(************** Content-type: application/mathematica ************** Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 382610, 10188]*) (*NotebookOutlinePosition[ 383265, 10211]*) (* CellTagsIndexPosition[ 383221, 10207]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Systems of differential equations", "Subtitle"], Cell["\<\ Ivan Cnop Vrije universiteit Brussel icnop@vub.ac.be\ \>", "Subsubtitle"], Cell[CellGroupData[{ Cell["Examples of systems of differential equations", "Subsection", CellAutoOverwrite->False], Cell["\<\ This notebook offers templates for solving linear systems of \ differential equations. The examples are classified by the entries of the coefficient matrix. Students should repeat the computations for many initial values and slightly \ different coefficient sets.\ \>", "Text", CellAutoOverwrite->False, FontWeight->"Bold"] }, Open ]], Cell[CellGroupData[{ Cell["Communicating containers with feedback", "Section", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell["Coefficient matrix", "Subsection"], Cell[BoxData[ \(a = \(-.4\); b = .1; c = .4; d = \(-.2\); \)], "Input", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"A", "=", RowBox[{"(", GridBox[{ {"a", "b"}, {"c", "d"} }], ")"}]}]], "Input", CellAutoOverwrite->False], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", StyleBox["0.4`", StyleBoxAutoDelete->True, PrintPrecision->1]}], ",", StyleBox["0.1`", StyleBoxAutoDelete->True, PrintPrecision->1]}], "}"}], ",", RowBox[{"{", RowBox[{ StyleBox["0.4`", StyleBoxAutoDelete->True, PrintPrecision->1], ",", RowBox[{"-", StyleBox["0.2`", StyleBoxAutoDelete->True, PrintPrecision->1]}]}], "}"}]}], "}"}]], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(A // MatrixForm\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ TagBox[ RowBox[{"(", GridBox[{ { RowBox[{"-", StyleBox["0.4`", StyleBoxAutoDelete->True, PrintPrecision->1]}], StyleBox["0.1`", StyleBoxAutoDelete->True, PrintPrecision->1]}, { StyleBox["0.4`", StyleBoxAutoDelete->True, PrintPrecision->1], RowBox[{"-", StyleBox["0.2`", StyleBoxAutoDelete->True, PrintPrecision->1]}]} }], ")"}], (MatrixForm[ #]&)]], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(Eigensystem[A]\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \({{\(-0.52360679774997898`\), \(-0.0763932022500210372`\)}, {{\(-0.628960169645094158`\), 0.777437524821135994`}, {\(-0.295241808844326314`\), \(-0.955422563220238529`\)}}}\)], "Output"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Solution", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(system = {\(x'\)[t] == First[A] . {x[t], y[t]}, \(y'\)[t] == Last[A] . {x[t], y[t]}}\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["x", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "==", RowBox[{ RowBox[{ RowBox[{"-", StyleBox["0.4`", StyleBoxAutoDelete->True, PrintPrecision->1]}], " ", \(x[t]\)}], "+", RowBox[{ StyleBox["0.1`", StyleBoxAutoDelete->True, PrintPrecision->1], " ", \(y[t]\)}]}]}], ",", RowBox[{ RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "==", RowBox[{ RowBox[{ StyleBox["0.4`", StyleBoxAutoDelete->True, PrintPrecision->1], " ", \(x[t]\)}], "-", RowBox[{ StyleBox["0.2`", StyleBoxAutoDelete->True, PrintPrecision->1], " ", \(y[t]\)}]}]}]}], "}"}]], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(initialValues = {x[0] == 10, y[0] == 0}\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \({x[0] == 10, y[0] == 0}\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(sol = DSolve[Flatten[{system, initialValues}], {x[t], y[t]}, t]\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \({{x[t] \[Rule] 0.25`15\ \(( 28.944271909999159220205911`15\ 2.71828182845904523536028747`15 \^\(\(-0.52360679774997898050514777`15\)\ t\) + 11.055728090000840779794089`15\ 2.71828182845904523536028747`15 \^\(\(-0.076393202250021038923755157`15\)\ t\))\), y[t] \[Rule] \(-8.944271909999159220205911`15\)\ 2.71828182845904523536028747`15 \^\(\(-0.52360679774997898050514777`15\)\ t\) + 8.944271909999159220205911`15\ 2.71828182845904523536028747`15 \^\(\(-0.076393202250021038923755157`15\)\ t\)}}\)], "Output"] }, Closed]], Cell["\<\ Except for the number e in the solution, we cannot guess what \ the solution looks like.\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Plane plot", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(ParametricPlot[{x[t] /. First[First[sol]], y[t] /. Last[First[sol]]}, { t, 0, 50}, PlotRange -> All, AxesLabel -> {vat1, vat2}]\)], "Input", CellAutoOverwrite->False], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0180002 0.0958191 0.0147142 0.107045 [ [.20964 .00221 -3 -9 ] [.20964 .00221 3 0 ] [.40128 .00221 -3 -9 ] [.40128 .00221 3 0 ] [.59291 .00221 -3 -9 ] [.59291 .00221 3 0 ] [.78455 .00221 -3 -9 ] [.78455 .00221 3 0 ] [.97619 .00221 -6 -9 ] [.97619 .00221 6 0 ] [1.025 .01471 0 -7 ] [1.025 .01471 28 7 ] [.0055 .12176 -6 -4.5 ] [.0055 .12176 0 4.5 ] [.0055 .2288 -6 -4.5 ] [.0055 .2288 0 4.5 ] [.0055 .33585 -6 -4.5 ] [.0055 .33585 0 4.5 ] [.0055 .44289 -6 -4.5 ] [.0055 .44289 0 4.5 ] [.0055 .54994 -6 -4.5 ] [.0055 .54994 0 4.5 ] [.018 .64303 -14 0 ] [.018 .64303 14 14 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .20964 .01471 m .20964 .02096 L s [(2)] .20964 .00221 0 1 Mshowa .40128 .01471 m .40128 .02096 L s [(4)] .40128 .00221 0 1 Mshowa .59291 .01471 m .59291 .02096 L s [(6)] .59291 .00221 0 1 Mshowa .78455 .01471 m .78455 .02096 L s [(8)] .78455 .00221 0 1 Mshowa .97619 .01471 m .97619 .02096 L s [(10)] .97619 .00221 0 1 Mshowa .125 Mabswid .06591 .01471 m .06591 .01846 L s .11382 .01471 m .11382 .01846 L s .16173 .01471 m .16173 .01846 L s .25755 .01471 m .25755 .01846 L s .30546 .01471 m .30546 .01846 L s .35337 .01471 m .35337 .01846 L s .44919 .01471 m .44919 .01846 L s .4971 .01471 m .4971 .01846 L s .54501 .01471 m .54501 .01846 L s .64082 .01471 m .64082 .01846 L s .68873 .01471 m .68873 .01846 L s .73664 .01471 m .73664 .01846 L s .83246 .01471 m .83246 .01846 L s .88037 .01471 m .88037 .01846 L s .92828 .01471 m .92828 .01846 L s .25 Mabswid 0 .01471 m 1 .01471 L s gsave 1.025 .01471 -61 -11 Mabsadd m 1 1 Mabs scale currentpoint translate 0 22 translate 1 -1 scale 63.000000 15.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (vat1) show 1.000000 setlinewidth %%DocumentNeededResources: font Courier %%DocumentSuppliedResources: %%DocumentNeededFonts: Courier %%DocumentSuppliedFonts: %%DocumentFonts: font Courier grestore .018 .12176 m .02425 .12176 L s [(1)] .0055 .12176 1 0 Mshowa .018 .2288 m .02425 .2288 L s [(2)] .0055 .2288 1 0 Mshowa .018 .33585 m .02425 .33585 L s [(3)] .0055 .33585 1 0 Mshowa .018 .44289 m .02425 .44289 L s [(4)] .0055 .44289 1 0 Mshowa .018 .54994 m .02425 .54994 L s [(5)] .0055 .54994 1 0 Mshowa .125 Mabswid .018 .03612 m .02175 .03612 L s .018 .05753 m .02175 .05753 L s .018 .07894 m .02175 .07894 L s .018 .10035 m .02175 .10035 L s .018 .14317 m .02175 .14317 L s .018 .16458 m .02175 .16458 L s .018 .18599 m .02175 .18599 L s .018 .2074 m .02175 .2074 L s .018 .25021 m .02175 .25021 L s .018 .27162 m .02175 .27162 L s .018 .29303 m .02175 .29303 L s .018 .31444 m .02175 .31444 L s .018 .35726 m .02175 .35726 L s .018 .37867 m .02175 .37867 L s .018 .40008 m .02175 .40008 L s .018 .42149 m .02175 .42149 L s .018 .4643 m .02175 .4643 L s .018 .48571 m .02175 .48571 L s .018 .50712 m .02175 .50712 L s .018 .52853 m .02175 .52853 L s .018 .57135 m .02175 .57135 L s .018 .59276 m .02175 .59276 L s .018 .61417 m .02175 .61417 L s .25 Mabswid .018 0 m .018 .61803 L s gsave .018 .64303 -75 -4 Mabsadd m 1 1 Mabs scale currentpoint translate 0 22 translate 1 -1 scale 63.000000 15.