rhs(DDsol2)=0, \+ t );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "which can be simplified by hand to" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Tinfle ction := 1/k * ln( A/P[0] - 1 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 278 "This time is the location of a possible inflection point. The complete verificatio n that an inflection point does occur at this time is left as an exerc ise. (Note that the assumptions on the parameters ensure that this tim e is real and positive.) At this time, the population is" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "simp lify( eval( sol2, t=Tinflection ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "That is, when 0 < " }{XPPEDIT 18 0 "P[0]" "6#&%\"PG6#\"\"!" }{TEXT -1 3 " < " } {XPPEDIT 18 0 "A/2" "6#*&%\"AG\"\"\"\"\"#!\"\"" }{TEXT -1 357 " the so lution changes concavity when the population is exactly half of the ca rrying capacity. The significance of this result is that it provides a means to estimate the carrying capacity for a population from data be fore the population actually reaches the carrying capacity. The only r equirement is to be able to locate the inflection point from the data. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "25.C" {TEXT -1 19 "25.C Go mpertz Model" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "Another attempt t o eliminate the unbounded growth of solutions to the Malthusian model \+ is the Gompertz model." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "ode3 := diff( p(t), t ) = k * ( 1 - ln(p(t))/a ) * p(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "ic; " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "The main difference is the ap pearance of " }{XPPEDIT 18 0 "ln(p(t))" "6#-%#lnG6#-%\"pG6#%\"tG" } {TEXT -1 81 " in the factor that modifies the net growth rate. With th is change, and assuming " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 35 " \+ > 0, the effective growth rate is " }{XPPEDIT 18 0 "k" "6#%\"kG" } {TEXT -1 6 " when " }{XPPEDIT 18 0 "p=1" "6#/%\"pG\"\"\"" }{TEXT -1 38 ", then decreases (logarithmically) as " }{XPPEDIT 18 0 "p" "6#%\"p G" }{TEXT -1 21 " increases. (For 0 < " }{XPPEDIT 18 0 "p" "6#%\"pG" } {TEXT -1 61 " < 1, the effective growth rate exceeds the net growth ra te.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "T he solution to this IVP is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "q4 := simplify( dsolve( \{ ode3, ic \}, p(t) ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "which simplifies (manually) to" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "sol3 := p(t) = exp( a ) * exp( (ln(P[0])-a)*exp(-k*t/a) );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "To begin the qualitative analysis of solutions to the Gom pertz model, observe that when " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 35 " are nonzero a nd have the same sign" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "assume( k/a > 0 ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "limit( sol3, t=infinity );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "unassign( 'k', 'a' ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "A representat ive direction field for this situation is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "param3 := \{ k=1, a= ln(4.) \}:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "DEplot( eval( ode3, p aram3 ), p(t), t=1..10, p=1..10 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "When " } {XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "a" "6# %\"aG" }{TEXT -1 22 " have different signs," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "assume( k/a < 0 ) :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "limit( sol3, t=infinity );" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "unassign( 'k', 'a' ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "The reason for Maple not simplifying this limit is that the val ue depends on the sign of " }{XPPEDIT 18 0 "ln(P[0])-a" "6#,&-%#lnG6#& %\"PG6#\"\"!\"\"\"%\"aG!\"\"" }{TEXT -1 56 ". A representative directi on field for this situation is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "param3 := \{ k=-1, a=ln(4.) \+ \}:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "DEplot( eval( ode3, param3 ) , p(t), t=1..10, p=1..10 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Note that solutions with \+ " }{XPPEDIT 18 0 "ln(P[0])" "6#-%#lnG6#&%\"PG6#\"\"!" }{TEXT -1 3 " < \+ " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 21 " decay to zero while " } {XPPEDIT 18 0 "ln(P[0])" "6#-%#lnG6#&%\"PG6#\"\"!" }{TEXT -1 3 " > " } {XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 174 " gives rise to solutions th at are unbounded as time increases without bound. A key difference fro m the logistic equation is that these unbounded solutions exist for al l time." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "The direction fields suggest that some solutions have a change in concavity. The second derivative of the solution is" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "DDsol3 := diff( sol3, t$2 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "which is zero when" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Tinfle ct3 := solve( rhs(DDsol3)=0, t );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 174 "After it has been confirmed that this time actually is an inflection point for the solution, it is interesting to note that the inflection point occurs \+ when the population is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "simplify( eval( sol3, t=Tinflect3 ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 92 "Thus, the limiting population for the Gompertz mod el occurs when the population is a factor " }{XPPEDIT 18 0 "exp(1)" "6 #-%$expG6#\"\"\"" }{TEXT -1 157 " larger than the population at the in flection point. (Recall that the logistic model's carrying capacity is twice the population at the change in concavity.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "[Back to " }{HYPERLNK 17 "ODE Power tool Table of Contents" 1 "unit00.mws" "" }{TEXT -1 1 "]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }