ࡱ; ~  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}Root EntryFgBook ]8CompObj^SummaryInformation( FMicrosoft Excel 5.0 WorksheetBiff5Excel.Sheet.5;  Oh+'Oh+'0TH (0HG*^. C     ``iݜ*o?H6A^BbzbO/(fvEE>j(9jJ2 """)))000___UUUMMMBBB999 ```( +9%I;/]E:IS+!!lYQGj2ga1aS {Cg..&YQFh.IRj#SjJu3lJA7e ,N, QdoV YC6r3w_GqC-}zn#&s[ Rp$L 6n{DuqVJ4H2Aph6B"}3%6Z\H"BMXR ӕ$sVo<gX .; ; ;,;;;@.!;L; ;)@;L;;; ;ʈ;,Curve fitting--4NRE 545 S. ArlinghausLogistic curve fitting.  ZR The exponential function leads to a situation of unbounded population growth.'Assumption for the exponential:XP The rate of population growth or decay at any time t is proportional to the$ size of the population at t.` XSymbolically: let Y_t [N from last time] represent the size of a population at time t.9 1The rate of growth of Y_t is proportional to Y_t:  dY_t / dt = kY_t1)where k is a constant of proportionality.B:This assumption led to the curve Y_t = Y_(t0) e^(kt) ---- D<a situation in which growth is unbounded as t becomes large..&Bounded growth--the logistic function.^V Assume now that in reality, when the population gets large, environmental factors<4dampen growth --- a population-environment dynamic.*"To express this idea symbolically:bZ assume that eventually the growth rate decreases ---- dY_t / dt eventually decreases.H@ assume that population size is limited to some maximum, q, e] l \ sandy B LOGISTIC/ __123Graph_A;I' ;__123Graph_AHORNBAR1.PIC;!J' ;__123Graph_AHORNBAR2.PIC;gJ'  ?;__123Graph_AHORNBAR3.PIC;J' ;__123Graph_AHORNBAR4.PIC;J' ;__123Graph_AHORNBAR5.PIC;9K' / __123Graph_B;sK' ;__123Graph_BHORNBAR1.PIC;K' ;__123Graph_BHORNBAR2.PIC;K' V;__123Graph_BHORNBAR3.PIC;EL' ;__123Graph_BHORNBAR4.PIC;L' ;__123Graph_BHORNBAR5.PIC;L' / __123Graph_C; M' ;__123Graph_CHORNBAR3.PIC;QM' ;__123Graph_CHORNBAR4.PIC;M' ;__123Graph_CHORNBAR5.PIC;M' / __123Graph_D;N' ;__123Graph_DHORNBAR3.PIC;]N' ;__123Graph_DHORNBAR4.PIC;N' ;__123Graph_DHORNBAR5.PIC;N' / __123Graph_X;#O' ;__123Graph_XHORNBAR1.PIC;iO' ;__123Graph_XHORNBAR2.PIC;O' Ry;__123Graph_XHORNBAR3.PIC;O' ;__123Graph_XHORNBAR4.PIC;;P' ;__123Graph_XHORNBAR5.PIC;P'  _Regression_Int/_Regression_Out:P' 50 _Regression_X;Q' #30 _Regression_Y;YQ' 6F$ ;R'  0 Print_Area_MI; =.8X@"1Helv1Arial1Arial1Arial1Arial1Arial1Arial"$"#,##0_);\("$"#,##0\)"$"#,##0_);[Red]\("$"#,##0\) "$"#,##0.00_);\("$"#,##0.00\)%""$"#,##0.00_);[Red]\("$"#,##0.00\)5*2_("$"* #,##0_);_("$"* \(#,##0\);_("$"* "-"_);_(@_),))_(* #,##0_);_(* \(#,##0\);_(* "-"_);_(@_)=,:_("$"* #,##0.00_);_("$"* \(#,##0.00\);_("$"* "-"??_);_(@_)4+1_(* #,##0.00_);_(* \(#,##0.00\);_(* "-"??