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Here we are interested in the \ question: how does the number of mates affect the proportion of resources \ that should be allocated by a hermaphrodite to male and females function in \ order to maximize individual fitness? prepared by Curt Lively, Indiana University. \ \>", "Text", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ "DEFINING THE VARIABLES...\na", StyleBox["i", FontVariations->{"CompatibilityType"->"Subscript"}], " = allocation to male function by the ith hermaphrodite in the population. \ \na = average allocation to male function in the remaining part of the \ population, excluding our target individual\nR = Total resource base for \ reproduction\nk = the number of mates\nV = The reproductive value though male \ fuction\nW", StyleBox["m", FontVariations->{"CompatibilityType"->"Subscript"}], " = Fitness through male function\nW", StyleBox["f", FontVariations->{"CompatibilityType"->"Subscript"}], " = Fitness though female function (assuming Bateman's principle).\nW", StyleBox["i", FontVariations->{"CompatibilityType"->"Subscript"}], " = Fitness of the ith individual (Wf + Wm)" }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[{ \(k =. \), "\n", \(a =. \), "\n", \(ai =. \), "\n", \(R =. \), "\n", \(V =. \)}], "Input"], Cell[TextData[{ "\nFIRST WE DEFINE FITNESS THROUGH FEMALE FUNCTION, W", StyleBox["f.", FontVariations->{"CompatibilityType"->"Subscript"}], "\n" }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ StyleBox["(", FontFamily->"Helvetica", FontColor->RGBColor[0, 0, 1]], StyleBox[ "Put the cursor anywhere in the bracket below and press \"SHIFT-RETURN.\" \ This will store W", FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 1]], StyleBox["f", FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 1], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" in memory)", FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell["Wf = R*(1 - ai)", "Input", AspectRatioFixed->True], Cell[BoxData[ \(\((1 - ai)\)\ R\)], "Output"] }, Open ]], Cell[TextData[{ "NEXT DEFINE FITNESS THROUGH MALE FUNCTION, W", StyleBox["m. ", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[ "V is the reproductive value of male function, which will be defined \ shortly.", FontColor->RGBColor[1, 0, 0]], "\n" }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ StyleBox["(", FontFamily->"Helvetica", FontColor->RGBColor[0, 0, 1]], StyleBox[ "Put the cursor anywhere in the bracket below and press \"SHIFT-RETURN.\")", FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell["Wm = V*R*ai", "Input", AspectRatioFixed->True], Cell[BoxData[ \(ai\ R\ V\)], "Output"] }, Open ]], Cell[TextData[ "\nNOW DEFINE THE VARIABLE, V, AS THE RATIO OF EGGS IN THE POPULATION TO \ SPERM IN THE POPULATION.\n"], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ StyleBox["(", FontFamily->"Helvetica", FontColor->RGBColor[0, 0, 1]], StyleBox[ "Put the cursor anywhere in the bracket below and press \"SHIFT-RETURN.\")", FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell["V = (k*(1 - a))/((k - 1)*a + ai)", "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(\((1 - a)\)\ k\)\/\(ai + a\ \((\(-1\) + k)\)\)\)], "Output"] }, Open ]], Cell[TextData[ "\nTOTAL FITNESS OF THE INDIVIDUAL IS THE SUM OF FITNESS THROUGH FEMALE + \ MALE FUNCTION.\n"], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ StyleBox["(", FontFamily->"Helvetica", FontColor->RGBColor[0, 0, 1]], StyleBox[ "Put the cursor anywhere in the bracket below and press \"SHIFT-RETURN.\")", FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell["Wi = Wf + Wm", "Input", AspectRatioFixed->True], Cell[BoxData[ \(\((1 - ai)\)\ R + \(\((1 - a)\)\ ai\ k\ R\)\/\(ai + a\ \((\(-1\) + \ k)\)\)\)], "Output"] }, Open ]], Cell[TextData[ "\nNOW CALCULATE THE FIRST DERIVATIVE OF INDIVIDUAL FITNESS WITH RESPECT TO \ ALLOCATION TO MALE FUNCTION.\n"], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ StyleBox["(", FontFamily->"Helvetica", FontColor->RGBColor[0, 0, 1]], StyleBox[ "Put the cursor anywhere in the bracket below and press \"SHIFT-RETURN.