(* Content-type: application/vnd.wolfram.cdf.text *) (*** Wolfram CDF File ***) (* http://www.wolfram.com/cdf *) (* CreatedBy='Mathematica 8.0' *) (*************************************************************************) (* *) (* The Mathematica License under which this file was created prohibits *) (* restricting third parties in receipt of this file from republishing *) (* or redistributing it by any means, including but not limited to *) (* rights management or terms of use, without the express consent of *) (* Wolfram Research, Inc. *) (* *) (*************************************************************************) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 835, 17] NotebookDataLength[ 71291, 1679] NotebookOptionsPosition[ 71375, 1666] NotebookOutlinePosition[ 71795, 1682] CellTagsIndexPosition[ 71752, 1679] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["The replicator-mutator dynamics with 3 strategies", "Title", CellChangeTimes->{{3.517752365493416*^9, 3.5177523895997944`*^9}, { 3.5177525177231226`*^9, 3.5177525185701714`*^9}}], Cell[TextData[{ "\nBy ", ButtonBox["Luis R. Izquierdo", BaseStyle->"Hyperlink", ButtonData->{ URL["http://luis.izquierdo.name"], None}, ButtonNote->"http://luis.izquierdo.name"], " and ", ButtonBox["Segismundo S. Izquierdo", BaseStyle->"Hyperlink", ButtonData->{ URL["http://segis.name"], None}, ButtonNote->"http://segis.name"] }], "Subtitle", CellChangeTimes->{{3.5177524023105216`*^9, 3.5177524217326326`*^9}, { 3.517752471921503*^9, 3.5177524892894964`*^9}}], Cell[BoxData[Cell[TextData[{ "This demonstration shows the phase portrait of the replicator-mutator \ dynamics with 3 strategies, in the 2-dimensional simplex. Parameters ", Cell[BoxData[ FormBox[ SubscriptBox["a", "ij"], TraditionalForm]], "InlineMath"], " \[GreaterEqual] 0 denote the payoff that an ", StyleBox["i", FontSlant->"Italic"], "-strategist obtains in an interaction with a ", StyleBox["j", FontSlant->"Italic"], "-strategist. Parameter \[Mu] denotes the total fraction of mutants or \ entrants in the population. Parameters ", Cell[BoxData[ FormBox[ SubscriptBox["m", "i"], TraditionalForm]], "InlineMath"], " \[GreaterEqual] 0 denote the weight given by mutants to strategy ", StyleBox["i", FontSlant->"Italic"], ", so ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SubscriptBox["m", "i"], "/", " ", RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "1"}], "3"], SubscriptBox["m", "k"]}]}], ")"}], TraditionalForm]], "InlineMath"], " is the fraction of mutants that adopt strategy ", StyleBox["i", FontSlant->"Italic"], ". The program also calculates the critical points of the system, and their \ corresponding eigenvalues (which are helpful to assess the dynamic stability \ of the critical point)." }], "ManipulateCaption", CellChangeTimes->{{3.5177293167771044`*^9, 3.517729329448829*^9}, { 3.5177298041399803`*^9, 3.517729942883916*^9}, {3.5177299794400063`*^9, 3.51773001376897*^9}, {3.517730075818519*^9, 3.517730094328578*^9}, { 3.5177302522836123`*^9, 3.517730313329104*^9}, {3.517730460187504*^9, 3.517730529818486*^9}, {3.5177451453814487`*^9, 3.517745167202697*^9}, { 