_);_(@_) General_)                + ) , *  !   !  @A !  LOGISTIC# HORNBAR5.PIC# HORNBAR4.PIC| HORNBAR3.PIC HORNBAR2.PIC+ HORNBAR1.PIC l  XG!'H+1}9@OW^l*f4  dMbP?_*+%;d LB&?'?(?)?MHP LaserJet III 8DR "d??U} } } }  }  G ;;- ;L; ;/N; ;; ;  ;  where 0 < Y_t < q. As Y_t approaches q, dY_t / dt approaches 0, so that populationG? size tends to become stable as t approaches infinity.\TThe result is a model that is exponential in shape initially and includes effects of6# ^+\(d=5FH2b@.fLiK ;!;- %;L0; 2;/N3; 5;7;8; 9;:;;; <;,=;>;?;@.!` Xenvironmental resistance in larger populations. A typical geometric characterization is! shown below.% 0te2]The various coordinates in this figure are derived from expressing the assumptions of bounded63.growth algebraically. One such expression is:$5dY_t / dt = kY_t * (q-Y_t)/q\7TThis equation meets the conditions of bounded growth because in the factor (q-Y_t)/q8when Y_t is smalli9a(q-Y_t)/q is close to 1 and the growth is therefore close to the exponential in the early stages;0:(when Y_t is large--that is, close to q--f;^(q-Y_t)/q is close to 0, and the growth rate dy_t / dt tapers off to the horizontal asymptote.j<bThe factor (q-Y_t)/q, which is what alters the symbolic form of the logistic from the exponential,0=(acts as a damper to growth--as required.u>m*************************************************************************************************************f?^To understand where the various quatities in the graph above come from, consider the following$,d i:(`m4jn4y@;B;- D;LF; G;/NI; K;M;O; Q;S;T; V;,X;Z;\;@.!^;L@set of manipulations.AB9Replace k/q by K so that the logistic equation now reads:DdY_t / dt = KY_t(q-Y_t)aFYin which the rate of growth is proportional to the product of the population size and theEG= difference between the maximum size and the population size.eI]Solve this latter equation for Y_t ---- separate the variables of the differential equation"K(dY_t)/(Y_t(q-Y_t)) = K dtMto yield+O#INT (dY_t)/(Y_t(q-Y_t)) = INT K dt.PQHUse a Table of INtegrals on the rational form in the left-hand integral:+S#1/q * ln|(Y_t)/(q-Y_t)| = Kt +C or,"Tln|Y_t/(q-Y_t)| = qKt +qC.&VBecause Y_t > 0 and q-Y_t > 0,#Xln (Y_t/(q-Y_t)) = qKt +qC.Z Therefore,$\Y_t/(q-Y_t) = e^(qKt)e^(qC).%^Replace e^qC by A. Therefore&@!E#eIi&/T/&*'(`;a;- b;Lc; e;/Nf; h;j;l; n;o;q; s;,u;w;y;@.!