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (vat2) show 1.000000 setlinewidth %%DocumentNeededResources: font Courier %%DocumentSuppliedResources: %%DocumentNeededFonts: Courier %%DocumentSuppliedFonts: %%DocumentFonts: font Courier grestore .5 Mabswid .97619 .01472 m .67704 .33167 L .48454 .50369 L .41969 .54994 L .36517 .58073 L .33934 .59166 L .3187 .59816 L .30961 .60028 L .30503 .60115 L .30032 .60191 L .29633 .60243 L .29425 .60266 L .29206 .60286 L .28828 .60313 L .28484 .60327 L .28291 .60331 L .28111 .60332 L .27783 .60326 L .27438 .6031 L .27248 .60297 L .27075 .60281 L .26488 .60208 L .26202 .60159 L .25897 .60098 L .25279 .59942 L .24653 .59737 L .2361 .5928 L .22652 .58719 L .21827 .58113 L .2046 .56822 L .19279 .55373 L .18193 .53728 L .16568 .50638 L .14066 .4432 L .12084 .38154 L .10555 .32952 L .09205 .28188 L .08112 .24271 L .07215 .21042 L .06402 .18105 L .05734 .15694 L .05131 .13513 L .04637 .11729 L .04232 .10263 L .03864 .08932 L .03562 .07842 L .03314 .06944 L .03088 .06126 L .02902 .05455 L .02734 .04847 L Mistroke .02596 .0435 L .02483 .0394 L .02381 .03572 L Mfstroke 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{290.625, 179.5}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgGoo001KOol00`00Oomoo`11 Ool60010Ool00`00Oomoo`02Ool00`00Oomoo`0nOol00`00Oomoo`02Ool00`00Oomoo`0lOol00`00 Oomoo`03Ool00`00Oomoo`02Ool00`00Oomoo`0fOol005aoo`03001oogoo041oo`05001oogooOol0 0011Ool00`00Oomoo`02Ool00`00Oomoo`0nOol00`00Oomoo`02Ool00`00Oomoo`0lOol00`00Oomo o`03Ool00`00Oomoo`02Ool00`00Oomoo`0fOol005eoo`03001oogoo041oo`04001oogoo0011Ool2 0003Ool00`00Oomoo`0oOol4000oOol00`00Oomoo`03Ool00`00Oomoo`02Ool00`00Oomoo`0fOol0 05ioo`03001oogoo041oo`03001oo`00045oo`03001oo`000080045oo`03001oogoo009oo`03001o ogoo03aoo`03001oogoo00=oo`03001oogoo009oo`03001oogoo03Ioo`00GWoo00<007ooOol0@7oo 00<007oo0000@Woo00<007ooOol0@Woo00<007ooOol00Woo00<007ooOol0?7oo00<007ooOol00goo 00<007ooOol00Woo00<007ooOol0=Woo001IOol00`00Oomoo`02Ool00`00Oomoo`11Ool20012Ool2 0013Ool00`00Oomoo`02Ool00`00Oomoo`0jOol30006Ool01000Oomoo`00>Goo001JOol40015Ool0 0`00Oomoo`12Ool30011Ool4000nOol20007Ool2000jOol001Moo`03001oogoo0?moogAoo`03001o ogoo00Aoo`<00003Ool0000000Aoo`@000Aoo`D000Qoo`005goo00<007ooOol0ogooLgoo00<007oo 000017oo00@007ooOomoo`8000Aoo`03001oogoo009oo`03001oogoo00=oo`03001oogoo00Qoo`00 5goo00<007ooOol0ogooLgoo00<007oo000017oo00<007ooOol00Woo00<007ooOol00Woo00<007oo Ool027oo00<007ooOol027oo000GOol00`00Oomoo`3oOombOol01@00Oomoogoo000017oo1@0017oo 00<007ooOol027oo00<007ooOol027oo000GOol00`00Oomoo`3oOombOol01@00Oomoogoo000027oo 00<007ooOol00Woo00<007ooOol027oo00<007ooOol027oo000@Oooo001_000Ool0 01Moo`03001oogoo009oo`03001oogoo0?mood=oo`03001oogoo04moo`005goo00<007ooOol00Woo 00<007ooOol0ogoo@Woo00<007ooOol0D7oo000GOol00`00Oomoo`03Ool00`00Oomoo`3oOom0Ool0 0`00Oomoo`1AOol001Moo`8000Aoo`03001oogoo0?moocmoo`03001oogoo059oo`005goo00<007oo Ool00goo00<007ooOol0ogoo?Woo00<007ooOol0Dgoo000GOol00`00Oomoo`03Ool00`00Oomoo`3o OolmOol00`00Oomoo`1DOol001Moo`03001oogoo00Aoo`03001oogoo0?mooc]oo`03001oogoo05Eo o`005goo00<007ooOol017oo00<007ooOol0ogoo>Woo00<007ooOol0EWoo000GOol00`00Oomoo`04 Ool00`00Oomoo`3oOoliOol00`00Oomoo`1GOol001Moo`03001oogoo00Eoo`03001oogoo0?moocMo o`03001oogoo05Qoo`005goo00<007ooOol01Goo00<007ooOol0ogoo=Woo00<007ooOol0FGoo000G Ool20007Ool00`00Oomoo`3oOoldOol00`00Oomoo`1JOol001Moo`03001oogoo00Ioo`03001oogoo 0?mooc=oo`03001oogoo05]oo`005goo00<007ooOol01Woo00<007ooOol0ogooOol00`00Oomoo`3oOol@Ool00`00Oomoo`1fOol001Moo`8000moo`03 001oogoo0?moo`moo`03001oogoo07Moo`005goo00<007ooOol03goo00<007ooOol0ogoo3Goo00<0 07ooOol0N7oo000GOol00`00Oomoo`0?Ool00`00Oomoo`3oOolaoo`03001oogoo095oo`005goo00<007ooOol05Woo00<0 07ooOol0jgoo00<007ooOol0TWoo000GOol00`00Oomoo`0GOol00`00Oomoo`3YOol00`00Oomoo`2C Ool001Moo`8001Qoo`03001oogoo0>Qoo`03001oogoo09Aoo`005goo00<007ooOol05goo00<007oo Ool0igoo00<007ooOol0UGoo000GOol00`00Oomoo`0GOol00`00Oomoo`3VOol00`00Oomoo`2FOol0 01Moo`03001oogoo01Qoo`03001oogoo0>Aoo`03001oogoo09Moo`005goo00<007ooOol067oo00<0 07ooOol0hgoo00<007ooOol0V7oo000GOol00`00Oomoo`0HOol00`00Oomoo`3ROol00`00Oomoo`2I Ool001Moo`03001oogoo01Qoo`03001oogoo0>9oo`03001oogoo09Uoo`005goo00<007ooOol06Goo 00<007ooOol0h7oo00<007ooOol0VWoo000GOol2000JOol00`00Oomoo`3OOol00`00Oomoo`2KOol0 01Moo`03001oogoo01Uoo`03001oogoo0=ioo`03001oogoo09aoo`005goo00<007ooOol06Woo00<0 07ooOol0g7oo00<007ooOol0WGoo000GOol00`00Oomoo`0JOol00`00Oomoo`3KOol00`00Oomoo`2N Ool001Moo`03001oogoo01Yoo`03001oogoo0=Yoo`03001oogoo09moo`005goo00<007ooOol06goo 00<007ooOol0f7oo00<007ooOol0X7oo000GOol00`00Oomoo`0KOol00`00Oomoo`3GOol00`00Oomo o`2QOol001Moo`03001oogoo01]oo`03001oogoo0=Ioo`03001oogoo0:9oo`005goo0P0077oo00<0 07ooOol0eGoo00<007ooOol0Xgoo000GOol00`00Oomoo`0LOol00`00Oomoo`3COol00`00Oomoo`2T Ool001Moo`03001oogoo01aoo`03001oogoo0=9oo`03001oogoo0:Eoo`005goo00<007ooOol077oo 00<007ooOol0dGoo00<007ooOol0YWoo000GOol00`00Oomoo`0MOol00`00Oomoo`3?Ool00`00Oomo o`2WOol000]oo`@000Qoo`03001oogoo01eoo`03001oogoo01oo`005goo00<007ooOol0;Goo00<007ooOol0QGoo00<007ooOol0hGoo000GOol00`00Oomo o`0^Ool00`00Oomoo`23Ool00`00Oomoo`3ROol001Moo`03001oogoo02ioo`03001oogoo089oo`03 001oogoo0>=oo`005goo00<007ooOol0;goo00<007ooOol0P7oo00<007ooOol0i7oo000GOol00`00 Oomoo`0_Ool00`00Oomoo`1nOol2003WOol001Moo`03001oogoo02moo`03001oogoo07eoo`03001o ogoo0>Moo`005goo0P00Yoo`005goo00<007ooOol0eoo`005goo00<007ooOol07oo00<007oo Ool0G7oo00<007ooOol0ogoo000GOol00`00Oomoo`0iOol00`00Oomoo`1IOol2003oOol3Ool001Mo o`03001oogoo03Uoo`03001oogoo05Qoo`03001oogoo0?moo`=oo`002goo100027oo00<007ooOol0 >Woo00<007ooOol0EWoo00<007ooOol0ogoo17oo000:Ool00`00Oomoo`02Ool00`00Oomoo`05Ool0 0`00Oomoo`0kOol00`00Oomoo`1COol2003oOol7Ool000moo`03001oogoo00Eoo`03001oogoo03]o o`03001oogoo059oo`03001oogoo0?moo`Moo`003goo00<007ooOol01Goo0`00?7oo00<007ooOol0 Cgoo0P00ogoo2Woo000:Ool00`00Oomoo`02Ool00`00Oomoo`05Ool00`00Oomoo`0mOol00`00Oomo o`1Ool2000LOol4003oOolZOol001Mo o`03001oogoo051oo`8001Moo`<00?moobioo`005goo00<007ooOol0DWoo1@0037oo1P00ogooOol0 01Uoo`03001oogoo0?mooiQoo`00ogoo]7oo003oOondOol00?mookAoo`00ogoo]7oo003oOondOol0 0?mookAoo`00\ \>"], ImageRangeCache->{{{0, 434.938}, {268.25, 0}} -> {-0.666941, -0.462588, 0.0429169, 0.0384161}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[\((y[t] /. Last[First[sol]])\), {t, 0, 50}, PlotRange -> All]\)], "Input", CellAutoOverwrite->False], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.0190476 0.0147142 0.107045 [ [.21429 .00221 -6 -9 ] [.21429 .00221 6 0 ] [.40476 .00221 -6 -9 ] [.40476 .00221 6 0 ] [.59524 .00221 -6 -9 ] [.59524 .00221 6 0 ] [.78571 .00221 -6 -9 ] [.78571 .00221 6 0 ] [.97619 .00221 -6 -9 ] [.97619 .00221 6 0 ] [.01131 .12176 -6 -4.5 ] [.01131 .12176 0 4.5 ] [.01131 .2288 -6 -4.5 ] [.01131 .2288 0 4.5 ] [.01131 .33585 -6 -4.5 ] [.01131 .33585 0 4.5 ] [.01131 .44289 -6 -4.5 ] [.01131 .44289 0 4.5 ] [.01131 .54994 -6 -4.5 ] [.01131 .54994 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .21429 .01471 m .21429 .02096 L s [(10)] .21429 .00221 0 1 Mshowa .40476 .01471 m .40476 .02096 L s [(20)] .40476 .00221 0 1 Mshowa .59524 .01471 m .59524 .02096 L s [(30)] .59524 .00221 0 1 Mshowa .78571 .01471 m .78571 .02096 L s [(40)] .78571 .00221 0 1 Mshowa .97619 .01471 m .97619 .02096 L s [(50)] .97619 .00221 0 1 Mshowa .125 Mabswid .0619 .01471 m .0619 .01846 L s .1 .01471 m .1 .01846 L s .1381 .01471 m .1381 .01846 L s .17619 .01471 m .17619 .01846 L s .25238 .01471 m .25238 .01846 L s .29048 .01471 m .29048 .01846 L s .32857 .01471 m .32857 .01846 L s .36667 .01471 m .36667 .01846 L s .44286 .01471 m .44286 .01846 L s .48095 .01471 m .48095 .01846 L s .51905 .01471 m .51905 .01846 L s .55714 .01471 m .55714 .01846 L s .63333 .01471 m .63333 .01846 L s .67143 .01471 m .67143 .01846 L s .70952 .01471 m .70952 .01846 L s .74762 .01471 m .74762 .01846 L s .82381 .01471 m .82381 .01846 L s .8619 .01471 m .8619 .01846 L s .9 .01471 m .9 .01846 L s .9381 .01471 m .9381 .01846 L s .25 Mabswid 0 .01471 m 1 .01471 L s .02381 .12176 m .03006 .12176 L s [(1)] .01131 .12176 1 0 Mshowa .02381 .2288 m .03006 .2288 L s [(2)] .01131 .2288 1 0 Mshowa .02381 .33585 m .03006 .33585 L s [(3)] .01131 .33585 1 0 Mshowa .02381 .44289 m .03006 .44289 L s [(4)] .01131 .44289 1 0 Mshowa .02381 .54994 m .03006 .54994 L s [(5)] .01131 .54994 1 0 Mshowa .125 Mabswid .02381 .03612 m .02756 .03612 L s .02381 .05753 m .02756 .05753 