\")", FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell["Firstder = D[Wi, ai]", "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(-R\) - \(\((1 - a)\)\ ai\ k\ R\)\/\((ai + a\ \((\(-1\) + k)\))\)\^2 + \ \(\((1 - a)\)\ k\ R\)\/\(ai + a\ \((\(-1\) + k)\)\)\)], "Output"] }, Open ]], Cell[TextData[ "\nCALCULATE THE SECOND DERIVATIVE OF INDIVIDUAL FITNESS AS A FUNCTION OF ITS \ ALLOCATION TO MALE FUNCTION. Remember, local stability requires a negative \ second derivative\n"], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ StyleBox["(", FontFamily->"Helvetica", FontColor->RGBColor[0, 0, 1]], StyleBox[ "Put the cursor anywhere in the bracket below and press \"SHIFT-RETURN.\")", FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell["Secondder = D[Firstder, ai]", "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(2\ \((1 - a)\)\ ai\ k\ R\)\/\((ai + a\ \((\(-1\) + k)\))\)\^3 - \(2\ \ \((1 - a)\)\ k\ R\)\/\((ai + a\ \((\(-1\) + k)\))\)\^2\)], "Output"] }, Open ]], Cell[TextData["\nSIMPLIFY THE SECOND DERIVATIVE. "], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ StyleBox["(", FontFamily->"Helvetica", FontColor->RGBColor[0, 0, 1]], StyleBox[ "Put the cursor anywhere in the bracket below and press \"SHIFT-RETURN.\")", FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell["Simplify [Secondder]", "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(2\ \((\(-1\) + a)\)\ a\ \((\(-1\) + k)\)\ k\ R\)\/\((ai + a\ \ \((\(-1\) + k)\))\)\^3\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(CSS = \[PartialD]\_\(a, ai\)Wi\)], "Input"], Cell[BoxData[ \(\(ai\ k\ R\)\/\((ai + a\ \((\(-1\) + k)\))\)\^2 - \(k\ R\)\/\(ai + a\ \ \((\(-1\) + k)\)\) + \(2\ \((1 - a)\)\ ai\ \((\(-1\) + k)\)\ k\ R\)\/\((ai + \ a\ \((\(-1\) + k)\))\)\^3 - \(\((1 - a)\)\ \((\(-1\) + k)\)\ k\ R\)\/\((ai + \ a\ \((\(-1\) + k)\))\)\^2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[CSS]\)], "Input"], Cell[BoxData[ \(\(-\(\(\((\(-1\) + k)\)\ k\ \((\(-ai\) + a\ \((\(-1\) + 2\ ai + k)\))\)\ R\)\/\((ai + a\ \((\(-1\) + k)\))\)\^3\)\)\)], \ "Output"] }, Open ]], Cell[TextData[{ "\nNote that the second derivative is less that zero when the numerator is \ negative, which is true whenever k > 1 AND a < 1. Hence for k > 1 (i.e., \ more than one mate) and a < 1 (i.e., less than 100% allocation to male \ function in the population), we can find a locally stable ESS by setting a", StyleBox["i", FontVariations->{"CompatibilityType"->"Subscript"}], " = a and solving for the case when the first derivative equal to zero.\n\n\ SET a", StyleBox["i", FontVariations->{"CompatibilityType"->"Subscript"}], " = a. REMEMBER WHEN THE POPULATION IS AT THE ESS, INDIVIDUAL FITNESS \ SHOULD BE MAXIMIZED AT THE ESS.\n\n" }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ StyleBox["(", FontFamily->"Helvetica", FontColor->RGBColor[0, 0, 1]], StyleBox[ "Put the cursor anywhere in the bracket below and press \"SHIFT-RETURN.\")", FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell["\<\ ai = astar a = astar\ \>", "Input", AspectRatioFixed->True], Cell[BoxData[ \(astar\)], "Output"], Cell[BoxData[ \(astar\)], "Output"] }, Open ]], Cell[TextData[ "\nSET THE FIRST DERIVATIVE EQUAL TO ZERO AND SOLVE FOR a. \n"], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ StyleBox["(", FontFamily->"Helvetica", FontColor->RGBColor[0, 0, 1]], StyleBox[ "Put the cursor anywhere in the bracket below and press \"SHIFT-RETURN.\")", FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell["Solve [Firstder == 0, astar]", "Input", AspectRatioFixed->True], Cell[BoxData[ \({{astar \[Rule] \(\(-1\) + k\)\/\(\(-1\) + 2\ k\)}}\)], "Output"] }, Open ]], Cell[TextData[{ "\nHence the allocation to male function, a, at \nthe ESS is:\n\n", StyleBox[" a = (k-1)/(2k - 1)", FontColor->RGBColor[1, 0, 0]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[StyleBox["Now set the population allocation, a, at the ESS", FontFamily->"Helvetica", FontColor->RGBColor[0, 0, 1]]], "Text", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ StyleBox["(", FontFamily->"Helvetica", FontColor->RGBColor[0, 0, 1]], StyleBox[ "Put the cursor anywhere in the bracket below and press \"SHIFT-RETURN.