`Y_t/(q-Y_t) = Ae^(qKt)aY_t=(q-Y_t)Ae^(qKt)!bY_t=qAe^(qKt)-Y_tAe^(qKt)(c Y_t = (qAe^(qKt)/(Ae^(qKt) + 1))de\Now divide top and bottom of the last equation by Ae^(qKt) [equivalent to multiplying by 1],fso that;h3Y_t = q/(1+(1/(Ae^(qKt)))) = q/(1+ 1/A * e^(-qKt)).XjPReplace 1/A by a and -qK by b producing a common form for the logistic function:lY_t = q/ (1 + ae^(bt))1n)with b < 0 because b = -qK, and q, K > 0.qoi*********************************************************************************************************7q/Facts about the graph of the logistic equation.Es=a. The line Y_t = q is a horizontal asymptote for the graph.)u! This is so because, for b<0,=w5lim (t --> inf) q/(1+ae^(bt)) ------> q/(1+a(0)) = q.dy\Can the logistic curve cross this horizontal asymptote to produce one or multiple crossings?$,"%,h?\"5u;I-A;;- ;L; ;/N; ;;; ;;; c[That is, can the curve approach the asymptote from above, or might the curve oscillate back@8and forth across the asymptote while closing down on it?f^This idea can be expressed in notation by setting Y_t = q in the logistic equation and solving^Vto see if there are any points that lie simultaneously on the curve and the asymptote.Can it be that: Y_t = Y_t / (1+ae^(bt))? Or, 1=1+ae^(bt)?NFOr, ae^(bt) =0--or that a=0--no, becasue a=1/A. Or that e^(bt)=0--no.c[Thus, the logistic growth curve cannot cross the horizontal asymptote so that it approaches<4it entirely from one side--in this case, from below.OGb. Find the coordinates of the inflection point of the logistic curve.gDjb$Rg@;;- ;L; ;/N; ;;; ;;; ;,;;;@.!;L; ;)@;Le]The inflection point is the point at which the curve changes from concave up to concave down.jbIt is also the point at which the growth rate is a maximum--to find this maximum differentiate one&form of the logistic equation:A9logistic equation: dY_t = dt = KY_t(q-Y_t)=KqY_t - KY_^2.&derivative: d^2Y_t /dt^2 = Kq - 2KY_t-%Set this last equation equal to zero.Kq - 2KY_t = 0.g_Therefore, Y_t = q/2. This is the vertical coordinate of the inflection point of the curve forjbY_t, the logistic curve -- dY_t /dt is increasing to the left of q/2 and increasing to the rightNF(from evidence of second derivatives). The value found is a maximum.Horizontal component:IA To find t, put Y_t = q/2 in the logistic equation and solve:q/2 = q/(1 + ae^(bt))F>Solving, 1 + ae^(bt) = 2; e^(bt)=1/a; e^(-bt)=a; -bt=ln a, andt = (ln a) / (-b)G?Thus, the labelling of the logistic curve is as given earlier--ZRthe height of the inflection point is half the height of the horizontal asymptote.LOGISTIC CURVEFIT USING ACTUAL DATA. EQUATION:,o|in*E21knR!M!JK^";;- ;L; ;/N; ;;; ;;; ;,;;;@.!;L; ;)@;L;;; ;ʈ;,;7t;;" y=q/(1+ae^(bt)), b<0.   