L s .02381 .07894 m .02756 .07894 L s .02381 .10035 m .02756 .10035 L s .02381 .14317 m .02756 .14317 L s .02381 .16458 m .02756 .16458 L s .02381 .18599 m .02756 .18599 L s .02381 .2074 m .02756 .2074 L s .02381 .25021 m .02756 .25021 L s .02381 .27162 m .02756 .27162 L s .02381 .29303 m .02756 .29303 L s .02381 .31444 m .02756 .31444 L s .02381 .35726 m .02756 .35726 L s .02381 .37867 m .02756 .37867 L s .02381 .40008 m .02756 .40008 L s .02381 .42149 m .02756 .42149 L s .02381 .4643 m .02756 .4643 L s .02381 .48571 m .02756 .48571 L s .02381 .50712 m .02756 .50712 L s .02381 .52853 m .02756 .52853 L s .02381 .57135 m .02756 .57135 L s .02381 .59276 m .02756 .59276 L s .02381 .61417 m .02756 .61417 L s .25 Mabswid .02381 0 m .02381 .61803 L s .5 Mabswid .02381 .01472 m .03279 .19022 L .04262 .33167 L .04758 .38695 L .05288 .43622 L .06244 .50369 L .06788 .53196 L .07379 .55589 L .07936 .573 L .08451 .58478 L .08914 .59255 L .09178 .59591 L .09425 .59843 L .09685 .60046 L .09923 .60182 L .10054 .60237 L .10123 .60261 L .10197 .60283 L .10268 .60299 L .10332 .60312 L .10458 .60327 L .10578 .60332 L .1069 .60328 L .10813 .60315 L .10945 .6029 L .11084 .60254 L .11215 .60209 L .1146 .60102 L .11902 .59836 L .12383 .59454 L .13464 .58307 L .14465 .56987 L .18391 .50676 L .22166 .44356 L .26187 .38186 L .30055 .3298 L .34169 .28212 L .38132 .24291 L .41943 .21059 L .45999 .18119 L .49903 .15707 L .54053 .13524 L .58051 .11738 L .61898 .10271 L .6599 .08939 L .6993 .07847 L .73719 .06949 L .77753 .0613 L .81635 .05459 L .85763 .0485 L Mistroke .89739 .04352 L .93563 .03943 L .97619 .03572 L Mfstroke 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{297.312, 183.625}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgOol0069oo`03001oogoo00Aoo`04001oogoo0014Ool01@00Oomo ogoo000017oo00@007ooOol0049oo`03001oogoo009oo`03001oogoo009oo`04001oogoo0017Ool0 0`00Oomoo`03Ool01000Oomoo`00@goo00<007ooOol00Woo00<007ooOol00Woo00@007ooOol000eo o`00HWoo00<007ooOol00goo00<007ooOol00Woo00<007ooOol0@Woo00<007ooOol017oo00<007oo Ool00Woo00<007ooOol0A7oo00D007ooOomoo`0000Aoo`03001oogoo041oo`H000=oo`03001oogoo 009oo`03001oogoo04Eoo`05001oogooOol00004Ool00`00Oomoo`0:Ool0069oo`03001oogoo00=o o`03001oogoo009oo`03001oogoo04=oo`03001oogoo00=oo`03001oogoo009oo`03001oogoo04Ao o`05001oogooOol00004Ool00`00Oomoo`10Ool01@00Oomoogoo000017oo00<007ooOol00Woo00<0 07ooOol0AGoo00D007ooOomoo`0000Aoo`03001oogoo00Yoo`00HWoo00<007ooOol00goo00<007oo Ool00Woo00<007ooOol0A7oo00<007ooOol00Woo00<007ooOol00Woo00<007ooOol0@Goo0`0017oo 00<007ooOol00Woo00<007ooOol0@Goo00@007ooOol000Aoo`03001oogoo009oo`03001oogoo041o o`03001oogoo009oo`05001oogooOol00004Ool00`00Oomoo`0:Ool0069oo`03001oogoo00=oo`03 001oogoo009oo`03001oogoo04Eoo`05001oogooOol00004Ool00`00Oomoo`14Ool01@00Oomoogoo 000017oo00<007ooOol0@Woo00<007oo000017oo00<007ooOol00Woo00<007ooOol0@7oo1@0017oo 00<007ooOol00Woo00<007ooOol02Woo001ROol00`00Oomoo`03Ool00`00Oomoo`02Ool00`00Oomo o`15Ool01@00Oomoogoo000017oo00<007ooOol0A7oo00D007ooOomoo`0000Aoo`03001oogoo049o o`03001oo`0000Aoo`03001oogoo009oo`03001oogoo041oo`03001oogoo00Ioo`03001oogoo009o o`03001oogoo00Yoo`00H7oo0`001Woo00@007ooOol004=oo`03001oogoo009oo`03001oogoo009o o`04001oogoo0012Ool00`00Oomoo`02Ool00`00Oomoo`02Ool01000Oomoo`00AWoo0P001Goo00@0 07ooOol004=oo`03001oogoo00Moo`04001oogoo000=Ool0065oo`8000Moo`8004Eoo`@000Ioo`80 04Aoo`@000Ioo`8004Qoo`03001oogoo00Aoo`8004Aoo`D000Ioo`8000ioo`005Woo00<007ooOol0 ogooYGoo000FOol00`00Oomoo`3oOonUOol001Ioo`03001oogoo0?moojEoo`005Woo00<007ooOol0 ogooYGoo000FOol00`00Oomoo`3oOonUOol001Ioo`03001oogoo0?moojEoo`0037ooo`00ZP0027oo 000FOol00`00Oomoo`0=Ool00`00Oomoo`0>Ool00`00Oomoo`0=Ool00`00Oomoo`0=Ool00`00Oomo o`0=Ool00`00Oomoo`0=Ool00`00Oomoo`0=Ool00`00Oomoo`0=Ool00`00Oomoo`0>Ool00`00Oomo o`0=Ool00`00Oomoo`0=Ool00`00Oomoo`0=Ool00`00Oomoo`0=Ool00`00Oomoo`0=Ool00`00Oomo o`0=Ool00`00Oomoo`0>Ool00`00Oomoo`0=Ool00`00Oomoo`0=Ool00`00Oomoo`0=Ool00`00Oomo o`0=Ool00`00Oomoo`0=Ool00`00Oomoo`0=Ool00`00Oomoo`0>Ool00`00Oomoo`0=Ool00`00Oomo o`0=Ool00`00Oomoo`0@Ool001Ioo`03001oogoo04ioo`03001oogoo04ioo`03001oogoo04eoo`03 001oogoo04ioo`03001oogoo04ioo`03001oogoo011oo`005Woo00<007ooOol0CWoo00<007ooOol0 CWoo00<007ooOol0CGoo00<007ooOol0CWoo00<007ooOol0CWoo00<007ooOol047oo000FOol00`00 Oomoo`3oOonUOol001Ioo`03001oogoo0?moojEoo`005Woo00<007ooOol0ogooYGoo000FOol00`00 Oomoo`3oOonUOol001Ioo`03001oogoo0?moojEoo`005Woo0`00ogooRWoo2@004Woo000FOol2003o OomnOol=000KOol001Ioo`800?moogEoo`T002Qoo`005Woo0P00ogooKGoo2000Ool001Ioo`800?moodaoo`H005Aoo`005Woo0`00ogooAWoo1@00FWoo000FOol2003o Oom2Ool5001OOol001Ioo`800?moocaoo`H006Aoo`005Woo0P00ogoo=Woo1P00JWoo000FOol2003o OolaOol5001`Ool001Ioo`800?moobaoo`D007Eoo`005Woo0P00ogoo:7oo1000NWoo000FOol2003o OolTOol4001nOol001Ioo`800?moob1oo`@0089oo`005Woo0`00ogoo6goo1000QWoo000FOol2003o OolIOol3002:Ool001Ioo`03001oo`000?mooaAoo`@008eoo`005Woo00<007oo0000ogoo4Goo0`00 TGoo000FOol00`00Ool0003oOol>Ool3002DOol001Ioo`03001oo`000?moo`Yoo`@009Moo`005Woo 00<007oo0000ogoo1goo0`00Vgoo000FOol00`00Ool0003oOol3Ool4002NOol001Ioo`03001oo`00 0?ioo`@00:9oo`005Woo0`00ngoo0`00YWoo000FOol00`00Ool0003hOol3002YOol001Ioo`03001o o`000?Eoo`<00:aoo`005Woo00<007oo0000lWoo0`00[goo000FOol00`00Ool0003`Ool2002bOol0 01Ioo`03001oo`000>eoo`<00;Aoo`005Woo00<007oo0000jWoo0`00]goo000:Ool50007Ool00`00 Ool0003XOol2002jOol000aoo`03001oogoo00Moo`03001oo`000>Ioo`800;aoo`0037oo00<007oo Ool01goo1000hgoo0P00_Woo000Ool00`00Oomoo`05 Ool00`00Oomoo`02Ool00`00Oomoo`2COol00`00Oomoo`3oOol:Ool000Uoo`03001oogoo009oo`03 001oogoo00Eoo`03001oogoo009oo`03001oogoo099oo`03001oogoo0?moo`]oo`002Woo100027oo 00<007ooOol00Woo00<007ooOol0TGoo00<007ooOol0ogoo37oo000FOol00`00Oomoo`03Ool00`00 Oomoo`2>Ool2003oOol?Ool001Ioo`03001oogoo00=oo`03001oogoo08eoo`03001oogoo0?moo`mo o`005Woo00<007ooOol00goo00<007ooOol0S7oo00<007ooOol0ogoo47oo000FOol30003Ool00`00 Oomoo`2;Ool00`00Oomoo`3oOolAOol001Ioo`03001oogoo00=oo`03001oogoo08Yoo`03001oogoo 0?mooa9oo`005Woo00<007ooOol00goo00<007ooOol0RGoo00<007ooOol0ogoo4goo000FOol00`00 Oomoo`03Ool00`00Oomoo`28Ool00`00Oomoo`3oOolDOol001Ioo`03001oogoo00=oo`03001oogoo 08Moo`03001oogoo0?mooaEoo`005Woo00<007ooOol00goo00<007ooOol0QWoo00<007ooOol0ogoo 5Woo000FOol00`00Oomoo`03Ool00`00Oomoo`25Ool00`00Oomoo`3oOolGOol001Ioo`03001oogoo 00=oo`03001oogoo08Aoo`03001oogoo0?mooaQoo`005Woo00<007ooOol00goo00<007ooOol0Pgoo 00<007ooOol0ogoo6Goo000FOol30003Ool00`00Oomoo`22Ool00`00Oomoo`3oOolJOol001Ioo`03 001oogoo00=oo`03001oogoo085oo`03001oogoo0?mooa]oo`005Woo00<007ooOol00goo00<007oo Ool0P7oo00<007ooOol0ogoo77oo000FOol00`00Oomoo`04Ool00`00Oomoo`1nOol00`00Oomoo`3o OolMOol001Ioo`03001oogoo00Aoo`03001oogoo07eoo`03001oogoo0?mooaioo`005Woo00<007oo Ool017oo00<007ooOol0OGoo00<007ooOol0ogoo7Woo000FOol00`00Oomoo`04Ool00`00Oomoo`1l Ool00`00Oomoo`3oOolOOol001Ioo`03001oogoo00Aoo`03001oogoo07]oo`03001oogoo0?moob1o o`005Woo00<007ooOol017oo00<007ooOol0NWoo00<007ooOol0ogoo8Goo000FOol30004Ool00`00 Oomoo`1iOol00`00Oomoo`3oOolROol001Ioo`03001oogoo00Aoo`03001oogoo07Qoo`03001oogoo 0?moob=oo`005Woo00<007ooOol017oo00<007ooOol0Mgoo00<007ooOol0ogoo97oo000FOol00`00 Oomoo`04Ool00`00Oomoo`1gOol00`00Oomoo`3oOolTOol001Ioo`03001oogoo00Aoo`03001oogoo 07Ioo`03001oogoo0?moobEoo`005Woo00<007ooOol017oo00<007ooOol0MGoo00<007ooOol0ogoo 9Woo000FOol00`00Oomoo`04Ool00`00Oomoo`1dOol00`00Oomoo`3oOolWOol001Ioo`03001oogoo 00Aoo`03001oogoo07=oo`03001oogoo0?moobQoo`005Woo00<007ooOol017oo00<007ooOol0LWoo 00<007ooOol0ogoo:Goo000FOol30005Ool00`00Oomoo`1aOol00`00Oomoo`3oOolYOol001Ioo`03 001oogoo00Eoo`03001oogoo071oo`03001oogoo0?moobYoo`005Woo00<007ooOol01Goo00<007oo Ool0Kgoo00<007ooOol0ogoo:goo000FOol00`00Oomoo`05Ool00`00Oomoo`1^Ool00`00Oomoo`3o Ool/Ool001Ioo`03001oogoo00Eoo`03001oogoo06eoo`03001oogoo0?moobeoo`005Woo00<007oo Ool01Goo00<007ooOol0K7oo00<007ooOol0ogoo;Woo000:Ool40008Ool00`00Oomoo`05Ool00`00 Oomoo`1[Ool00`00Oomoo`3oOol_Ool000Uoo`03001oogoo009oo`03001oogoo00Eoo`03001oogoo 00Eoo`03001oogoo06]oo`03001oogoo0?moobmoo`003Woo00<007ooOol01Goo00<007ooOol01Goo 00<007ooOol0JWoo00<007ooOol0ogoo<7oo000>Ool00`00Oomoo`05Ool40004Ool00`00Oomoo`1Y