3.517746642443076*^9, 3.517746713078116*^9}, {3.517746747979112*^9, 3.517746860605554*^9}, {3.5177469113844585`*^9, 3.517746916155731*^9}}]], "Text", CellChangeTimes->{{3.5177653878123274`*^9, 3.517765390591486*^9}, { 3.517765592084011*^9, 3.5177656106060705`*^9}, {3.5178485190824604`*^9, 3.5178485190824604`*^9}, {3.5180038880270824`*^9, 3.5180038929253626`*^9}}, FontWeight->"Bold", Background->GrayLevel[1]], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`a11$$ = 1, $CellContext`a12$$ = 1, $CellContext`a13$$ = 0.01, $CellContext`a21$$ = 1, $CellContext`a22$$ = 1, $CellContext`a23$$ = 0.99, $CellContext`a31$$ = 0.01, $CellContext`a32$$ = 0.99, $CellContext`a33$$ = 0.4, $CellContext`m1$$ = 1, $CellContext`m2$$ = 1, $CellContext`m3$$ = 1, $CellContext`update$$ = 0, $CellContext`\[Mu]$$ = 0.0001, Typeset`show$$ = True, Typeset`bookmarkList$$ = { "\"Fig. 2a\"" :> {$CellContext`a11$$ = 1, $CellContext`a12$$ = 1, $CellContext`a13$$ = 0.99, $CellContext`a21$$ = 1, $CellContext`a22$$ = 1, $CellContext`a23$$ = 0.01, $CellContext`a31$$ = 0.99, $CellContext`a32$$ = 0.01, $CellContext`a33$$ = 0.4, $CellContext`m1$$ = 1, $CellContext`m2$$ = 1, $CellContext`m3$$ = 1, $CellContext`update$$ = 100, $CellContext`\[Mu]$$ = 0.0001}, "\"Fig. 2b\"" :> {$CellContext`a11$$ = 1, $CellContext`a12$$ = 1, $CellContext`a13$$ = 0.5, $CellContext`a21$$ = 1, $CellContext`a22$$ = 1, $CellContext`a23$$ = 0.5, $CellContext`a31$$ = 0.5, $CellContext`a32$$ = 0.5, $CellContext`a33$$ = 0.4, $CellContext`m1$$ = 1, $CellContext`m2$$ = 1, $CellContext`m3$$ = 1, $CellContext`update$$ = 200, $CellContext`\[Mu]$$ = 0.0001}, "\"Fig. 2c\"" :> {$CellContext`a11$$ = 1, $CellContext`a12$$ = 1, $CellContext`a13$$ = 0.01, $CellContext`a21$$ = 1, $CellContext`a22$$ = 1, $CellContext`a23$$ = 0.99, $CellContext`a31$$ = 0.01, $CellContext`a32$$ = 0.99, $CellContext`a33$$ = 0.4, $CellContext`m1$$ = 1, $CellContext`m2$$ = 1, $CellContext`m3$$ = 1, $CellContext`update$$ = 300, $CellContext`\[Mu]$$ = 0.0001}}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`a11$$], 1}, 0., 10, Rational[1, 100]}, {{ Hold[$CellContext`a12$$], 1}, 0., 10, Rational[1, 100]}, {{ Hold[$CellContext`a13$$], 0.01}, 0., 10, Rational[1, 100]}, {{ Hold[$CellContext`a21$$], 1}, 0., 10, Rational[1, 100]}, {{ Hold[$CellContext`a22$$], 1}, 0., 10, Rational[1, 100]}, {{ Hold[$CellContext`a23$$], 0.99}, 0., 10, Rational[1, 100]}, {{ Hold[$CellContext`a31$$], 0.01}, 0., 10, Rational[1, 100]}, {{ Hold[$CellContext`a32$$], 0.99}, 0., 10, Rational[1, 100]}, {{ Hold[$CellContext`a33$$], 0.4}, 0., 10, Rational[1, 100]}, {{ Hold[$CellContext`update$$], 0}, 0, 100}, {{ Hold[$CellContext`\[Mu]$$], 0.01}, 0.01, 1, Rational[1, 100]}, {{ Hold[$CellContext`m1$$], 1}, 0., 5, Rational[1, 100]}, {{ Hold[$CellContext`m2$$], 1}, 0., 5, Rational[1, 100]}, {{ Hold[$CellContext`m3$$], 1}, 0., 5, Rational[1, 100]}, {{ Hold[$CellContext`a11$$], 1}, Dynamic[InputField[ Dynamic[$CellContext`a11$$, ($CellContext`a11$$ = If[ NumberQ[#], Clip[#, {0.001, Infinity}], 0.001])& ], Number, FieldSize -> 3]& ]}, {{ Hold[$CellContext`a12$$], 1}, Dynamic[InputField[ Dynamic[$CellContext`a12$$, ($CellContext`a12$$ = If[ NumberQ[#], Clip[#, {0, Infinity}], 0])& ], Number, FieldSize -> 3]& ]}, {{ Hold[$CellContext`a13$$], 0.01}, Dynamic[InputField[ Dynamic[$CellContext`a13$$, ($CellContext`a13$$ = If[ NumberQ[#], Clip[#, {0, Infinity}], 0])& ], Number, FieldSize -> 3]& ]}, {{ Hold[$CellContext`a21$$], 1}, Dynamic[InputField[ Dynamic[$CellContext`a21$$, ($CellContext`a21$$ = If[ NumberQ[#], Clip[#, {0, Infinity}], 0])& ], Number, FieldSize -> 3]& ]}, {{ Hold[$CellContext`a22$$], 1}, Dynamic[InputField[ Dynamic[$CellContext`a22$$, ($CellContext`a22$$ = If[ NumberQ[#], Clip[#, {0.001, Infinity}], 0.001])& ], Number, FieldSize -> 3]& ]}, {{ Hold[$CellContext`a23$$], 0.99}, Dynamic[InputField[ Dynamic[$CellContext`a23$$, ($CellContext`a23$$ = If[ NumberQ[#], Clip[#, {0, Infinity}], 0])& ], Number, FieldSize -> 3]& ]}, {{ Hold[$CellContext`a31$$], 0.01}, Dynamic[InputField[ Dynamic[$CellContext`a31$$, ($CellContext`a31$$ = If[ NumberQ[#], Clip[#, {0, Infinity}], 0])& ], Number, FieldSize -> 3]& ]}, {{ Hold[$CellContext`a32$$], 0.99}, Dynamic[InputField[ Dynamic[$CellContext`a32$$, ($CellContext`a32$$ = If[ NumberQ[#], Clip[#, {0, Infinity}], 0])& ], Number, FieldSize -> 3]& ]}, {{ Hold[$CellContext`a33$$], 0.4}, Dynamic[InputField[ Dynamic[$CellContext`a33$$, ($CellContext`a33$$ = If[ NumberQ[#], Clip[#, {0.001, Infinity}], 0.001])& ], Number, FieldSize -> 3]& ]}, {{ Hold[$CellContext`\[Mu]$$], 0.0001}, Rational[1, 100000], 1, 0.00001}, {{ Hold[$CellContext`m1$$], 1}, Dynamic[InputField[ Dynamic[$CellContext`m1$$, ($CellContext`m1$$ = If[ NumberQ[#], Clip[#, {0.001, Infinity}], 0.001])& ], Number, FieldSize -> 3]& ]}, {{ Hold[$CellContext`m2$$], 1}, Dynamic[InputField[ Dynamic[$CellContext`m2$$, ($CellContext`m2$$ = If[ NumberQ[#], Clip[#, {0.001, Infinity}], 0.001])& ], Number, FieldSize -> 3]& ]}, {{ Hold[$CellContext`m3$$], 1}, Dynamic[InputField[ Dynamic[$CellContext`m3$$, ($CellContext`m3$$ = If[ NumberQ[#], Clip[#, {0.001, Infinity}], 0.001])& ], Number, FieldSize -> 3]& ]}, { Hold[ Row[{"", Grid[{{ Style["payoffs", Bold]}, { Manipulate`Place[1], Manipulate`Place[2], Manipulate`Place[3]}, { Manipulate`Place[4], Manipulate`Place[5], Manipulate`Place[6]}, { Manipulate`Place[7], Manipulate`Place[8], Manipulate`Place[9]}}, Alignment -> Left, Spacings -> {2, 0}], Button["Update", Increment[$CellContext`update$$], ImageSize -> {70, 50}], Grid[{{ Style["mutation rates", Bold], SpanFromLeft}, {"", SpanFromLeft}, { Manipulate`Place[10], SpanFromLeft}, { Manipulate`Place[11], Manipulate`Place[12], Manipulate`Place[13]}}, Alignment -> Left, Spacings -> {2, 0}]}, Spacer[30]]], Manipulate`Dump`ThisIsNotAControl}}, Typeset`size$$ = { 400., {208.5, 213.5}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = False, $CellContext`a11$9964$$ = 0, $CellContext`a12$9965$$ = 0, $CellContext`a13$9966$$ = 0, $CellContext`a21$9967$$ = 0, $CellContext`a22$9968$$ = 0, $CellContext`a23$9969$$ = 0, $CellContext`a31$9970$$ = 0, $CellContext`a32$9971$$ = 0, $CellContext`a33$9972$$ = 0, $CellContext`update$9973$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 2, StandardForm, "Variables" :> {$CellContext`a11$$ = 1, $CellContext`a12$$ = 1, $CellContext`a13$$ = 0.01, $CellContext`a21$$ = 1, $CellContext`a22$$ = 1, $CellContext`a23$$ = 0.99, $CellContext`a31$$ = 0.01, $CellContext`a32$$ = 0.99, $CellContext`a33$$ = 0.4, $CellContext`m1$$ = 1, $CellContext`m2$$ = 1, $CellContext`m3$$ = 1, $CellContext`update$$ = 0, $CellContext`\[Mu]$$ = 0.0001}, "ControllerVariables" :> { Hold[$CellContext`a11$$, $CellContext`a11$9964$$, 0], Hold[$CellContext`a12$$, $CellContext`a12$9965$$, 0], Hold[$CellContext`a13$$, $CellContext`a13$9966$$, 0], Hold[$CellContext`a21$$, $CellContext`a21$9967$$, 0], Hold[$CellContext`a22$$, $CellContext`a22$9968$$, 0], Hold[$CellContext`a23$$, $CellContext`a23$9969$$, 0], Hold[$CellContext`a31$$, $CellContext`a31$9970$$, 0], Hold[$CellContext`a32$$, $CellContext`a32$9971$$, 0], Hold[$CellContext`a33$$, $CellContext`a33$9972$$, 0], Hold[$CellContext`update$$, $CellContext`update$9973$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> Module[{$CellContext`initialPoints$, $CellContext`mutationMatrix$, \ $CellContext`mutantsList$ = Rationalize[{$CellContext`m1$$, $CellContext`m2$$, \ $CellContext`m3$$}], $CellContext`A$ = Rationalize[{{$CellContext`a11$$, $CellContext`a12$$, \ $CellContext`a13$$}, {$CellContext`a21$$, $CellContext`a22$$, \ $CellContext`a23$$}, {$CellContext`a31$$, $CellContext`a32$$, \ $CellContext`a33$$}}], $CellContext`listOfProjectedTrajectories$, \ $CellContext`listOfArrows$, $CellContext`graphWithTrajectories$, \ $CellContext`background$, $CellContext`criticalPoints$}, \ $CellContext`update$$; $CellContext`mutationMatrix$ = (1 - Rationalize[$CellContext`\[Mu]$$]) IdentityMatrix[3] + (Rationalize[$CellContext`\[Mu]$$]/ Total[$CellContext`mutantsList$]) Map[Table[#, { 3}]& , $CellContext`mutantsList$]; \ $CellContext`listOfProjectedTrajectories$ = Flatten[ Table[ $CellContext`Simplex2DCoord[ $CellContext`trajectory[ N[{$CellContext`initx1, $CellContext`initx2, 1 - $CellContext`initx1 - $CellContext`initx2}], \ $CellContext`A$, $CellContext`mutationMatrix$, $CellContext`tMax]], \ {$CellContext`initx1, 0, 1 + $CellContext`step/ 2, $CellContext`step}, {$CellContext`initx2, 0, 1 - $CellContext`initx1, $CellContext`step}], 1]; $CellContext`listOfArrows$ = Flatten[ Map[Table[{ Arrowheads[0.03], Arrow[{ ReplaceAll[#, $CellContext`t -> $CellContext`i], ReplaceAll[#, $CellContext`t -> $CellContext`i + 0.1]}]}, {$CellContext`i, 0.2, 3, 2}]& , $CellContext`listOfProjectedTrajectories$], 1]; $CellContext`graphWithTrajectories$ = ParametricPlot[ Evaluate[$CellContext`listOfProjectedTrajectories$], \ {$CellContext`t, 0, $CellContext`tMax}, PlotStyle -> Black]; $CellContext`background$ = DensityPlot[ Norm[ $CellContext`Simplex2DCoord[ $CellContext`expectedChange[ $CellContext`Simplex2DInverseCoord[{$CellContext`y1, \ $CellContext`y2}], $CellContext`A$, $CellContext`mutationMatrix$]]], \ {$CellContext`y1, 0, 1}, {$CellContext`y2, 0, 3^Rational[1, 2]/2}, ImageSize -> 400, ColorFunction -> "Rainbow", RegionFunction -> Function[{$CellContext`y1, $CellContext`y2, $CellContext`z}, And[$CellContext`y2 < 3^Rational[1, 2] $CellContext`y1, $CellContext`y2 < 3^Rational[1, 2] (1 - $CellContext`y1)]], BoundaryStyle -> Directive[Black, Thick], Frame -> False, AspectRatio -> 3^Rational[1, 2]/2, PlotRange -> {{-0.1, 1 + 0.1}, {-0.1, 3^Rational[1, 2]/2 + 0.1}}]; $CellContext`criticalPoints$ = Reduce[{$CellContext`expectedChange[{$CellContext`x1, \ $CellContext`x2, $CellContext`x3}, $CellContext`A$, \ $CellContext`mutationMatrix$] == 0, 0 <= $CellContext`x1 <= 1, 0 <= $CellContext`x2 <= 1, 0 <= $CellContext`x3 <= 1, $CellContext`x1 + $CellContext`x2 + $CellContext`x3 == 1}, {$CellContext`x1, $CellContext`x2, $CellContext`x3}, Reals, Backsubstitution -> True]; Column[{ Show[{$CellContext`background$, \ $CellContext`graphWithTrajectories$, Graphics[$CellContext`listOfArrows$], Graphics[{ EdgeForm[Thick], Red, Map[Disk[ $CellContext`Simplex2DCoord[#], 0.03]& , ReplaceAll[{$CellContext`x1, $CellContext`x2, \ $CellContext`x3}, { ToRules[$CellContext`criticalPoints$]}]]}], Graphics[{ Inset[ Style[ "\!\(\*SubscriptBox[\(x\), \(3\)]\) = 1", Large, Bold], {0.5, 0.9}], Inset[ Style[ "\!\(\*SubscriptBox[\(x\), \(1\)]\) = 1", Large, Bold], { 0, -0.05}], Inset[ Style[ "\!\(\*SubscriptBox[\(x\), \(2\)]\) = 1", Large, Bold], { 1, -0.05}]}]}, PlotRange -> {{-0.1, 1 + 0.1}, {-0.1, 3^Rational[1, 2]/2 + 0.1}}, Frame -> False, Axes -> False], Column[{ Style["Critical points", Red, Bold, 14], Row[{ TableForm[ ReplaceRepeated[{$CellContext`x1, $CellContext`x2, \ $CellContext`x3}, { ToRules[ N[$CellContext`criticalPoints$]]}], TableHeadings -> {None, { Style["\!\(\*SubscriptBox[\(x\), \(1\)]\)", Bold, 14], Style["\!\(\*SubscriptBox[\(x\), \(2\)]\)", Bold, 14], Style["\!\(\*SubscriptBox[\(x\), \(3\)]\)", Bold, 14]}}, TableSpacing -> {2, 1}, TableAlignments -> {Left, Bottom}], TableForm[ Map[Eigenvalues[ $CellContext`Jacob[#, $CellContext`A$, \ $CellContext`mutationMatrix$]]& , ReplaceRepeated[{$CellContext`x1, $CellContext`x2}, { ToRules[ N[$CellContext`criticalPoints$]]}]], TableHeadings -> {None, { Style[ "\!\(\*SubscriptBox[\(\[Lambda]\), \(1\)]\)", Bold, 14], Style[ "\!\(\*SubscriptBox[\(\[Lambda]\), \(2\)]\)", Bold, 14]}}, TableSpacing -> {2, 1}, TableAlignments -> {Right, Bottom}]}, " "]}, Alignment -> Center, Spacings -> 1, Frame -> True]}, Center]], "Specifications" :> {{{$CellContext`a11$$, 1}, 0., 10, Rational[1, 100], ControlType -> None}, {{$CellContext`a12$$, 1}, 0., 10, Rational[1, 100], ControlType -> None}, {{$CellContext`a13$$, 0.01}, 0., 10, Rational[1, 100], ControlType -> None}, {{$CellContext`a21$$, 1}, 0., 10, Rational[1, 100], ControlType -> None}, {{$CellContext`a22$$, 1}, 0., 10, Rational[1, 100], ControlType -> None}, {{$CellContext`a23$$, 0.99}, 0., 10, Rational[1, 100], ControlType -> None}, {{$CellContext`a31$$, 0.01}, 0., 10, Rational[1, 100], ControlType -> None}, {{$CellContext`a32$$, 0.99}, 0., 10, Rational[1, 100], ControlType -> None}, {{$CellContext`a33$$, 0.4}, 0., 10, Rational[1, 100], ControlType -> None}, {{$CellContext`update$$, 0}, 0, 100, ControlType -> None}, {{$CellContext`\[Mu]$$, 0.01}, 0.01, 1, Rational[1, 100], ControlType -> None}, {{$CellContext`m1$$, 1}, 0., 5, Rational[1, 100], ControlType -> None}, {{$CellContext`m2$$, 