FIND:5- from the given data, values for a and b.E= Then, use the equation on a number of t-values to graph the equation.TO FIND a and b:H@ All that is required is two values of t, an upper bound q,H@ and the assumption that the distribution is to be logistic.EXAMPLE:80 Inland coal consumption in India--'000 tons4,Data of K. Hornbarger (545 student in 1992). t y least sqs year 0 is fit 1970; year 16is 1987projected beyond 1987@1(ȅ%@ |E^ӫ?Dt//n&@A?@ZH)@% |E^ӫ?Lt//n&@A@ @]?;@@a@)4\@@p@]@@@gj@@@2]$@@@^i@ @P@1 @"@`@pɘ@$@#@a?@<v& 9ILL<8/$!K^55555555;;- ;L; ;/N; ;;; ;;; ;,;;;@.!;L; ;)@;L;;; ;ʈ;,;7t;;;;\; &@AHYaA(@AR BkA*@A`͒NPdA,@pA(bGkA.@GAF8A0@AAF>Now suppose instead that estimates based on remaining reservesH@require that inland consumption never exceed 250000 tons of coalKCper year. Suppose also that the last year for which we have actualC;data is the year 1987. Assume a logistic representation in7/the form y=250000/(1+ae^(bt)) and find a and b."In 1970, when t=0, y=77435 Thus, 77435=250000/(1+a)"So that 1+a=250,000/77435=Thus, a=2.228514?7Now, find b. Use other actual data endpoint from 1987.$In 1987, when t=16, y=184790,$184790=250000/(1+2.228514*exp(16*b))-%1+2.228514*exp(16*b)=250,000/184,790=)"l?AA( exp(16*b)=(1.352887-1)/2.228514=.ZD?l?l%@16b=ln(0.158350)="g ųj? tV?AB7X555555JLOG;&$&C(0^^C;;- ;L; ;/N; ;;;  ; ; ;  ;, ;;;@.!;L; ;)@;L;;; ;ʈ;,;7t;;;;\; ;' b=(ln(0.158350))/16='I(|0*D?A0(Logistic equation fit using actual data:,$y=250,000/(1+2.228514e^(-0.11518*t)) tlogistic exponentialactual~ @ 6O[@M*Al%@\='o|D A (ȅ%@ %  |E^ӫ?Lt//n&@A~ @ ? D  m0գp@  4 *Al%@\='o|LA ZH)@  ~ @ @   L ^bE@   ]?;@  ~  @ @   F\?@   )4\@  ~ a@ @   }Ja^@   ]@  ~ p@@ :u@  gj@  ~ @@ @@ 2]$@ ~ @@ 뀑@ ^i@ ~ @ @ Aj,A 1 @ ~ P@"@ wƮ A pɘ@ ~ `@$@ lfA a?@ ~ #@&@ 9A HYaA ~ A(@ yoґA R BkA ~ A*@ n'J]A `͒NPdA ~ A,@ w!A (bGkA ~ pA.@ n{bA F8A ~ GA0@  A A%( |E^ӫ?Lt//n&@A~ A1@ ]c>B7A4)*Al%@\='o|LACH:A2@*L([AYGo"A3@!jA`K{ A4@xAL A5@y)w Ahh A6@ )'§ AH9 AD lK40Jkkkkkkkkkkkkks]]] ;!;- ";L#; $;/N%; &;';(; );*;+; ,;,-;.;/;@.!0;L1; 2;)@3;L4;5;6; 7;ʈ8;,9;7t:;;;<;=;\>; ?;'  7@! [\ A }JA!8@"!YEo A !4s33A "9@#"Q>u A!"mUʭ@A!#:@$#'Et A"#i5A"$;@%$'= A#$;(A#%<@&%FOQ A$%\UA$&=@'&|C{H A%&]MA%'>@('/ A&'N~-W*A&(?@)(~'~ A'(5uA')@@*); A()ODA()%)8 |E^ӫ?Lt//n&@A*@@+*~uߎ A)*4*9*Al%@\='o|LA*^7DA))+A@,++:L+sёu8 A**+ǏaA*),A@-+,]J4[ A+*,r/fA+)-B@.+-&*rz A,*- OT A,).B@/+.V#R" A-*."pM A-)/C@0+/`ݲ A.*/9r|o !A.)0C@1+0q]I A/*0&oE"A/)1D@2+1{· A0*1U ~#A0)2D@3+2+Z A1*2'=`%A1)3E@4+3% A2*3+p)5<&A2)4E@5+4lr6dk A3*4vz'A3)5F@6+5KhA4*5T(A4)6F@7+6@B$A5*66,*A5)7G@8+7!ug.A6*7 Rm+A6)8G@9+8c7A7*8P--A7)9H@:+9`C=?A8*9h.A89%9? |E^ӫ?Lt//n&@A:H@;+:;3\GA9:4:?*Al%@\='o|LA:R ͙C0A99;I@<;;?L;}MA::;CD_H$,1A:9<I@=;<leSA;:<V!2A;9=J@>;=OJ YA<:=oXn$3A<9>J@?;>x]A=:>lܡ64A=9?K@?;?qGbA>:?