Ool00`00Oomoo`3oOolaOol000]oo`<000Qoo`03001oogoo00Eoo`03001oogoo06Qoo`03001oogoo 0?mooc9oo`003Woo00<007ooOol01Goo00<007ooOol01Goo00<007ooOol0J7oo00<007ooOol0ogoo Ool00`00Oomoo`05Ool00`00Oomoo`05Ool00`00Oomoo`1WOol00`00Oomoo`3oOolcOol0 00Uoo`03001oogoo009oo`03001oogoo00Eoo`03001oogoo00Eoo`03001oogoo06Ioo`03001oogoo 0?moocAoo`002Woo100027oo00<007ooOol01Woo00<007ooOol0I7oo00<007ooOol0ogoo=Goo000F Ool00`00Oomoo`06Ool00`00Oomoo`1SOol00`00Oomoo`3oOolfOol001Ioo`03001oogoo00Ioo`03 001oogoo06=oo`03001oogoo0?moocIoo`005Woo00<007ooOol01Woo00<007ooOol0HWoo00<007oo Ool0ogoo=goo000FOol30006Ool00`00Oomoo`1QOol00`00Oomoo`3oOolhOol001Ioo`03001oogoo 00Ioo`03001oogoo061oo`03001oogoo0?moocUoo`005Woo00<007ooOol01Woo00<007ooOol0H7oo 00<007ooOol0ogoo>Goo000FOol00`00Oomoo`06Ool00`00Oomoo`1OOol00`00Oomoo`3oOoljOol0 01Ioo`03001oogoo00Ioo`03001oogoo05ioo`03001oogoo0?mooc]oo`005Woo00<007ooOol01Woo 00<007ooOol0GGoo00<007ooOol0ogoo?7oo000FOol00`00Oomoo`06Ool00`00Oomoo`1LOol00`00 Oomoo`3oOolmOol001Ioo`03001oogoo00Ioo`03001oogoo05aoo`03001oogoo0?mooceoo`005Woo 00<007ooOol01goo00<007ooOol0FWoo00<007ooOol0ogoo?Woo000FOol30007Ool00`00Oomoo`1I Ool00`00Oomoo`3oOoloOol001Ioo`03001oogoo00Moo`03001oogoo05Qoo`03001oogoo0?mood1o o`005Woo00<007ooOol01goo00<007ooOol0F7oo00<007ooOol0ogoo@7oo000FOol00`00Oomoo`07 Ool00`00Oomoo`1GOol00`00Oomoo`3oOom1Ool001Ioo`03001oogoo00Moo`03001oogoo05Ioo`03 001oogoo0?mood9oo`005Woo00<007ooOol01goo00<007ooOol0EWoo00<007ooOol0ogoo@Woo000F Ool00`00Oomoo`07Ool00`00Oomoo`1EOol00`00Oomoo`3oOom3Ool001Ioo`03001oogoo00Moo`03 001oogoo05Eoo`03001oogoo0?mood=oo`005Woo00<007ooOol027oo00<007ooOol0Dgoo00<007oo Ool0ogooA7oo000FOol30008Ool00`00Oomoo`1BOol00`00Oomoo`3oOom5Ool001Ioo`03001oogoo 00Qoo`03001oogoo059oo`03001oogoo0?moodEoo`005Woo00<007ooOol027oo00<007ooOol0DGoo 00<007ooOol0ogooAWoo000FOol00`00Oomoo`08Ool00`00Oomoo`1@Ool00`00Oomoo`3oOom7Ool0 01Ioo`03001oogoo00Qoo`03001oogoo051oo`03001oogoo0?moodMoo`005Woo00<007ooOol027oo 00<007ooOol0Cgoo00<007ooOol0ogooB7oo000FOol00`00Oomoo`09Ool00`00Oomoo`1=Ool00`00 Oomoo`3oOom9Ool001Ioo`03001oogoo00Uoo`03001oogoo04eoo`03001oogoo0?moodUoo`005Woo 00<007ooOol02Goo00<007ooOol0C7oo00<007ooOol0ogooBWoo000FOol30009Ool00`00Oomoo`1< Ool00`00Oomoo`3oOom:Ool001Ioo`03001oogoo00Uoo`03001oogoo04]oo`03001oogoo0?mood]o o`005Woo00<007ooOol02Goo00<007ooOol0BWoo00<007ooOol0ogooC7oo000FOol00`00Oomoo`09 Ool00`00Oomoo`1:Ool00`00Oomoo`3oOomOol000aoo`<000Moo`03001oogoo 00Yoo`03001oogoo04Ioo`03001oogoo0?moodmoo`003Goo00<007ooOol01Woo00<007ooOol02Woo 00<007ooOol0AWoo00<007ooOol0ogooCgoo0009Ool60007Ool40009Ool00`00Oomoo`15Ool00`00 Oomoo`3oOom@Ool000Uoo`05001oogooOol00008Ool00`00Oomoo`0:Ool00`00Oomoo`14Ool00`00 Oomoo`3oOomAOol000Yoo`04001oogoo0008Ool00`00Oomoo`0;Ool00`00Oomoo`13Ool00`00Oomo o`3oOomAOol000]oo`03001oo`0000Qoo`03001oogoo00]oo`03001oogoo049oo`03001oogoo0?mo oe9oo`002goo00<007oo000027oo00<007ooOol02goo00<007ooOol0@Goo00<007ooOol0ogooDgoo 000goo00<007ooOol0ogooF7oo000FOol00`00Oomo o`07oo00<007ooOol0ogooFWoo 000FOol00`00Oomoo`0=Ool00`00Oomoo`0gOol00`00Oomoo`3oOomKOol001Ioo`<000eoo`03001o ogoo03Moo`03001oogoo0?mooe]oo`005Woo00<007ooOol03Goo00<007ooOol0=Woo00<007ooOol0 ogooG7oo000FOol00`00Oomoo`0=Ool00`00Oomoo`0fOol00`00Oomoo`3oOomLOol001Ioo`03001o ogoo00eoo`03001oogoo03Eoo`03001oogoo0?mooeeoo`005Woo00<007ooOol03Goo00<007ooOol0 =7oo00<007ooOol0ogooGWoo000FOol00`00Oomoo`0>Ool00`00Oomoo`0cOol00`00Oomoo`3oOomN Ool001Ioo`03001oogoo00ioo`03001oogoo039oo`03001oogoo0?mooemoo`005Woo00<007ooOol0 3Woo00<007ooOol0Ool00`00Oomoo`0aOol0 0`00Oomoo`3oOomPOol001Ioo`03001oogoo00ioo`03001oogoo031oo`03001oogoo0?moof5oo`00 5Woo0`003Woo00<007ooOol0<7oo00<007ooOol0ogooHGoo000FOol00`00Oomoo`0>Ool00`00Oomo o`0_Ool00`00Oomoo`3oOomROol001Ioo`03001oogoo00moo`03001oogoo02ioo`03001oogoo0?mo of9oo`005Woo00<007ooOol03goo00<007ooOol0;Goo00<007ooOol0ogooHgoo000FOol00`00Oomo o`0?Ool00`00Oomoo`0/Ool00`00Oomoo`3oOomTOol001Ioo`03001oogoo00moo`03001oogoo02ao o`03001oogoo0?moofAoo`005Woo00<007ooOol03goo00<007ooOol0:goo00<007ooOol0ogooIGoo 000FOol00`00Oomoo`0?Ool00`00Oomoo`0[Ool00`00Oomoo`3oOomUOol001Ioo`03001oogoo011o o`03001oogoo02Uoo`03001oogoo0?moofIoo`005Woo0`0047oo00<007ooOol0:7oo00<007ooOol0 ogooIgoo000FOol00`00Oomoo`0@Ool00`00Oomoo`0XOol00`00Oomoo`3oOomWOol001Ioo`03001o ogoo011oo`03001oogoo02Moo`03001oogoo0?moofQoo`005Woo00<007ooOol047oo00<007ooOol0 9goo00<007ooOol0ogooJ7oo000FOol00`00Oomoo`0AOol00`00Oomoo`0UOol00`00Oomoo`3oOomY Ool001Ioo`03001oogoo015oo`03001oogoo02Aoo`03001oogoo0?moofYoo`002Woo100027oo00<0 07ooOol04Goo00<007ooOol097oo00<007ooOol0ogooJWoo0009Ool00`00Oomoo`02Ool00`00Oomo o`05Ool00`00Oomoo`0AOol00`00Oomoo`0SOol00`00Oomoo`3oOom[Ool000ioo`03001oogoo00Eo o`03001oogoo015oo`03001oogoo02=oo`03001oogoo0?moof]oo`003Woo00<007ooOol01Goo1000 4Goo00<007ooOol08Goo00<007ooOol0ogooK7oo0009Ool00`00Oomoo`02Ool00`00Oomoo`05Ool0 0`00Oomoo`0BOol00`00Oomoo`0POol00`00Oomoo`3oOom]Ool000Uoo`D000Qoo`03001oogoo019o o`03001oogoo021oo`03001oogoo0?moofeoo`002Goo00<007ooOol02Woo00<007ooOol04Woo00<0 07ooOol07goo00<007ooOol0ogooKWoo0009Ool00`00Oomoo`0:Ool00`00Oomoo`0COol00`00Oomo o`0NOol00`00Oomoo`3oOom^Ool000Uoo`D000Qoo`03001oogoo01=oo`03001oogoo01eoo`03001o ogoo0?moofmoo`005Woo00<007ooOol057oo00<007ooOol06goo00<007ooOol0ogooL7oo000FOol0 0`00Oomoo`0DOol00`00Oomoo`0KOol00`00Oomoo`3oOom`Ool001Ioo`03001oogoo01Aoo`03001o ogoo01Yoo`03001oogoo0?moog5oo`005Woo0`005Goo00<007ooOol06Goo00<007ooOol0ogooLGoo 000FOol00`00Oomoo`0EOol00`00Oomoo`0HOol00`00Oomoo`3oOombOol001Ioo`03001oogoo01Io o`03001oogoo01Ioo`03001oogoo0?moog=oo`005Woo00<007ooOol05Woo00<007ooOol05Goo00<0 07ooOol0ogooM7oo000FOol00`00Oomoo`0FOol00`00Oomoo`0DOol00`00Oomoo`3oOomeOol001Io o`03001oogoo01Moo`03001oogoo019oo`03001oogoo0?moogIoo`005Woo00<007ooOol05goo00<0 07ooOol04Woo00<007ooOol0ogooMWoo000FOol00`00Oomoo`0HOol00`00Oomoo`0@Ool00`00Oomo o`3oOomgOol001Ioo`03001oogoo01Qoo`03001oogoo00moo`03001oogoo0?moogQoo`005Woo0`00 6Goo00<007ooOol03Goo00<007ooOol0ogooNGoo000FOol00`00Oomoo`0JOol00`00Oomoo`0:Ool2 003oOomlOol001Ioo`03001oogoo01]oo`03001oogoo00Moo`800?moogioo`005Woo00<007ooOol0 77oo00<007ooOol01Goo00<007ooOol0ogooOWoo000FOol00`00Oomoo`0MOol7003oOon1Ool001Io o`03001oogoo0?moojEoo`005Woo00<007ooOol0ogooYGoo000FOol00`00Oomoo`3oOonUOol001Io o`03001oogoo0?moojEoo`005Woo0`00ogooYGoo000FOol00`00Oomoo`3oOonUOol00001\ \>"], ImageRangeCache->{{{0, 444.938}, {274.438, 0}} -> {-2.82517, -0.41448, 0.185854, 0.0330709}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Closed]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Competing species", "Section", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell["Coefficient matrix", "Subsection"], Cell[BoxData[ \(a = .1; b = \(-.1\); c = \(-.2\); d = .4; \)], "Input", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"A", "=", RowBox[{"(", GridBox[{ {"a", "b"}, {"c", "d"} }], ")"}]}]], "Input", CellAutoOverwrite->False], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ StyleBox["0.1`", StyleBoxAutoDelete->True, PrintPrecision->1], ",", RowBox[{"-", StyleBox["0.1`", StyleBoxAutoDelete->True, PrintPrecision->1]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", StyleBox["0.2`", StyleBoxAutoDelete->True, PrintPrecision->1]}], ",", StyleBox["0.4`", StyleBoxAutoDelete->True, PrintPrecision->1]}], "}"}]}], "}"}]], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(A // MatrixForm\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ TagBox[ RowBox[{"(", GridBox[{ { StyleBox["0.1`", StyleBoxAutoDelete->True, PrintPrecision->1], RowBox[{"-", StyleBox["0.1`", StyleBoxAutoDelete->True, PrintPrecision->1]}]}, { RowBox[{"-", StyleBox["0.2`", StyleBoxAutoDelete->True, PrintPrecision->1]}], StyleBox["0.4`", StyleBoxAutoDelete->True, PrintPrecision->1]} }], ")"}], (MatrixForm[ #]&)]], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(Eigensystem[A]\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \({{0.456155281280883073`, 0.0438447187191169351`}, {{0.270323012706148047`, \(-0.962769686270538649`\)}, {\(-0.871928209278037336`\), \(-0.489633738487448333`\)}}}\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(system = {\(x'\)[t] == First[A] . {x[t], y[t]}, \(y'\)[t] == Last[A] . {x[t], y[t]}}\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["x", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "==", RowBox[{ RowBox[{ StyleBox["0.1`", StyleBoxAutoDelete->True, PrintPrecision->1], " ", \(x[t]\)}], "-", RowBox[{ StyleBox["0.1`", StyleBoxAutoDelete->True, PrintPrecision->1], " ", \(y[t]\)}]}]}], ",", RowBox[{ RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "==", RowBox[{ RowBox[{ RowBox[{"-", StyleBox["0.2`", StyleBoxAutoDelete->True, PrintPrecision->1]}], " ", \(x[t]\)}], "+", RowBox[{ StyleBox["0.4`", StyleBoxAutoDelete->True, PrintPrecision->1], " ", \(y[t]\)}]}]}]}], "}"}]], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(initialValues = {x[0] == 150, y[0] == 200}\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \({x[0] == 150, y[0] == 200}\)], "Output"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Solution", "Subsection"], Cell[BoxData[ \(\(sol = DSolve[Flatten[{system, initialValues}], {x[t], y[t]}, t];\)\)], "Input", CellAutoOverwrite->False] }, Closed]], Cell[CellGroupData[{ Cell["Plane plot", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(ParametricPlot[{x[t] /. First[First[sol]], y[t] /. Last[First[sol]]}, { t, 0, 5}, PlotRange -> All, AspectRatio -> Automatic]\)], "Input", CellAutoOverwrite->False], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 4.44812 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.272437 0.00469169 -0.832431 0.00469169 [ [.03785 .09341 -9 -9 ] [.03785 .09341 9 0 ] [.50702 .09341 -6 -9 ] [.50702 .09341 6 0 ] [.74161 .09341 -9 -9 ] [.74161 .09341 9 0 ] [.97619 .09341 -9 -9 ] [.97619 .09341 9 0 ] [.25994 1.04425 -18 -4.5 ] [.25994 1.04425 0 4.5 ] [.25994 1.98258 -18 -4.5 ] [.25994 1.98258 0 4.5 ] [.25994 2.92092 -18 -4.5 ] [.25994 2.92092 0 4.5 ] [.25994 3.85926 -24 -4.5 ] [.25994 3.85926 0 4.5 ] [ 0 0 0 0 ] [ 1 4.44812 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .03785 .10591 m .03785 .11216 L s [(-50)] .03785 .09341 0 1 Mshowa .50702 .10591 m .50702 .11216 L s [(50)] .50702 .09341 0 1 Mshowa .74161 .10591 m .74161 .11216 L s [(100)] .74161 .09341 0 1 Mshowa .97619 .10591 m .97619 .11216 L s [(150)] .97619 .09341 0 1 Mshowa .125 Mabswid .08477 .10591 m .08477 .10966 L s .13169 .10591 m .13169 .10966 L s .1786 .10591 m .1786 .10966 L s .22552 .10591 m .22552 .10966 L s .31935 .10591 m .31935 .10966 L s .36627 .10591 m .36627 .10966 L s .41319 .10591 m .41319 .10966 L s .4601 .10591 m .4601 .10966 L s .55394 .10591 m .55394 .10966 L s .60086 .10591 m .60086 .10966 L s .64777 .10591 m .64777 .10966 L s .69469 .10591 m .69469 .10966 L s .78852 .10591 m .78852 .10966 L s .83544 .10591 m .83544 .10966 L s .88236 .10591 m .88236 .10966 L s .92927 .10591 m .92927 .10966 L s .25 Mabswid 0 .10591 m 1 .10591 L s .27244 1.04425 m .27869 1.04425 L s [(400)] .25994 1.04425 1 0 Mshowa .27244 1.98258 m .27869 1.98258 L s [(600)] .25994 1.98258 1 0 Mshowa .27244 2.92092 m .27869 2.92092 L s [(800)] .25994 2.92092 1 0 Mshowa .27244 3.85926 m .27869 3.85926 L s [(1000)] .25994 3.85926 1 0 Mshowa .125 Mabswid .27244 .34049 m .27619 .34049 L s .27244 .57508 m .27619 .57508 L s .27244 .80966 m .27619 .80966 L s .27244 1.27883 m .27619 1.27883 L s .27244 1.51341 m .27619 1.51341 L s .27244 1.748 m .27619 1.748 L s .27244 2.21717 m .27619 2.21717 L s .27244 2.45175 m .27619 2.45175 L s .27244 2.68634 m .27619 2.68634 L s .27244 3.15551 m .27619 3.15551 L s .27244 3.39009 m .27619 3.39009 L s .27244 3.62468 m .27619 3.62468 L s .27244 4.09384 m .27619 4.09384 L s .27244 4.32843 m .27619 4.32843 L s .25 Mabswid .27244 0 m .27244 4.44812 L s .5 Mabswid .97619 .10591 m .97088 .15558 L .96376 .21483 L .95566 .2758 L .94645 .33976 L .93495 .41417 L .92208 .49254 L .90624 .58383 L .88866 .68036 L .86938 .78196 L .84605 .90038 L .82057 1.02548 L .793 1.15701 L .75999 1.31038 L .72432 1.4722 L .68176 1.6611 L .63579 1.86112 L .5865 2.07188 L .52803 2.31793 L .4653 2.5781 L .39099 2.88214 L .31119 3.2047 L .22602 3.54523 L .12551 3.94316 L .02381 4.34221 L s 0 0 m 1 0 L 1 4.44812 L 0 4.44812 L closepath clip newpath % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{114, 507.062}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgOol0 035oo`03001oogoo06Ioo`03001oogoo00ioo`00Ool0035oo`03001oogoo06Ioo`03001oogoo00ioo`00Ool0035oo`03 001oogoo06Ioo`03001oogoo00ioo`00Ool00`00Oomoo`0VOol0035oo`03001oogoo04ioo`03001oogoo02Ioo`007oo000aOol00`00Oomoo`0lOol00`00Oomoo`0hOol0035oo`03 001oogoo03aoo`03001oogoo03Qoo`00goo00<007ooOol0>Goo000aOol00`00 Oomoo`0kOol00`00Oomoo`0iOol0035oo`03001oogoo03]oo`03001oogoo03Uoo`00goo00<007ooOol0>Goo000aOol00`00Oomoo`0jOol00`00Oomoo`0jOol0035oo`03001oogoo 03Yoo`03001oogoo03Yoo`00Woo00<007ooOol0>Woo000aOol00`00Oomoo`0j Ool00`00Oomoo`0jOol0035oo`03001oogoo03Uoo`03001oogoo03]oo`00Goo 00<007ooOol0>goo000aOol00`00Oomoo`0iOol00`00Oomoo`0kOol0035oo`03001oogoo03Uoo`03 001oogoo03]oo`00Goo00<007ooOol0>goo000aOol00`00Oomoo`0hOol00`00 Oomoo`0lOol0035oo`03001oogoo03Qoo`03001oogoo03aoo`007oo00<007oo Ool0?7oo000aOol00`00Oomoo`0hOol00`00Oomoo`0lOol0035oo`03001oogoo03Moo`03001oogoo 03eoo`00Ool0035oo`03001oogoo02Ioo`03001o ogoo04ioo`00Ool00`00Oomoo`1VOol001Eoo`03001oogoo009oo`03001oogoo009oo`04001oogoo0005 Ool01000Oomoo`001Goo00<007ooOol03Woo00<007ooOol0IWoo000EOol00`00Oomoo`02Ool01@00 Oomoogoo000017oo00D007ooOomoo`0000Aoo`03001oogoo009oo`03001oogoo00ioo`03001oogoo 06Ioo`005Goo00<007ooOol00Woo00D007ooOomoo`0000Aoo`05001oogooOol00004Ool00`00Oomo o`02Ool2000?Ool00`00Oomoo`1VOol001Ioo`@000Aoo`03001oogoo009oo`05001oogooOol00004 Ool00`00Oomoo`02Ool00`00Oomoo`0=Ool00`00Oomoo`1WOol001Eoo`03001oogoo009oo`05001o ogooOol00004Ool01@00Oomoogoo000017oo00<007ooOol00Woo00<007ooOol03Goo00<007ooOol0 Igoo000EOol00`00Oomoo`02Ool01@00Oomoogoo000017oo00D007ooOomoo`0000Aoo`03001oogoo 009oo`03001oogoo00eoo`03001oogoo06Moo`005Goo00<007ooOol00Woo00<007ooOol00Woo00@0 07ooOol000Eoo`04001oogoo0005Ool00`00Oomoo`0=Ool00`00Oomoo`1WOol001Ioo`@000Ioo`80 00Moo`8000Ioo`03001oogoo00aoo`03001oogoo06Qoo`00Ool00`00Oomoo`1gOol0021oo`03001oogoo00ioo`03001oogoo 07Moo`007goo00<007ooOol03goo00<007ooOol0Mgoo000OOol00`00Oomoo`0?Ool00`00Oomoo`1g Ool001moo`03001oogoo00moo`03001oogoo07Moo`007goo00<007ooOol03goo00<007ooOol0Mgoo 000=Ool50005Ool20005Ool01000Ool000001goo0P001Woo00<007ooOol0Mgoo000?Ool00`00Oomo o`04Ool01000Oomoo`0017oo0P000Woo00<007ooOol00goo00@007ooOol000Eoo`03001oogoo07Mo o`003goo00<007ooOol00goo00<007ooOol00Woo00D007ooOomoo`0000Aoo`05001oogooOol00004 Ool00`00Oomoo`02Ool00`00Oomoo`1gOol000moo`03001oogoo00=oo`03001oogoo009oo`05001o ogooOol00004Ool01@00Oomoogoo000017oo00<007ooOol00Woo0P00N7oo000?Ool00`00Oomoo`03 Ool00`00Oomoo`02Ool00`00Oomoo`020004Ool01@00Oomoogoo000017oo00<007ooOol00Woo00<0 07ooOol0Mgoo000?Ool00`00Oomoo`03Ool00`00Oomoo`02Ool00`00Oomoo`020004Ool01@00Oomo ogoo000017oo00<007ooOol00Woo00<007ooOol0Mgoo000?Ool00`00Oomoo`03Ool00`00Oomoo`02 Ool00`00Oomoo`020004Ool01@00Oomoogoo000017oo00<007ooOol00Woo00<007ooOol0Mgoo000= Ool30006Ool01000Oomoo`000Woo00@007ooOol0009oo`03001oogoo00=oo`04001oogoo0005Ool0 0`00Oomoo`1gOol000ioo`8000Moo`8000=oo`04001oogooOol20007Ool20006Ool00`00Oomoo`1g Ool001aoo`03001oogoo019oo`03001oogoo07Moo`0077oo00<007ooOol04Woo00<007ooOol0Mgoo 000KOol00`00Oomoo`0COol00`00Oomoo`1gOol001]oo`03001oogoo01=oo`03001oogoo07Moo`00 6goo00<007ooOol04goo00<007ooOol0Mgoo000KOol00`00Oomoo`0COol00`00Oomoo`1gOol001Yo o`03001oogoo01Aoo`03001oogoo07Moo`006Woo00<007ooOol057oo00<007ooOol0Mgoo000JOol0 0`00Oomoo`0DOol00`00Oomoo`1gOol001Yoo`03001oogoo01Aoo`03001oogoo07Moo`006Goo00<0 07ooOol05Goo00<007ooOol0Mgoo000IOol00`00Oomoo`0EOol00`00Oomoo`1gOol001Uoo`03001o ogoo01Eoo`03001oogoo07Moo`006Goo00<007ooOol05Goo00<007ooOol0Mgoo000HOol00`00Oomo o`0FOol00`00Oomoo`1gOol001Qoo`03001oogoo01Ioo`03001oogoo07Moo`0067oo00<007ooOol0 5Woo00<007ooOol0Mgoo000HOol00`00Oomoo`0FOol00`00Oomoo`1gOol001Moo`03001oogoo01Mo