1}, 0., 5, Rational[1, 100], ControlType -> None}, {{$CellContext`m3$$, 1}, 0., 5, Rational[1, 100], ControlType -> None}, {{$CellContext`a11$$, 1}, Dynamic[InputField[ Dynamic[$CellContext`a11$$, ($CellContext`a11$$ = If[ NumberQ[#], Clip[#, {0.001, Infinity}], 0.001])& ], Number, FieldSize -> 3]& ], ControlPlacement -> 1}, {{$CellContext`a12$$, 1}, Dynamic[InputField[ Dynamic[$CellContext`a12$$, ($CellContext`a12$$ = If[ NumberQ[#], Clip[#, {0, Infinity}], 0])& ], Number, FieldSize -> 3]& ], ControlPlacement -> 2}, {{$CellContext`a13$$, 0.01}, Dynamic[InputField[ Dynamic[$CellContext`a13$$, ($CellContext`a13$$ = If[ NumberQ[#], Clip[#, {0, Infinity}], 0])& ], Number, FieldSize -> 3]& ], ControlPlacement -> 3}, {{$CellContext`a21$$, 1}, Dynamic[InputField[ Dynamic[$CellContext`a21$$, ($CellContext`a21$$ = If[ NumberQ[#], Clip[#, {0, Infinity}], 0])& ], Number, FieldSize -> 3]& ], ControlPlacement -> 4}, {{$CellContext`a22$$, 1}, Dynamic[InputField[ Dynamic[$CellContext`a22$$, ($CellContext`a22$$ = If[ NumberQ[#], Clip[#, {0.001, Infinity}], 0.001])& ], Number, FieldSize -> 3]& ], ControlPlacement -> 5}, {{$CellContext`a23$$, 0.99}, Dynamic[InputField[ Dynamic[$CellContext`a23$$, ($CellContext`a23$$ = If[ NumberQ[#], Clip[#, {0, Infinity}], 0])& ], Number, FieldSize -> 3]& ], ControlPlacement -> 6}, {{$CellContext`a31$$, 0.01}, Dynamic[InputField[ Dynamic[$CellContext`a31$$, ($CellContext`a31$$ = If[ NumberQ[#], Clip[#, {0, Infinity}], 0])& ], Number, FieldSize -> 3]& ], ControlPlacement -> 7}, {{$CellContext`a32$$, 0.99}, Dynamic[InputField[ Dynamic[$CellContext`a32$$, ($CellContext`a32$$ = If[ NumberQ[#], Clip[#, {0, Infinity}], 0])& ], Number, FieldSize -> 3]& ], ControlPlacement -> 8}, {{$CellContext`a33$$, 0.4}, Dynamic[InputField[ Dynamic[$CellContext`a33$$, ($CellContext`a33$$ = If[ NumberQ[#], Clip[#, {0.001, Infinity}], 0.001])& ], Number, FieldSize -> 3]& ], ControlPlacement -> 9}, {{$CellContext`\[Mu]$$, 0.0001}, Rational[1, 100000], 1, 0.00001, Appearance -> "Labeled", ControlPlacement -> 10}, {{$CellContext`m1$$, 1}, Dynamic[InputField[ Dynamic[$CellContext`m1$$, ($CellContext`m1$$ = If[ NumberQ[#], Clip[#, {0.001, Infinity}], 0.001])& ], Number, FieldSize -> 3]& ], ControlPlacement -> 11}, {{$CellContext`m2$$, 1}, Dynamic[InputField[ Dynamic[$CellContext`m2$$, ($CellContext`m2$$ = If[ NumberQ[#], Clip[#, {0.001, Infinity}], 0.001])& ], Number, FieldSize -> 3]& ], ControlPlacement -> 12}, {{$CellContext`m3$$, 1}, Dynamic[InputField[ Dynamic[$CellContext`m3$$, ($CellContext`m3$$ = If[ NumberQ[#], Clip[#, {0.001, Infinity}], 0.001])& ], Number, FieldSize -> 3]& ], ControlPlacement -> 13}, Row[{"", Grid[{{ Style["payoffs", Bold]}, { Manipulate`Place[1], Manipulate`Place[2], Manipulate`Place[3]}, { Manipulate`Place[4], Manipulate`Place[5], Manipulate`Place[6]}, { Manipulate`Place[7], Manipulate`Place[8], Manipulate`Place[9]}}, Alignment -> Left, Spacings -> {2, 0}], Button["Update", Increment[$CellContext`update$$], ImageSize -> {70, 50}], Grid[{{ Style["mutation rates", Bold], SpanFromLeft}, { "", SpanFromLeft}, { Manipulate`Place[10], SpanFromLeft}, { Manipulate`Place[11], Manipulate`Place[12], Manipulate`Place[13]}}, Alignment -> Left, Spacings -> {2, 0}]}, Spacer[30]]}, "Options" :> { SynchronousUpdating -> False, Alignment -> Center, TrackedSymbols :> {$CellContext`\[Mu]$$, $CellContext`update$$}}, "DefaultOptions" :> {}], ImageSizeCache->{673., {275., 280.