*W5A>9Dl]]]]]]]]]s]]]]]]]]]]]]]s]]]A;B;- C;LD; E;/NF; G;H;I; J;K;L; M;,N;O;P;@.!Q;LR; S;)@T;LU;V;W; X;ʈY;,Z;7t[;\;];^;\_; IAASuppose instead, that the 1987 data was suspect and that you felt+B#more confident using the 1980 data.DStill, a=2.228514)E!But, in 1980--when t=9, y=108040./G'So, 108,040=250,000/(1+2.228514*e^(9b)),H$(1+2.228512*e^(9b))=250,000/108,040=)H0(M@?A`@+J#Thus, e^(9b)=(2.313957-1)/2.228512=.J]?H[:@u@LSo, 9b=ln(0.589611)"Lԓ7J SK?A Mb='M\a LSK?A 2O*So, the logistic equation in this case is:4P, y=250,000/(1+2.228514e^(-0.05869914*t)) RtR logistic 2R logistic 1R exponentialRactual~ S@S6O[@*Al%@4M DSA@S6O[@ *Al%@\='o|DSA1S(ȅ%@ |E^ӫ?DSt//n&@A~ S@T?UDSTd@ST4Tc*Al%@4M LATm0գp@ST4Tc*Al%@\='o|LATZH)@ST%Tc |E^ӫ?Lt//n&@A~ T@U@VUUdLUi<\x@TTU^bE@TTU]?;@TT~ U @V@WUVFF@UTVF\?@UTV)4\@UT~ Va@W@XUW @VTW }Ja^@VTW]@VT~ Wp@X@YUX2ܹP@WTX:u@WTXgj@WT~ X@Y@ZUY @XTY@@XTY2]$@XT~ Y@Z@[UZPl@YTZ뀑@YTZ^i@YT~ Z@[ @\U[ oc@ZT[Aj,AZT[1 @ZT~ [P@\"@]U\<, `@[T\wƮ A[T\pɘ@[T~ \`@]$@^U]8zIiB@\T]lfA\T]a?@\T~ ]#@^&@_U^ĭ%@]T^9A]T^HYaA]T~ ^A_(@`U_sd @^T_yoґA^T_R BkA^T~ _ABt XM/-3]aE968b&`;a;- b;Lc; d;/Ne; f;g;h; i;j;k; l;,m;n;o;@.!p;Lq; r;)@s;Lt;u;v; w;ʈx;,y;7tz;{;|;};\~; ;' `*@aU`\|@_T`n'J]A_T``͒NPdA_T~ `Aa,@bUa?x[@d@`Taw!A`Ta(bGkA`T~ apAb.@cUb%MΆ@aTbn{bAaTbF8AaT~ bGAc0@dUcOAbTc AbTcAbT~ cAd1@eUdTAcd4ds*Al%@4M LAd]c>B7Acd4ds*Al%@\='o|LAdCH:Acd%ds |E^ӫ?Lt//n&@Ae2@feetLetS2Adde([AddeYGo"Addf3@gefkmDqAedf!jAedf`K{ Aedg4@heg@AfdgxAfdgL Afdh5@iehRAgdhy)w Agdhhh Agdi6@jeiSAhdi)'§ AhdiH9 Ahdj7@kej,XWAidj[\ Aidj}JAidk8@lek=ȶ6{AjdkYEo Ajdk4s33Ajdl9@mel֧)->)AkdlQ>u AkdlmUʭ@Akdm:@nemuAldm'Et Aldmi5Aldn;@oenpJAmdn'= Amdn;(Amdo<@peokqTAndoFOQ Ando\UAndp=@qepKͳAodp|C{H Aodp]MAodq>@req Apdq/ ApdqN~-W*Apdr?@serqkAqdr~'~ Aqdr5uAqds@@tesArds; ArdsODArdt@@uetn @Ast4t*Al%@4M LAt~uߎ Ast4t*Al%@\='o|LAt^7DAst%t |E^ӫ?Lt//n&@AuA@vuuLu֥ZlAttusёu8 AttuǏaAttvA@wuvAutv]J4[ Autvr/fAutwB@xuw Avtw&*rz Avtw OT AvtxB@yuxV4VAwtxV#R" Awtx"pM AwtyC@zuy;YAxty`ݲ Axty9r|o !AxtzC@{uz//Aytzq]I Aytz&oE"Ayt{D@|u{fRj( Azt{{· Azt{U ~#Azt|D@}u|)i A{t|+Z A{t|'=`%A{t}E@~u}L. A|t}% A|t}+p)5<&A|t~E@u~5kǿ A}t~lr6dk A}t~vz'A}tF@uG4 A~tKhA~tT(A~tDl|||||||||||||||||||||||;;- ;L; ;/N; ;;; ;;; ;,;;;@.!;L; ;)@;L;;; ;ʈ;,;7t;;;;\; F@uSYJU At@B$At6,*AtG@ut At!ug.At Rm+AtG@uF͝ѽ Atc7AtP--AtH@u@s At`C=?Ath.AtH@u=; A4*Al%@4M LA;3\GA4*Al%@\='o|LAR ͙C0A% |E^ӫ?Lt//n&@AI@Lj `I A}MACD_H$,1AI@08s AleSAV!2AJ@GE AOJ YAoXn$3AJ@Z. Ax]Alܡ64AK@6H AqGbA*W5ABX||||||||;;- ;L; ;/N; ;;; ;;; ;,;;;@.!;L; ;)@;L;;; ;ʈ;,;7t;;;;\; ;' Dl;;- ;L; ;/N; ;;; ;;; ;,;;;@.!;L; ;)@;L;;; ;ʈ;,;7t;;;;\; ;' Dl;;- ;L; ;/N; ;;; ;;; ;,;;;@.!;L; ;)@;L;;; ;ʈ;,;7t;;;;\; ;' Dl;;- ;L; ;/N; ;;;  ; ; ;  ;, ;;;@.!;L; ;)@;L;;; ;ʈ;,;7t;:#;$;- %;L&; ';/N(; );*;+; ,;-;.; /;,0 ;1 ;2 ;@.!3 ;L4 ; 5 ;)@6 ;L7 ;8 ;9 ; : ;ʈ; ;,< ;7t= ;> ;? ;>D0@ ;A ;- B ;LC ; D ;/NE ; F ;x]*#z0@]*R/zs/z@],) .A@@],$12)A@@]*!