o`03001oogoo07Moo`005goo00<007ooOol05goo00<007ooOol0Mgoo000GOol00`00Oomoo`0GOol0 0`00Oomoo`1gOol001Moo`03001oogoo01Moo`03001oogoo07Moo`005Woo00<007ooOol067oo00<0 07ooOol0Mgoo000FOol00`00Oomoo`0HOol00`00Oomoo`1gOol001Ioo`03001oogoo01Qoo`03001o ogoo07Moo`005Woo00<007ooOol067oo00<007ooOol0Mgoo000EOol00`00Oomoo`0IOol00`00Oomo o`1gOol001Eoo`03001oogoo01Uoo`03001oogoo07Moo`005Goo00<007ooOol06Goo00<007ooOol0 Mgoo000EOol00`00Oomoo`0IOol00`00Oomoo`1gOol001Aoo`03001oogoo01Yoo`03001oogoo07Mo o`0057oo00<007ooOol06Woo00<007ooOol0Mgoo000DOol00`00Oomoo`0JOol00`00Oomoo`1gOol0 01Aoo`03001oogoo01Yoo`03001oogoo07Moo`004goo00<007ooOol06goo00<007ooOol0Mgoo000C Ool00`00Oomoo`0KOol00`00Oomoo`1gOol001=oo`03001oogoo01]oo`03001oogoo07Moo`004goo 00<007ooOol06goo00<007ooOol0Mgoo000BOol00`00Oomoo`0LOol00`00Oomoo`1gOol0019oo`03 001oogoo01aoo`03001oogoo07Moo`004Woo00<007ooOol077oo00<007ooOol0Mgoo000BOol00`00 Oomoo`0LOol00`00Oomoo`1gOol0019oo`03001oogoo01aoo`03001oogoo07Moo`004Goo00<007oo Ool07Goo00<007ooOol0Mgoo000AOol00`00Oomoo`0MOol00`00Oomoo`1gOol0015oo`03001oogoo 01eoo`03001oogoo07Moo`004Goo00<007ooOol07Goo00<007ooOol0Mgoo000@Ool00`00Oomoo`0N Ool00`00Oomoo`1gOol0011oo`03001oogoo01ioo`03001oogoo07Moo`0047oo00<007ooOol07Woo 00<007ooOol0Mgoo000@Ool00`00Oomoo`0NOol00`00Oomoo`1gOol000moo`03001oogoo01moo`03 001oogoo07Moo`003goo00<007ooOol07goo00<007ooOol0Mgoo000?Ool00`00Oomoo`0OOol00`00 Oomoo`1gOol000moo`03001oogoo01moo`03001oogoo07Moo`003Woo00<007ooOol087oo00<007oo Ool0Mgoo000>Ool00`00Oomoo`0POol00`00Oomoo`1gOol000ioo`03001oogoo021oo`03001oogoo 07Moo`003Woo00<007ooOol087oo00<007ooOol0Mgoo000=Ool00`00Oomoo`0QOol00`00Oomoo`1g Ool000eoo`03001oogoo025oo`03001oogoo07Moo`003Goo00<007ooOol08Goo00<007ooOol0Mgoo 000=Ool00`00Oomoo`0QOol00`00Oomoo`1gOol000aoo`03001oogoo029oo`03001oogoo07Moo`00 37oo00<007ooOol08Woo00<007ooOol0Mgoo000"], ImageRangeCache->{{{0, 170}, {759.562, 0}} -> {-68.8829, 120.3, 2.09793, 2.09793}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(ParametricPlot[{x[t] /. First[First[sol]], y[t] /. Last[First[sol]]}, { t, 0, 20}, PlotRange -> All]\)], "Input", CellAutoOverwrite->False], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.975635 3.70515e-06 0.0145866 6.42229e-07 [ [.04935 .00209 -21 -9 ] [.04935 .00209 21 0 ] [.23461 .00209 -21 -9 ] [.23461 .00209 21 0 ] [.41986 .00209 -21 -9 ] [.41986 .00209 21 0 ] [.60512 .00209 -21 -9 ] [.60512 .00209 21 0 ] [.79038 .00209 -18 -9 ] [.79038 .00209 18 0 ] [.96313 .14303 -36 -4.5 ] [.96313 .14303 0 4.5 ] [.96313 .27148 -36 -4.5 ] [.96313 .27148 0 4.5 ] [.96313 .39992 -36 -4.5 ] [.96313 .39992 0 4.5 ] [.96313 .52837 -36 -4.5 ] [.96313 .52837 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .04935 .01459 m .04935 .02084 L s [(-250000)] .04935 .00209 0 1 Mshowa .23461 .01459 m .23461 .02084 L s [(-200000)] .23461 .00209 0 1 Mshowa .41986 .01459 m .41986 .02084 L s [(-150000)] .41986 .00209 0 1 Mshowa .60512 .01459 m .60512 .02084 L s [(-100000)] .60512 .00209 0 1 Mshowa .79038 .01459 m .79038 .02084 L s [(-50000)] .79038 .00209 0 1 Mshowa .125 Mabswid .0864 .01459 m .0864 .01834 L s .12345 .01459 m .12345 .01834 L s .1605 .01459 m .1605 .01834 L s .19755 .01459 m .19755 .01834 L s .27166 .01459 m .27166 .01834 L s .30871 .01459 m .30871 .01834 L s .34576 .01459 m .34576 .01834 L s .38281 .01459 m .38281 .01834 L s .45691 .01459 m .45691 .01834 L s .49397 .01459 m .49397 .01834 L s .53102 .01459 m .53102 .01834 L s .56807 .01459 m .56807 .01834 L s .64217 .01459 m .64217 .01834 L s .67922 .01459 m .67922 .01834 L s .71627 .01459 m .71627 .01834 L s .75333 .01459 m .75333 .01834 L s .82743 .01459 m .82743 .01834 L s .86448 .01459 m .86448 .01834 L s .90153 .01459 m .90153 .01834 L s .93858 .01459 m .93858 .01834 L s .0123 .01459 m .0123 .01834 L s .25 Mabswid 0 .01459 m 1 .01459 L s .97563 .14303 m .98188 .14303 L s [(200000)] .96313 .14303 1 0 Mshowa .97563 .27148 m .98188 .27148 L s [(400000)] .96313 .27148 1 0 Mshowa .97563 .39992 m .98188 .39992 L s [(600000)] .96313 .39992 1 0 Mshowa .97563 .52837 m .98188 .52837 L s [(800000)] .96313 .52837 1 0 Mshowa .125 Mabswid .97563 .0467 m .97938 .0467 L s .97563 .07881 m .97938 .07881 L s .97563 .11092 m .97938 .11092 L s .97563 .17514 m .97938 .17514 L s .97563 .20726 m .97938 .20726 L s .97563 .23937 m .97938 .23937 L s .97563 .30359 m .97938 .30359 L s .97563 .3357 m .97938 .3357 L s .97563 .36781 m .97938 .36781 L s .97563 .43204 m .97938 .43204 L s .97563 .46415 m .97938 .46415 L s .97563 .49626 m .97938 .49626 L s .97563 .56048 m .97938 .56048 L s .97563 .59259 m .97938 .59259 L s .25 Mabswid .97563 0 m .97563 .61803 L s .5 Mabswid .97619 .01472 m .97617 .01475 L .97612 .0148 L .97604 .01486 L .97592 .01495 L .97573 .0151 L .97544 .01529 L .97498 .0156 L .9743 .01604 L .97335 .01665 L .97185 .0176 L .96972 .01894 L .96674 .02081 L .96211 .02369 L .95559 .02775 L .94539 .03407 L .9309 .04305 L .91066 .05557 L .8792 .07503 L .83481 .10247 L .7653 .14541 L .66631 .20656 L .52755 .29226 L .31132 .42579 L .02381 .60332 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{290.25, 179.375}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgOol60004Ool40006Ool20007Ool20007Ool20007Ool2000LOol6 0005Ool20007Ool20007Ool20007Ool20007Ool2000MOol50004Ool40006Ool20007Ool20007Ool2 0007Ool2000LOol50005Ool20007Ool20007Ool20007Ool20007Ool2000QOol40006Ool20007Ool2 0007Ool20007Ool20016Ool000moo`05001oogooOol00003Ool00`00Oomoo`02Ool00`00Oomoo`02 Ool01000Oomoo`001Goo00@007ooOol000Eoo`04001oogoo0005Ool01000Oomoo`0077oo00D007oo Oomoo`0000Aoo`04001oogoo0005Ool01000Oomoo`001Goo00@007ooOol000Eoo`04001oogoo0005 Ool01000Oomoo`007Woo00<007ooOol00goo00<007ooOol00Woo00<007ooOol00Woo00@007ooOol0 00Eoo`04001oogoo0005Ool01000Oomoo`001Goo00@007ooOol001eoo`03001oogoo00Aoo`04001o ogoo0005Ool01000Oomoo`001Goo00@007ooOol000Eoo`04001oogoo0005Ool01000Oomoo`007goo 00<007ooOol00Woo00<007ooOol00Woo00@007ooOol000Eoo`04001oogoo0005Ool01000Oomoo`00 1Goo00@007ooOol004Eoo`0047oo00<007ooOol02Goo00D007ooOomoo`0000Aoo`05001oogooOol0 0004Ool01@00Oomoogoo000017oo00D007ooOomoo`0000Aoo`03001oogoo01Yoo`03001oogoo00Ao o`03001oogoo009oo`05001oogooOol00004Ool01@00Oomoogoo000017oo00D007ooOomoo`0000Ao o`05001oogooOol00004Ool00`00Oomoo`0KOol00`00Oomoo`08Ool01@00Oomoogoo000017oo00D0 07ooOomoo`0000Aoo`05001oogooOol00004Ool01@00Oomoogoo000017oo00<007ooOol06Woo00<0 07ooOol00goo00<007ooOol00Woo00D007ooOomoo`0000Aoo`05001oogooOol00004Ool01@00Oomo ogoo000017oo00D007ooOomoo`0000Aoo`03001oogoo025oo`05001oogooOol00004Ool01@00Oomo ogoo000017oo00D007ooOomoo`0000Aoo`05001oogooOol00004Ool00`00Oomoo`12Ool000Eoo`L0 00Eoo`03001oogoo00Qoo`05001oogooOol00004Ool01@00Oomoogoo000017oo00D007ooOomoo`00 00Aoo`05001oogooOol00004Ool00`00Oomoo`0?Ool70005Ool00`00Oomoo`03Ool00`00Oomoo`02 Ool01@00Oomoogoo000017oo00D007ooOomoo`0000Aoo`05001oogooOol00004Ool01@00Oomoogoo 000017oo00<007ooOol03goo1`001Goo00<007ooOol027oo00D007ooOomoo`0000Aoo`05001oogoo Ool00004Ool01@00Oomoogoo000017oo00D007ooOomoo`0000Aoo`03001oogoo00ioo`L000Eoo`03 001oogoo00=oo`03001oogoo009oo`05001oogooOol00004Ool01@00Oomoogoo000017oo00D007oo Oomoo`0000Aoo`05001oogooOol00004Ool00`00Oomoo`0COol70007Ool01@00Oomoogoo000017oo 00D007ooOomoo`0000Aoo`05001oogooOol00004Ool01@00Oomoogoo000017oo00<007ooOol0@Woo 000BOol00`00Oomoo`02Ool00`00Oomoo`02Ool01@00Oomoogoo000017oo00D007ooOomoo`0000Ao o`05001oogooOol00004Ool01@00Oomoogoo000017oo00<007ooOol077oo00<007ooOol00Woo00<0 07ooOol00Woo00D007ooOomoo`0000Aoo`05001oogooOol00004Ool01@00Oomoogoo000017oo00D0 07ooOomoo`0000Aoo`03001oogoo01]oo`03001oogoo00=oo`03001oogoo009oo`05001oogooOol0 0004Ool01@00Oomoogoo000017oo00D007ooOomoo`0000Aoo`05001oogooOol00004Ool00`00Oomo o`0JOol00`00Oomoo`03Ool00`00Oomoo`02Ool01@00Oomoogoo000017oo00D007ooOomoo`0000Ao o`05001oogooOol00004Ool01@00Oomoogoo000017oo00<007ooOol077oo00<007ooOol00Woo00D0 07ooOomoo`0000Aoo`05001oogooOol00004Ool01@00Oomoogoo000017oo00D007ooOomoo`0000Ao o`03001oogoo049oo`004goo00@007ooOomoo`D000Aoo`03001oogoo009oo`05001oogooOol00004 Ool01@00Oomoogoo000017oo00D007ooOomoo`0000Aoo`03001oogoo01eoo`05001oogooOol00004 