\")", FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell["\<\ astar = (k - 1)/(2*k -1)\ \>", "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(\(-1\) + k\)\/\(\(-1\) + 2\ k\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[Secondder]\)], "Input"], Cell[BoxData[ \(\(\((2 - 4\ k)\)\ R\)\/\(\((\(-1\) + k)\)\ k\)\)], "Output"] }, Open ]], Cell[TextData[StyleBox["\nNote that the second derivative is negative, so our \ solution for a* is evolutionarily stable\n", FontFamily->"Helvetica", FontWeight->"Bold"]], "Text", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 1]], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[CSS]\)], "Input"], Cell[BoxData[ \(\(\((\(-3\) + 8\ k - 4\ k\^2)\)\ R\)\/\(\((\(-1\) + k)\)\ k\)\)], \ "Output"] }, Open ]], Cell[TextData[StyleBox["\nThe solution is also convergence stable. (don't \ worry if this does not make sense. I did not cover it.)\nNow let the \ individual allocation, ai, be a variable again. So (ai) is no longer equal \ to (a). The purpose is to graph individual fitness as a function of ai when \ the population is at the ESS. We would expect that Wi is maximized under \ this condition when ai converges on a.\n\nI'm also going to set the Resources \ equal to 1 unit.\n", FontFamily->"Helvetica", FontWeight->"Bold"]], "Text", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 1]], Cell[TextData[{ StyleBox["(", FontFamily->"Helvetica", FontColor->RGBColor[0, 0, 1]], StyleBox[ "Put the cursor anywhere in the bracket below and press \"SHIFT-RETURN.\")", FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell["\<\ ai = . R=1\ \>", "Input", AspectRatioFixed->True], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell[TextData[StyleBox[ "\nNow, for the crux of the bisquit. Let there be two mates (k = 2).\n", FontFamily->"Helvetica", FontWeight->"Bold"]], "Text", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 1]], Cell[TextData[{ StyleBox["(", FontFamily->"Helvetica", FontColor->RGBColor[0, 0, 1]], StyleBox[ "Put the cursor anywhere in the bracket below and press \"SHIFT-RETURN.\")", FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell["k=2", "Input", AspectRatioFixed->True], Cell[BoxData[ \(2\)], "Output"] }, Open ]], Cell[TextData[ "The ESS can be found for two mates by simplifying the population allocation, \ a."], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0, 0, 1], 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Linear increase in fitness with increasing allocation to female \ function. (This is a consequence of our assumtion that female function is \ limited only by resources.) REMEMBER: allocation to female fitness increases \ from right to left; it's equal to 1 - ai. Bueno, Now let's plot the relationship between total fitness and the \ allocation by the ith individual to male function. 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That is when the individual allocation is at the ESS.\n\n\n\n\ Now let's let the local mating population be large, say 100 mates." }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ StyleBox["(", FontFamily->"Helvetica", FontColor->RGBColor[0, 0, 1]], StyleBox[ "Put the cursor anywhere in the bracket below and press \"SHIFT-RETURN.\")", FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell["k=100", "Input", AspectRatioFixed->True], Cell[BoxData[ \(100\)], "Output"] }, Open ]], Cell[TextData[ "The new ESS for 100 mates is found by simplifying the parameter, a."], "Text",\ Evaluatable->False, AspectRatioFixed->True, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ StyleBox["(", FontFamily->"Helvetica", FontColor->RGBColor[0, 0, 1]], StyleBox[ "Put the cursor anywhere in the bracket below and press \"SHIFT-RETURN.\")", FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell["Simplify [a]", "Input", AspectRatioFixed->True], Cell[BoxData[ \(99\/199\)], "Output"] }, Open ]], Cell[TextData[{ "\nHence, the ESS is at 99/199, which is about 1/2.\n\nNow, as previously, \ let's first plot the relationship between fitness through ", StyleBox["male", FontVariations->{"Underline"->True}], " function and the allocation by the ith individual to male function." }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ StyleBox["(", FontFamily->"Helvetica", FontColor->RGBColor[0, 0, 1]], StyleBox[ "Put the cursor anywhere in the bracket below and press \"SHIFT-RETURN.\")", FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell[TextData[{ "Plot [{Wm}, {ai, 0, 1}, \nAxesLabel->{\"a", StyleBox["i", FontVariations->{"CompatibilityType"->"Subscript"}], "\",\"W", StyleBox["m", 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