#@]*""@], ()1 @]h @(s) @((ln a)/(-b), q/2)g]*  ..@]b  ,. @ (0, q/(1+a))M ], .n. @]^ !U"n @Y_t = q11]Z ?"O# @̵Y_t ]*zn@]*?s@],UA@@],}bwA@@]*| |@],w{b}bA@@],{z{A@@]>sU@A@' nS'  l &APage &P"s??333QQ;_#QE43QQ;QE4D FA?u 3O 3 43*4%=b 3Oa&Q Year, t=1 is 1971'4%1 3O&Q !inland coal consumption, India'4523  43"  ;^3O @%3OQ44$%3O&Q43      4444 ". ]> @A@n' S'  l &APage &P" ??3`3773QQ;_# ?QE4773QQ;5_# ?QE4773QQ; ?QE4D FAvz 3Ob 3 43*4523  43" | &3O| %3OQ44$%3O&Q43      4444 nn ]>S%b@A@' S'  l &APage &P"S??3H3773QQ;7_#SQE4773QQ;S_#SQE4773QQ;o_#SQE4773QQ;SQE4D FAi 3O]{ 3 43*4523  43"  3O %3OQ44$%3O&Q43      4444 n > ????m9@L@A1980 logistic 1987 logistic exponential actual 1971-87 qq     }HORNBAR1.PIC@Aqq}HORNBAR2.PICqq}HORNBAR3.PIC9@L@A1980 logistic 1987 logistic exponential actual 1971-87 qq}HORNBAR4.PIC9@L@A1980 logistic 1987 logistic exponential actual 1971-87 qq}HORNBAR5.PICqq"   l &APage &P"??q3 LOGISTICK!3AA3QQ;y_#Q;z_#3    4E4AA3QQ;+z_#Q;Gz_#3    4E4AA3QQ;cz_#Q;z_#3    4E4AA3QQ;z_#Q;3    4E4DFAK 3O4k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ȅ%@A@B @BxYgH@B ]e@BC]:GJ@Ct '@C ȥ@CD[{@D4 @Dj@DEd@Em0գp@EZH)@E@Fa(bC)@FS@Frbl@FGd@G:@Gpg@GH7l`E@H$Dq@HZ@HIi<\x@I^bE@I]?;@I @Jt@J"lq@J{@JKS>R@K݈[@KJXM@KLmlg@LkhC@L8lJ@LMFF@MF\?@M)4\@Ma@NwGWPz@NVXk@N-!R@NOP T@O->@Ok:&@OPf@Pda@P{@@PQ @Q }Ja^@Q]@Qp@R3?M@RɩP>:@R@RSм'@ST.BA:@Si8,@ST.@T&g@TC_ 4%~@TU2ܹP@U:u@Ugj@U@V]~1$@V_1׆@V#@VWECZ@Wc@WYx@WXAф@X|5f@XRv @XY @Y@@Y2]$@Y@ZS1@ZV!yF@Z@ڢ|@Z[3r[4@[s,Ӷ@[7@[\JXOk@\_>#C'@\eL0@\]Pl@]뀑@]^i@]@^'@^E4@^nb@^_@_Qx@_G@_`JŏH@`}@`#!(@`a oc@aAj,Aa1 @aP@bǟ:@bdAbL t)j@bc@ 3@ckhGtfAcm@cdIWJ(@d=Ad=2@de<, `@ewƮ Aepɘ@e`@f Ә@f} CAfSn@fgww"A@gu7zAgO~i@ghc @ha&AhwP@hi8zIiB@ilfAia?@i#@jc{@jWܥ9=Aj6 d@jk?d@kqmZTAk W)AklK@l|c(AlR1UbFAlmĭ%@m9AmHYaAmAn0AQ^@nI@AnT6c'AnoVڷЗ@oZN)AoNdAopdU)@p]ApYh3/Apqsd @qyoґAqR BkAqArU[qOC@r:\Ar!TBArsI|@soq)*vAs{?RAst cӵ@tJuM+AtY$Atu\|@un'J]Au`͒NPdAuAv_m(@vvDAv/"Avwsa@w|%3Aw$U\AwxF)@xRAx[{0(Axy?x[@d@yw!Ay(bGkAypAzMk;# @z!dPAz"HAz{%@>G@{iyA{[k7A{|yF@|cA|X}:A|}%MΆ@}n{bA}F8A}GA~@~t\l A~oA~;JAo6A_ۃA2z2AbA   l &APage &P"A??q3 LOGISTICK!3773Q  1980 logisticQ;z_#Q;z_#3    4E4773Q  1987 logisticQ; {_#Q;'{_#3    4E4773Q  exponentialQ;C{_#Q;_{_#3    4E43Q actual 1971-87Q;{{_#Q;3    4E4DFAK 3O4k9 3*9@43*L@A#4523   43"  3O%3OQ44444 n7????@@@@@@@@@@@@@@@@@@@@@@@@ @ @ @ @ "@ "@ "@ "@ $@ $@ $@ $@ &@ &@ &@ &@ (@ (@ (@ (@ *@ *@ *@ *@,@,@,@,@.