(************** Content-type: application/mathematica **************
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Notebook[{
Cell[BoxData[{
    \(ai =. \), "\n", 
    \(apop =. \), "\n", 
    \(R =. \), "\n", 
    \(Wi =. \), "\[IndentingNewLine]", 
    \(Cd =. \)}], "Input",
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Cell[TextData[StyleBox["This is Fisher's sex ratio problem.  We what to show \
that the solution apop=1/2 is a CSS.",
  FontSize->18]], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Wi = \(R\ \((1 - ai)\)\)\/Cd + \(\((R\ ai)\)\ \((R\ \((1 - apop)\))\)\)\
\/\(\(Cs\ Cd\ \((R\ apop)\)\)\/Cs\)\)], "Input",
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Cell[BoxData[
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"Output"]
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    \(Firstder = \[PartialD]\_ai Wi\)], "Input",
  ImageRegion->{{-0, 1}, {0, 1}}],

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Cell[BoxData[
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Cell[TextData[StyleBox["Note that, in the graph below, the first derivative \
is equal to zero and apop = 1/2, and that the slope is negative at that \
point.  Hence, the solution apop=1/2 is convergence stable.  That means that \
if the population is away from apop=1/2, mutants that are closer to 1/2 will \
be favored by selection, and the population will converge on the equilibrium \
of one half.  ",
  FontSize->18]], "Text"],

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Cell[BoxData[{
    \(\(R = 1;\)\), "\[IndentingNewLine]", 
    \(\(\(Cd = 0.1;\)\(\[IndentingNewLine]\)
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    \(Plot[Firstder, \ {apop, \ 0.2,  .8}]\)}], "Input"],

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