Ool01@00Oomoogoo000017oo00D007ooOomoo`0000Aoo`05001oogooOol00004Ool01@00Oomoogoo 000017oo00<007ooOol06goo00<007ooOol00goo1@0017oo00<007ooOol00Woo00D007ooOomoo`00 00Aoo`05001oogooOol00004Ool01@00Oomoogoo000017oo00<007ooOol06Woo00<007ooOol00goo 00<007ooOol00Woo00D007ooOomoo`0000Aoo`05001oogooOol00004Ool01@00Oomoogoo000017oo 00D007ooOomoo`0000Aoo`03001oogoo01aoo`D000Aoo`03001oogoo009oo`05001oogooOol00004 Ool01@00Oomoogoo000017oo00D007ooOomoo`0000Aoo`03001oogoo049oo`004goo00D007ooOomo o`0000Qoo`03001oogoo009oo`05001oogooOol00004Ool01@00Oomoogoo000017oo00D007ooOomo o`0000Aoo`03001oogoo01eoo`05001oogooOol00004Ool01@00Oomoogoo000017oo00D007ooOomo o`0000Aoo`05001oogooOol00004Ool01@00Oomoogoo000017oo00<007ooOol06goo00<007ooOol0 0goo00<007ooOol01Woo00<007ooOol00Woo00D007ooOomoo`0000Aoo`05001oogooOol00004Ool0 1@00Oomoogoo000017oo00<007ooOol06Woo00<007ooOol00goo00<007ooOol00Woo00D007ooOomo o`0000Aoo`05001oogooOol00004Ool01@00Oomoogoo000017oo00D007ooOomoo`0000Aoo`03001o ogoo01aoo`03001oogoo00Ioo`03001oogoo009oo`05001oogooOol00004Ool01@00Oomoogoo0000 17oo00D007ooOomoo`0000Aoo`03001oogoo049oo`003Woo00<007ooOol00Woo00D007ooOomoo`00 00Uoo`04001oogoo0005Ool01000Oomoo`001Goo00@007ooOol000Eoo`04001oogoo000KOol00`00 Oomoo`02Ool00`00Oomoo`02Ool01000Oomoo`001Goo00@007ooOol000Eoo`04001oogoo0005Ool0 1000Oomoo`001Goo00@007ooOol001aoo`<000Eoo`03001oogoo00Moo`04001oogoo0005Ool01000 Oomoo`001Goo00@007ooOol000Eoo`04001oogoo000KOol30006Ool01000Oomoo`001Goo00@007oo Ool000Eoo`04001oogoo0005Ool01000Oomoo`001Goo00@007ooOol001moo`03001oogoo00Moo`04 001oogoo0005Ool01000Oomoo`001Goo00@007ooOol000Eoo`04001oogoo0015Ool000moo`@000Ao o`D000Ioo`8000Moo`8000Moo`8000Moo`8001eoo`@000Ioo`8000Moo`8000Moo`8000Moo`8000Mo o`8001ioo`8000Eoo`D000Ioo`8000Moo`8000Moo`8000Moo`8001eoo`8000Moo`8000Moo`8000Mo o`8000Moo`8000Moo`80021oo`D000Ioo`8000Moo`8000Moo`8000Moo`8004Ioo`00ogooY7oo00<0 07ooOol03Goo003oOonTOol00`00Oomoo`0=Ool00?moojAoo`03001oogoo00eoo`00ogooY7oo00<0 07ooOol03Goo003oOonTOol00`00Oomoo`0=Ool00?moojAoo`03001oogoo00eoo`0047ooo`00W`00 1Goo000EOol00`00Oomoo`0Goo00<007ooOol03Goo003oOomVOol2000lOol00`00Oomoo`0=Ool00?moofAoo`8003ioo`03001o ogoo00eoo`00ogooHgoo00<007ooOol0?Woo0P003Woo003oOomQOol20011Ool00`00Oomoo`0=Ool0 0?moof1oo`03001oogoo045oo`03001oogoo00eoo`00ogooGWoo0P00A7oo00<007ooOol03Goo003o OomLOol20016Ool00`00Oomoo`0=Ool00?mooe]oo`03001oogoo04Ioo`03001oogoo00eoo`00ogoo FGoo0P00BGoo00<007ooOol03Goo003oOomGOol2001;Ool00`00Oomoo`0=Ool00?mooeIoo`03001o ogoo04]oo`03001oogoo00eoo`00ogooE7oo0P00CWoo00<007ooOol03Goo003oOomCOol00`00Oomo o`1>Ool00`00Oomoo`0=Ool00?mooe5oo`8001Moo`H000Eoo`8000Moo`8000Moo`8000Moo`8000Mo o`8000Uoo`03001oogoo00eoo`00ogooCgoo0P006Woo00D007ooOomoo`0000Aoo`04001oogoo0005 Ool01000Oomoo`001Goo00@007ooOol000Eoo`04001oogoo0005Ool01000Oomoo`0027oo00<007oo Ool03Goo003oOom>Ool00`00Oomoo`0KOol00`00Oomoo`04Ool00`00Oomoo`02Ool01@00Oomoogoo 000017oo00D007ooOomoo`0000Aoo`05001oogooOol00004Ool01@00Oomoogoo000017oo00<007oo Ool01Goo0`003Goo003oOomOol00?moob1oo`80089oo`03001oogoo00eoo`00ogoo7Woo0P00Q7oo00<007ooOol0 3Goo003oOolMOol00`00Oomoo`24Ool00`00Oomoo`0=Ool00?mooa]oo`8008Moo`03001oogoo00eo o`00ogoo6Woo00<007ooOol0Qgoo00<007ooOol03Goo003oOolHOol2002:Ool00`00Oomoo`0=Ool0 0?mooaMoo`03001oogoo08Yoo`03001oogoo00eoo`00ogoo5Goo0P00SGoo00<007ooOol03Goo003o OolCOol2002?Ool00`00Oomoo`0=Ool00?mooa9oo`03001oogoo08moo`03001oogoo00eoo`00ogoo 47oo0P00TWoo00<007ooOol03Goo003oOol?Ool00`00Oomoo`2BOol00`00Oomoo`0=Ool00?moo`eo o`8009Eoo`8000ioo`00ogoo2goo0P00Ugoo00<007ooOol03Goo003oOol:Ool00`00Oomoo`2GOol0 0`00Oomoo`0=Ool00?moo`Qoo`8009Yoo`03001oogoo00eoo`00ogoo1goo00<007ooOol0VWoo00<0 07ooOol03Goo003oOol5Ool2002MOol00`00Oomoo`0=Ool00?moo`Aoo`03001oogoo09eoo`03001o ogoo00eoo`00ogoo0Woo0P00X7oo00<007ooOol03Goo003oOol2002ROol00`00Oomoo`0=Ool00?io o`03001oogoo0:9oo`03001oogoo00eoo`00o7oo0P00YGoo00<007ooOol03Goo003kOol00`00Oomo o`1^Ool30005Ool20007Ool20007Ool20007Ool20007Ool20009Ool00`00Oomoo`0=Ool00?Uoo`80 079oo`03001oogoo00=oo`04001oogoo0005Ool01000Oomoo`001Goo00@007ooOol000Eoo`04001o ogoo0005Ool01000Oomoo`0027oo00<007ooOol03Goo003gOol2001`Ool60003Ool00`00Oomoo`02 Ool01@00Oomoogoo000017oo00D007ooOomoo`0000Aoo`05001oogooOol00004Ool01@00Oomoogoo 000017oo00<007ooOol01Goo0`003Goo003fOol00`00Oomoo`1`Ool01@00Oomoogoo000017oo00<0 07ooOol00Woo00D007ooOomoo`0000Aoo`05001oogooOol00004Ool01@00Oomoogoo000017oo00D0 07ooOomoo`0000Aoo`03001oogoo00Eoo`03001oogoo00eoo`00m7oo0P00M7oo00@007ooOol000Ao o`03001oogoo009oo`05001oogooOol00004Ool01@00Oomoogoo000017oo00D007ooOomoo`0000Ao o`05001oogooOol00004Ool00`00Oomoo`05Ool00`00Oomoo`0=Ool00?=oo`03001oogoo07Eoo`03 001oo`0000Aoo`03001oogoo009oo`05001oogooOol00004Ool01@00Oomoogoo000017oo00D007oo Oomoo`0000Aoo`05001oogooOol00004Ool00`00Oomoo`05Ool00`00Oomoo`0=Ool00?5oo`8007Qo o`03001oo`0000Aoo`03001oogoo009oo`05001oogooOol00004Ool01@00Oomoogoo000017oo00D0 07ooOomoo`0000Aoo`05001oogooOol00004Ool00`00Oomoo`05Ool00`00Oomoo`0=Ool00?1oo`03 001oogoo07Uoo`8000Eoo`04001oogoo0005Ool01000Oomoo`001Goo00@007ooOol000Eoo`04001o ogoo0005Ool01000Oomoo`0027oo00<007ooOol03Goo003^Ool2001mOol00`00Oomoo`04Ool20007 Ool20007Ool20007Ool20007Ool20009Ool00`00Oomoo`0=Ool00>aoo`800;Eoo`03001oogoo00eo o`00jgoo00<007ooOol0]Goo00<007ooOol03Goo003YOol2002hOol00`00Oomoo`0=Ool00>Qoo`03 001oogoo0;Qoo`03001oogoo00eoo`00iWoo0P00^goo00<007ooOol03Goo003TOol2002mOol00`00 Oomoo`0=Ool00>=oo`03001oogoo0;eoo`03001oogoo00eoo`00hGoo0P00`7oo0P003Woo003OOol2 0032Ool00`00Oomoo`0=Ool00=ioo`03001oogoo0<9oo`03001oogoo00eoo`00g7oo0P00aGoo00<0 07ooOol03Goo003JOol20037Ool00`00Oomoo`0=Ool00=Uoo`03001oogoo0Ool005oo`03 001oogoo00eoo`00_goo00<007ooOol0hGoo00<007ooOol03Goo002mOol2003TOol00`00Oomoo`0= Ool00;]oo`800>Ioo`03001oogoo00eoo`00^Woo00<007ooOol0iWoo00<007ooOol03Goo002hOol2 003YOol00`00Oomoo`0=Ool00;Ioo`800>]oo`8000ioo`00]Goo00<007ooOol0jgoo00<007ooOol0 3Goo002cOol2003^Ool00`00Oomoo`0=Ool00;9oo`03001oogoo0>ioo`03001oogoo00eoo`00/7oo 0P00lGoo00<007ooOol03Goo002^Ool2003cOol00`00Oomoo`0=Ool00:eoo`03001oogoo0?=oo`03 001oogoo00eoo`00Zgoo0P00mWoo00<007ooOol03Goo002YOol2003hOol00`00Oomoo`0=Ool00:Qo o`03001oogoo0?Qoo`03001oogoo00eoo`00YWoo0P00ngoo00<007ooOol03Goo002TOol20034Ool4 0006Ool20007Ool20007Ool20007Ool20007Ool20009Ool00`00Oomoo`0=Ool00:=oo`03001oogoo 0<=oo`03001oogoo009oo`03001oogoo009oo`04001oogoo0005Ool01000Oomoo`001Goo00@007oo Ool000Eoo`04001oogoo0005Ool01000Oomoo`0027oo00<007ooOol03Goo002QOol20036Ool00`00 Oomoo`02Ool01@00Oomoogoo000017oo00D007ooOomoo`0000Aoo`05001oogooOol00004Ool01@00 Oomoogoo000017oo00D007ooOomoo`0000Aoo`03001oogoo00Eoo`<000eoo`00X7oo00<007ooOol0 aWoo00<007ooOol00Woo00D007ooOomoo`0000Aoo`05001oogooOol00004Ool01@00Oomoogoo0000 17oo00D007ooOomoo`0000Aoo`05001oogooOol00004Ool00`00Oomoo`05Ool00`00Oomoo`0=Ool0 09ioo`800Ool005ioo`800?moodAoo`03001oogoo00eoo`00 GGoo00<007ooOol0ogooA7oo00<007ooOol03Goo001KOol2003oOom7Ool00`00Oomoo`0=Ool005Uo o`800?moodUoo`03001oogoo00eoo`00F7oo00<007ooOol0ogooBGoo00<007ooOol03Goo001FOol2 003oOomGoo00<0 07ooOol0ogooJ7oo00<007ooOol03Goo000gOol2003oOom[Ool00`00Oomoo`0=Ool003Ioo`03001o ogoo0?moof]oo`03001oogoo00eoo`00=7oo0P00ogooKWoo0P003Woo000bOol2003oOom`Ool00`00 Oomoo`0=Ool0035oo`03001oogoo0?moog1oo`03001oogoo00eoo`00;goo0P00ogooLgoo00<007oo Ool03Goo000]Ool2003oOomeOol00`00Oomoo`0=Ool002aoo`03001oogoo0?moogEoo`03001oogoo 00eoo`00:Woo0P00ogooN7oo00<007ooOol03Goo000XOol2003oOomjOol00`00Oomoo`0=Ool002Mo o`03001oogoo0?moogYoo`03001oogoo00eoo`009Goo0P00ogooOGoo00<007ooOol03Goo000TOol0 0`00Oomoo`3oOommOol00`00Oomoo`0=Ool0029oo`800?mooh1oo`03001oogoo00eoo`0087oo0P00 ogooPWoo00<007ooOol03Goo000OOol00`00Oomoo`3oOon2Ool2000>Ool001eoo`800?moohEoo`03 001oogoo00eoo`006goo0P00ogooQgoo00<007ooOol03Goo000JOol00`00Oomoo`3oOon7Ool00`00 Oomoo`0=Ool001Uoo`03001oogoo0?moohQoo`03001oogoo00eoo`00ogooY7oo00<007ooOol03Goo 003oOonTOol00`00Oomoo`0=Ool00?moojAoo`03001oogoo00eoo`00ogooY7oo00<007ooOol03Goo 003oOonTOol00`00Oomoo`0=Ool00?moojAoo`03001oogoo00eoo`00\ \>"], ImageRangeCache->{{{0, 434.375}, {268.062, 0}} -> {-273830, -70324.8, 979.566, 