@.@.@.@0@0@0@0@1@1@1@2@2@2@3@3@3@4@4@4@5@5@5@6@6@6@7@7@7@8@8@8@9@9@9@:@:@:@;@;@;@<@<@<@=@=@=@>@>@>@?@?@?@ @@ @@ @@!@@!@@!@@"A@"A@"A@#A@#A@#A@$B@$B@$B@%B@%B@%B@&C@&C@&C@'C@'C@'C@(D@(D@(D@)D@)D@)D@*E@*E@*E@+E@+E@+E@,F@,F@,F@-F@-F@-F@.G@.G@.G@/G@/G@/G@0H@0H@0H@1H@1H@1H@2I@2I@2I@3I@3I@3I@4J@4J@4J@5J@5J@5J@6K@6K@6K@76O[@76O[@7(ȅ%@7@8d@8m0գp@8ZH)@8@9i<\x@9^bE@9]?;@9 @:FF@:F\?@:)4\@:a@; @; }Ja^@;]@;p@<2ܹP@<:u@<gj@<@= @=@@=2]$@=@>Pl@>뀑@>^i@>@? oc@?Aj,A?1 @?P@@<, `@@wƮ A@pɘ@@`@A8zIiB@AlfAAa?@A#@Bĭ%@B9ABHYaABACsd @CyoґACR BkACAD\|@Dn'J]AD`͒NPdADAE?x[@d@Ew!AE(bGkAEpAF%MΆ@Fn{bAFF8AFGAGOAG AGAGAHTAH]c>B7AHCH:AItS2AI([AIYGo"AJkmDqAJ!jAJ`K{ AK@AKxAKL ALRALy)w ALhh AMSAM)'§ AMH9 AN,XWAN[\ AN}JAO=ȶ6{AOYEo AO4s33AP֧)->)APQ>u APmUʭ@AQuAQ'Et AQi5ARpJAR'= AR;(ASkqTASFOQ AS\UATKͳAT|C{H AT]MAU AU/ AUN~-W*AVqkAV~'~ AV5uAWAW; AWODAXn @AX~uߎ AX^7DAY֥ZlAYsёu8 AYǏaAZAZ]J4[ AZr/fA[ A[&*rz A[ OT A\V4VA\V#R" A\"pM A];YA]`ݲ A]9r|o !A^//A^q]I A^&oE"A_fRj( A_{· A_U ~#A`)i A`+Z A`'=`%AaL. Aa% Aa+p)5<&Ab5kǿ Ablr6dk Abvz'AcG4 AcKhAcT(AdSYJU Ad@B$Ad6,*Aet Ae!ug.Ae Rm+AfF͝ѽ Afc7AfP--Ag@s Ag`C=?Agh.Ah=; Ah;3\GAhR ͙C0Aij `I Ai}MAiCD_H$,1Aj08s AjleSAjV!2AkGE AkOJ YAkoXn$3AlZ. Alx]Allܡ64Am6H AmqGbAm*W5A> m*  l &APage &P"5A??q3 LOGISTICK!3773Q  1980 logisticQ;{_#Q;{_#3    4E4773Q  1987 logisticQ;{_#Q;|_#3    4E4773Q  exponentialQ;#|_#Q;?|_#3    4E43Q actual 1971-87Q;[|_#Q;3    4E4DFAK 3O4k9 3*9@43*L@A#4523   43"  3O%3OQ44444 nP????@@@@@@@@@@@@@@@@@@@@@@@@ @ @ @ @ "@ "@ "@ "@ $@ $@ $@ $@ &@ &@ &@ &@ (@ (@ (@ (@ *@ *@ *@ *@,@,@,@,@.@.@.@.@0@0@0@0@1@1@1@2@2@2@3@3@3@4@4@4@5@5@5@6@6@6@7@7@7@8@8@8@9@9@9@:@:@:@;@;@;@<@<@<@=@=@=@>@>@>@?@?@?@ @@ @@ @@!@@!@@!@@"A@"A@"A@#A@#A@#A@$B@$B@$B@%B@%B@%B@&C@&C@&C@'C@'C@'C@(D@(D@(D@)D@)D@)D@*E@*E@*E@+E@+E@+E@,F@,F@,F@-F@-F@-F@.G@.G@.G@/G@/G@/G@0H@0H@0H@1H@1H@1H@2I@2I@2I@3I@3I@3I@4J@4J@4J@5J@5J@5J@6K@6K@6K@76O[@76O[@7(ȅ%@7@8d@8m0գp@8ZH)@8@9i<\x@9^bE@9]?;@9 @:FF@:F\?@:)4\@:a@; @; }Ja^@;]@;p@<2ܹP@<:u@<gj@<@= @=@@=2]$@=@>Pl@>뀑@>^i@>@? oc@?Aj,A?1 @?P@@<, `@@wƮ A@pɘ@@`@A8zIiB@AlfAAa?@A#@Bĭ%@B9ABHYaABACsd @CyoґACR BkACAD\|@Dn'J]AD`͒NPdADAE?x[@d@Ew!AE(bGkAEpAF%MΆ@Fn{bAFF8AFGAGOAG AGAGAHTAH]c>B7AHCH:AItS2AI([AIYGo"AJkmDqAJ!jAJ`K{ AK@AKxAKL ALRALy)w ALhh AMSAM)'§ AMH9 AN,XWAN[\ AN}JAO=ȶ6{AOYEo AO4s33AP֧)->)APQ>u APmUʭ@AQuAQ'Et AQi5ARpJAR'= AR;(ASkqTASFOQ AS\UATKͳAT|C{H AT]MAU AU/ AUN~-W*AVqkAV~'~ AV5uAWAW; AWODAXn @AX~uߎ AX^7DAY֥ZlAYsёu8 AYǏaAZAZ]J4[ AZr/fA[ A[&*rz A[ OT A\V4VA\V#R" A\"pM A];YA]`ݲ A]9r|o !A^//A^q]I A^&oE"A_fRj( A_{· A_U ~#A`)i A`+Z A`'=`%AaL. Aa% Aa+p)5<&Ab5kǿ Ablr6dk Abvz'AcG4 AcKhAcT(AdSYJU Ad@B$Ad6,*Aet Ae!ug.Ae Rm+AfF͝ѽ Afc7AfP--Ag@s Ag`C=?Agh.Ah=; Ah;3\GAhR ͙C0Aij `I Ai}MAiCD_H$,1Aj08s AjleSAjV!2AkGE AkOJ YAkoXn$3AlZ. Alx]Allܡ64Am6H AmqGbAm*W5A> m*  l &APage &P"5A??q3 LOGISTICK!3773QQ;|_# ?Q;|_#Ry3    4E4773QQ;|_#VQ;Ry3    4E4DFAo 3O4k93*43*#4523   43" 444 ni??