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, Initialization:>({$CellContext`Simplex2DCoord[{ Pattern[$CellContext`x1, Blank[]], Pattern[$CellContext`x2, Blank[]], Pattern[$CellContext`x3, Blank[]]}] := $CellContext`x1 {0, 0} + $CellContext`x2 {1, 0} + $CellContext`x3 {1/2, Sqrt[3]/2}, $CellContext`trajectory[ Pattern[$CellContext`initialPoint, Blank[List]], Pattern[$CellContext`A, Blank[]], Pattern[$CellContext`M, Blank[]], Pattern[$CellContext`time, Blank[]]] := Module[{$CellContext`expChange, $CellContext`phaseSolutions}, \ $CellContext`expChange = $CellContext`expectedChange[{ $CellContext`x1[$CellContext`t], $CellContext`x2[$CellContext`t], $CellContext`x3[$CellContext`t]}, $CellContext`A, \ $CellContext`M]; $CellContext`phaseSolutions = NDSolve[{ Derivative[1][$CellContext`x1][$CellContext`t] == Part[$CellContext`expChange, 1], Derivative[1][$CellContext`x2][$CellContext`t] == Part[$CellContext`expChange, 2], Derivative[1][$CellContext`x3][$CellContext`t] == Part[$CellContext`expChange, 3], $CellContext`x1[0] == Part[$CellContext`initialPoint, 1], $CellContext`x2[0] == Part[$CellContext`initialPoint, 2], $CellContext`x3[0] == Part[$CellContext`initialPoint, 3]}, {$CellContext`x1, $CellContext`x2, $CellContext`x3}, \ {$CellContext`t, 0, $CellContext`time}]; Join[ ReplaceAll[ $CellContext`x1[$CellContext`t], $CellContext`phaseSolutions], ReplaceAll[ $CellContext`x2[$CellContext`t], $CellContext`phaseSolutions], ReplaceAll[ $CellContext`x3[$CellContext`t], $CellContext`phaseSolutions]]], \ $CellContext`expectedChange[ Pattern[$CellContext`x, Blank[]], Pattern[$CellContext`A, Blank[]], Pattern[$CellContext`M, Blank[]]] := Dot[$CellContext`M, $CellContext`x Dot[$CellContext`A, $CellContext`x]] - Dot[$CellContext`x, $CellContext`A, $CellContext`x] $CellContext`x, Attributes[Derivative] = {NHoldAll, ReadProtected}, $CellContext`tMax = 10, $CellContext`step = 0.1, $CellContext`Simplex2DInverseCoord[{ Pattern[$CellContext`y1, Blank[]], Pattern[$CellContext`y2, Blank[]]}] := {1, 0, 0} + $CellContext`y1 {-1, 1, 0} + $CellContext`y2 {(-1)/Sqrt[3], (-1)/Sqrt[3], 2/Sqrt[3]}, Attributes[Thick] = {ReadProtected}, Attributes[PlotRange] = {ReadProtected}, $CellContext`Jacob[{ Pattern[$CellContext`x, Blank[]], Pattern[$CellContext`y, Blank[]]}, Pattern[$CellContext`A, Blank[]], Pattern[$CellContext`M, Blank[]]] := Module[{$CellContext`x1, $CellContext`x2}, ReplaceAll[ Evaluate[ D[ Most[ $CellContext`expectedChange[{$CellContext`x1, $CellContext`x2, 1 - $CellContext`x1 - $CellContext`x2}, $CellContext`A, \ $CellContext`M]], {{$CellContext`x1, $CellContext`x2}}]], {$CellContext`x1 -> \ $CellContext`x, $CellContext`x2 -> $CellContext`y}]]}; Typeset`initDone$$ = True), SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], 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