5651.32}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Closed]], Cell["The population x dies out first.", "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[{x[t] /. First[First[sol]], y[t] /. Last[First[sol]]}, {t, 0, 15}, PlotRange -> {0, 1000}, PlotStyle -> {Hue[0], Hue[.7]}]\)], "Input", CellAutoOverwrite->False], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.0634921 0 0.000618034 [ [.15079 -0.0125 -3 -9 ] [.15079 -0.0125 3 0 ] [.27778 -0.0125 -3 -9 ] [.27778 -0.0125 3 0 ] [.40476 -0.0125 -3 -9 ] [.40476 -0.0125 3 0 ] [.53175 -0.0125 -3 -9 ] [.53175 -0.0125 3 0 ] [.65873 -0.0125 -6 -9 ] [.65873 -0.0125 6 0 ] [.78571 -0.0125 -6 -9 ] [.78571 -0.0125 6 0 ] [.9127 -0.0125 -6 -9 ] [.9127 -0.0125 6 0 ] [.01131 .12361 -18 -4.5 ] [.01131 .12361 0 4.5 ] [.01131 .24721 -18 -4.5 ] [.01131 .24721 0 4.5 ] [.01131 .37082 -18 -4.5 ] [.01131 .37082 0 4.5 ] [.01131 .49443 -18 -4.5 ] [.01131 .49443 0 4.5 ] [.01131 .61803 -24 -4.5 ] [.01131 .61803 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .15079 0 m .15079 .00625 L s [(2)] .15079 -0.0125 0 1 Mshowa .27778 0 m .27778 .00625 L s [(4)] .27778 -0.0125 0 1 Mshowa .40476 0 m .40476 .00625 L s [(6)] .40476 -0.0125 0 1 Mshowa .53175 0 m .53175 .00625 L s [(8)] .53175 -0.0125 0 1 Mshowa .65873 0 m .65873 .00625 L s [(10)] .65873 -0.0125 0 1 Mshowa .78571 0 m .78571 .00625 L s [(12)] .78571 -0.0125 0 1 Mshowa .9127 0 m .9127 .00625 L s [(14)] .9127 -0.0125 0 1 Mshowa .125 Mabswid .05556 0 m .05556 .00375 L s .0873 0 m .0873 .00375 L s .11905 0 m .11905 .00375 L s .18254 0 m .18254 .00375 L s .21429 0 m .21429 .00375 L s .24603 0 m .24603 .00375 L s .30952 0 m .30952 .00375 L s .34127 0 m .34127 .00375 L s .37302 0 m .37302 .00375 L s .43651 0 m .43651 .00375 L s .46825 0 m .46825 .00375 L s .5 0 m .5 .00375 L s .56349 0 m .56349 .00375 L s .59524 0 m .59524 .00375 L s .62698 0 m .62698 .00375 L s .69048 0 m .69048 .00375 L s .72222 0 m .72222 .00375 L s .75397 0 m .75397 .00375 L s .81746 0 m .81746 .00375 L s .84921 0 m .84921 .00375 L s .88095 0 m .88095 .00375 L s .94444 0 m .94444 .00375 L s .97619 0 m .97619 .00375 L s .25 Mabswid 0 0 m 1 0 L s .02381 .12361 m .03006 .12361 L s [(200)] .01131 .12361 1 0 Mshowa .02381 .24721 m .03006 .24721 L s [(400)] .01131 .24721 1 0 Mshowa .02381 .37082 m .03006 .37082 L s [(600)] .01131 .37082 1 0 Mshowa .02381 .49443 m .03006 .49443 L s [(800)] .01131 .49443 1 0 Mshowa .02381 .61803 m .03006 .61803 L s [(1000)] .01131 .61803 1 0 Mshowa .125 Mabswid .02381 .0309 m .02756 .0309 L s .02381 .0618 m .02756 .0618 L s .02381 .09271 m .02756 .09271 L s .02381 .15451 m .02756 .15451 L s .02381 .18541 m .02756 .18541 L s .02381 .21631 m .02756 .21631 L s .02381 .27812 m .02756 .27812 L s .02381 .30902 m .02756 .30902 L s .02381 .33992 m .02756 .33992 L s .02381 .40172 m .02756 .40172 L s .02381 .43262 m .02756 .43262 L s .02381 .46353 m .02756 .46353 L s .02381 .52533 m .02756 .52533 L s .02381 .55623 m .02756 .55623 L s .02381 .58713 m .02756 .58713 L s .25 Mabswid .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath 1 0 0 r .5 Mabswid .02381 .09271 m .04262 .09163 L .06244 .09013 L .08255 .08815 L .10458 .08537 L .12579 .08198 L .14509 .0782 L .18408 .06805 L .20323 .0616 L .22401 .05326 L .26488 .03192 L .30423 .00346 L s .30423 .00346 m .30776 0 L s .2 0 1 r .02381 .12361 m .06244 .14505 L .10458 .17576 L .14415 .21388 L .18221 .26181 L .22272 .32891 L .26171 .41427 L .30316 .5348 L s .30316 .5348 m .32461 .61803 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{292.625, 180.75}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHggoo000^Ool00`00Oomoo`15 Ool3O03oOolnOol002ioo`8004Aoo`9l0?mood5oo`00;Woo00<007ooOol0@7oo0g`0ogoo@goo000^ Ool00`00Oomoo`0mOol3O03oOom6Ool002ioo`03001oogoo03Yoo`=l0?moodUoo`00;Woo00<007oo Ool0=Woo17`0ogooC7oo000^Ool00`00Oomoo`0cOol3O03oOom@Ool002ioo`03001oogoo02moo`Al 0?mooe=oo`00;Woo00<007ooOol0:Goo1W`0ogooEgoo000^Ool00`00Oomoo`0SOol6O03oOomMOol0 02ioo`03001oogoo01moo`Al0?moof=oo`00;Woo00<007ooOol06Goo1W`0ogooIgoo000^Ool00`00 Oomoo`0@Ool9O03oOom]Ool002ioo`04001oogooOol?O03oOomfOol002ioo`Al0?moohEoo`00;Woo 00<007ooOol0ogooQWoo000^Ool00`00Oomoo`3oOon6Ool002ioo`03001oogoo0?moohIoo`00;Woo 00<007ooOol0ogooQWoo000^Ool00`00Oomoo`3oOon6Ool002ioo`03001oogoo0?moohIoo`00;Woo 00<007ooOol0ogooQWoo000^Ool00`00Oomoo`3oOon6Ool000moo`H000Eoo`8000Moo`8000Uoo`03 001oogoo0?moohIoo`0047oo00D007ooOomoo`0000Aoo`04001oogoo0005Ool01000Oomoo`0027oo 00<007ooOol0ogooQWoo000AOol00`00Oomoo`04Ool00`00Oomoo`02Ool01@00Oomoogoo000017oo 00<007ooOol01Goo00<007ooOol0ogooQWoo000BOol00`00Oomoo`03Ool00`00Oomoo`02Ool01@00 Oomoogoo000017oo00<007ooOol01Goo0QPO00<007ooOol0ogooQ7oo000COol00`00Oomoo`02Ool0 0`00Oomoo`02Ool01@00Oomoogoo000017oo00<007ooOol01Goo00@007oo61lH7omoohEoo`0057oo 00D007ooOomoo`0000Aoo`05001oogooOol00004Ool00`00Oomoo`05Ool01000Oomoogoo0QPOogoo Pgoo000DOol01@00Oomoogoo000017oo00D007ooOomoo`0000Aoo`03001oogoo00Eoo`03001oogoo 00=oo`8H7omooh5oo`003goo00<007ooOol00Woo00<007ooOol00Woo00@007ooOol000Eoo`04001o ogoo0008Ool00`00Oomoo`05Ool00aPOOomoo`3oOomnOol0011oo`@000Ioo`8000Moo`8000Uoo`03 001oogoo00Ioo`8H7omoogioo`00;Woo00<007ooOol027oo0QPOogooO7oo000^Ool00`00Oomoo`0: Ool261ooOomjOol002ioo`03001oogoo00aoo`8H7omoogQoo`00;Woo00<007ooOol03Woo00Goo00goo00goo000^Ool00`00Oomoo`19Ool00aPOOomo o`3oOoljOol002ioo`03001oogoo04Uoo`0361moogoo0?moocYoo`00;Woo00<007ooOol0BWoo00Goo000^Ool00`00Oomoo`1;Ool00aPOOomoo`3oOolhOol002ioo`03001oogoo04]o o`0361moogoo0?moocQoo`00;Woo00<007ooOol0C7oo00Ool00aPOOomoo`3oOoleOol002ioo`03001o ogoo04ioo`0361moogoo0?moocEoo`00;Woo00<007ooOol0Cgoo00Ool000Uoo`03001oogoo00=oo`03001oogoo009oo`05001oogooOol0 0004Ool01@00Oomoogoo000017oo00<007ooOol0ogooSWoo0007Ool30006Ool01000Oomoo`001Goo 00@007ooOol000Eoo`04001oogoo003oOonAOol000Qoo`8000Moo`8000Moo`8000Moo`800?mooi9o o`00ogoo]goo0000\ \>"], ImageRangeCache->{{{0, 437.938}, {270.125, 0}} -> {-1.82894, -75.3505, 0.0596155, 6.12444}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Closed]], Cell["Now x is going to zero rapidly.", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Different initial values", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(initialValues = {x[0] == 400, y[0] == 200}\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \({x[0] == 400, y[0] == 200}\)], "Output"] }, Closed]], Cell[BoxData[ \(\(sol = DSolve[Flatten[{system, initialValues}], {x[t], y[t]}, t];\)\)], "Input", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell["Plane plot", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(ParametricPlot[{x[t] /. First[First[sol]], y[t] /. Last[First[sol]]}, { t, 0, 10}, AxesOrigin -> {0, 0}]\)], "Input", CellAutoOverwrite->False], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations -0.463015 0.00121706 0.541131 0.00031094 [ [.02381 .52863 -9 -9 ] [.02381 .52863 9 0 ] [.26722 .52863 -9 -9 ] [.26722 .52863 9 0 ] [.51063 .52863 -9 -9 ] [.51063 .52863 9 0 ] [.75405 .52863 -12 -9 ] [.75405 .52863 12 0 ] [.99746 .52863 -12 -9 ] [.99746 .52863 12 0 ] [-0.47551 .07472 -30 -4.5 ] [-0.47551 .07472 0 4.5 ] [-0.47551 .15246 -30 -4.5 ] [-0.47551 .15246 0 4.5 ] [-0.47551 .23019 -30 -4.5 ] [-0.47551 .23019 0 4.5 ] [-0.47551 .30793 -24 -4.5 ] [-0.47551 .30793 0 4.5 ] [-0.47551 .38566 -24 -4.5 ] [-0.47551 .38566 0 4.5 ] [-0.47551 .4634 -24 -4.5 ] [-0.47551 .4634 0 4.5 ] [ -0.46301 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .02381 .54113 m .02381 .54738 L s [(400)] .02381 .52863 0 1 Mshowa .26722 .54113 m .26722 .54738 L s [(600)] .26722 .52863 0 1 Mshowa .51063 .54113 m .51063 .54738 L s [(800)] .51063 .52863 0 1 Mshowa .75405 .54113 m .75405 .54738 L s [(1000)] .75405 .52863 0 1 Mshowa .99746 .54113 m .99746 .54738 L s [(1200)] .99746 .52863 0 1 Mshowa .125 Mabswid .08466 .54113 m .08466 .54488 L s .14552 .54113 m .14552 .54488 L s .20637 .54113 m .20637 .54488 L s .32807 .54113 m .32807 .54488 L s .38893 .54113 m .38893 .54488 L s .44978 .54113 m .44978 .54488 L s .57149 .54113 m .57149 .54488 L s .63234 .54113 m .63234 .5