@@@@@@@@@@@@ @ @ "@ "@ $@ $@ &@ &@ (@ (@ *@ *@,@,@.@.@0@0@1@1@2@2@3@3@4@4@5@5@6@6@7@7@8@8@9@9@:@:@;@;@<@<@=@=@>@>@?@?@ @@ @@!@@!@@"A@"A@#A@#A@$B@$B@%B@%B@&C@&C@'C@'C@(D@(D@)D@)D@*E@*E@+E@+E@,F@,F@-F@-F@.G@.G@/G@/G@0H@0H@1H@1H@2I@2I@3I@3I@4J@4J@5J@5J@6K@6K@76O[@7(ȅ%@8m0գp@8ZH)@9^bE@9]?;@:F\?@:)4\@; }Ja^@;]@<:u@<gj@=@@=2]$@>뀑@>^i@?Aj,A?1 @@wƮ A@pɘ@AlfAAa?@B9ABHYaACyoґACR BkADn'J]AD`͒NPdAEw!AE(bGkAFn{bAFF8AG AGAH]c>B7AHCH:AI([AIYGo"AJ!jAJ`K{ AKxAKL ALy)w ALhh AM)'§ AMH9 AN[\ AN}JAOYEo AO4s33APQ>u APmUʭ@AQ'Et AQi5AR'= AR;(ASFOQ AS\UAT|C{H AT]MAU/ AUN~-W*AV~'~ AV5uAW; AWODAX~uߎ AX^7DAYsёu8 AYǏaAZ]J4[ AZr/fA[&*rz A[ OT A\V#R" A\"pM A]`ݲ A]9r|o !A^q]I A^&oE"A_{· A_U ~#A`+Z A`'=`%Aa% Aa+p)5<&Ablr6dk Abvz'AcKhAcT(Ad@B$Ad6,*Ae!ug.Ae Rm+Afc7AfP--Ag`C=?Agh.Ah;3\GAhR ͙C0Ai}MAiCD_H$,1AjleSAjV!2AkOJ YAkoXn$3Alx]Allܡ64AmqGbAm*W5A> m*  l &APage &P"5A??q3 LOGISTICK!33QQ;}_#Q;}_#3     4E4773QQ;;}_#Q;3    4E4DFAK 3O4k93*43*@A#4523   43" 444 nz|??@@@@@@@@@@@@ @ @ "@ "@ $@ $@ &@ &@ (@ (@ *@ *@,@,@.@.@0@0@ !"#$%&'()*+,-./01234567@7(ȅ%@8@8ZH)@9 @9]?;@:a@:)4\@;p@;]@<@<gj@=@=2]$@>@>^i@?P@?1 @@`@@pɘ@A#@Aa?@BABHYaACACR BkADAD`͒NPdAEpAE(bGkAFGAFF8AGAGAHCH:AIYGo"AJ`K{ AKL ALhh AMH9 AN}JAO4s33APmUʭ@AQi5AR;(AS\UAT]MAUN~-W*AV5uAWODAX^7DAYǏaAZr/fA[ OT A\"pM A]9r|o !A^&oE"A_U ~#A`'=`%Aa+p)5<&Abvz'AcT(Ad6,*Ae Rm+AfP--Agh.AhR ͙C0AiCD_H$,1AjV!2AkoXn$3Allܡ64Am*W5A> m* ࡱ; ~ #-'- -  -- !-- !-'sandyhMicrosoft Excelࡱ; ~-""- !"-**#- !#*-??#- !#?-TT#- !#T-ii#- !#i-~~#- !#~-#- !#-#- !#-#- !#-#- !#-- !-ࡱ; ~ Oh+'Oh+'0TH (0HG*^. C     ``iݜ*o?H6A^BbzbO/(fvEE>j(9jJ2 """)))000___UUUMMMBBB999 ```( +9%I;/]E:IS+!!lYQGj2ga1aS {Cg..&YQFh.IRj#SjJu3lJA7e ,N, QdoV YC6r3w_GqC-}zn#&s[ Rp$L 6n{DuqVJ4H2Aph6B"}3%6Z\H"BMXR ӕ$sVo<gX .YLg:}ߚ=VCq8SeFMjLZc"LeN?Pppiiiwwwfxю7Tvpxd?2}Dx#$_`,19م>w؁V!0ȳypQit|45'  ' - -""- !"-**#- !#*-??#- !#?-TT#- !#T-ii#- !#i-~~#- !#~-#- !#-#- !#-#- !#-#- !#-- !-- !-- !-- !-'- #-'- "Helv-  2 &" - -"- ! -!!- !!-"- !-  - ! --  2  "*1 2 , ,"?2 2 A A"T3 2 V V"i4 2 k k"~5 2 "6 2 "7 2 "8 2 "9 2 V$A 2 B 2 UC "System-'- #"Helv- 2 &Curve fitting--4    2 ,&NRE 545  2 A& S. Arlinghaus   *2 V&Logistic curve fitting.     2 k& 2 &R The exponential function leads to a situation of unbounded population growth.                   62 &Assumption for the exponential:        2 &P The rate of population growth or decay at any time t is proportional to the