(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 40937, 1373] NotebookOptionsPosition[ 36106, 1217] NotebookOutlinePosition[ 37489, 1259] CellTagsIndexPosition[ 37446, 1256] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Exercises lesson 3", "Title", CellChangeTimes->{{3.4323368555625*^9, 3.4323368600625*^9}, { 3.432915856640625*^9, 3.43291586221875*^9}, {3.432918644203125*^9, 3.432918649390625*^9}, {3.43353945928125*^9, 3.433539468421875*^9}}], Cell[CellGroupData[{ Cell["\<\ 1. Modify the sex ratio model developed in class to determine the optimum sex \ ratio from the perspective of either (1) the mother queen or (2) the workers \ in haplodiploid social Hymenoptera. Calculate the worker optimum both for the \ case where the mother queen mates with a single male or with two males. \ Assume that all males are produced by the mother queen.\ \>", "Subsection", CellChangeTimes->{{3.432915853421875*^9, 3.432915919265625*^9}, { 3.43291600403125*^9, 3.432916004546875*^9}, {3.432916133125*^9, 3.43291613521875*^9}, {3.432925063828125*^9, 3.4329250735625*^9}, { 3.432953114453125*^9, 3.432953122984375*^9}, {3.432954159359375*^9, 3.432954159609375*^9}, 3.432971627*^9, {3.43299095840625*^9, 3.43299096078125*^9}, {3.433539514875*^9, 3.433539666703125*^9}, { 3.433540884421875*^9, 3.433540894984375*^9}, {3.4335414224375*^9, 3.433541430328125*^9}}, ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell["\<\ 1.1 Optimum sex ratio from the perspective of the mother queen\ \>", "Subsubsection", CellChangeTimes->{{3.4335397096875*^9, 3.43353981459375*^9}, 3.4335408436875*^9}], Cell["\<\ aa = wild type, queen produces sex ratio (female investment) F Aa = rare mutant, queen produces sex ratio (female investment) F+d, with d \ small n = number of offspring produced We assume that the males are produced by the queen\ \>", "Text", CellChangeTimes->{{3.433541548234375*^9, 3.433541573703125*^9}, { 3.43354326240625*^9, 3.433543271359375*^9}}], Cell["\<\ When A gene is rare we need to consider 3 types of colonies: wild type colonies : aa x a freq 1-2pf-pm M type colonies : Aa x a freq 2pf P type colonies : aa x A freq pm If the frequency of the A gene in females and males is pf and pm, then when \ rare its frequency in wild type, M type and P type colonies are approx. \ 1-2pf-pm, 2pf and pm :\ \>", "Text", CellChangeTimes->{{3.433540561796875*^9, 3.4335405649375*^9}, { 3.43354093525*^9, 3.4335409745*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"Wfreq", "=", RowBox[{"1", "-", RowBox[{"2", "*", "pf"}], "-", "pm"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Mfreq", "=", RowBox[{"2", "*", "pf"}]}], ";"}], " "}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Pfreq", "=", "pm"}], ";"}], " ", "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"AmalesM", "=", RowBox[{ RowBox[{"(", RowBox[{"n", "/", "2"}], ")"}], RowBox[{"(", RowBox[{"1", "-", RowBox[{"(", RowBox[{"F", "+", "d"}], ")"}]}], ")"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"AmalesP", "=", "0"}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"AafemalesM", "=", RowBox[{ RowBox[{"(", RowBox[{"n", "/", "2"}], ")"}], RowBox[{"(", RowBox[{"F", "+", "d"}], ")"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"AafemalesP", "=", RowBox[{"n", "*", "F"}]}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"totmalesW", "=", RowBox[{"n", "*", RowBox[{"(", RowBox[{"1", "-", "F"}], ")"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"totfemalesW", "=", RowBox[{"n", "*", "F"}]}], ";"}]}], "Input", CellChangeTimes->{{3.433540513078125*^9, 3.43354051725*^9}, { 3.433541069171875*^9, 3.43354108*^9}, {3.43354115825*^9, 3.433541184765625*^9}, {3.433541227390625*^9, 3.433541228703125*^9}, { 3.433542786609375*^9, 3.433542788640625*^9}, {3.4335430905625*^9, 3.43354309734375*^9}, {3.4335431513125*^9, 3.4335431515*^9}, { 3.433543219546875*^9, 3.433543246796875*^9}}], Cell["\<\ Now let's calculate the frequency of the A gene in the next generation in \ males and females. For males: number of A males produced\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"pmng", "=", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"Mfreq", "*", "AmalesM"}], "+", RowBox[{"Pfreq", "*", "AmalesP"}]}], ")"}], "/", "totmalesW"}], "]"}]}]], "Input"], Cell[BoxData[ FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "d", "+", "F"}], ")"}], " ", "pf"}], RowBox[{ RowBox[{"-", "1"}], "+", "F"}]]], "Output", CellChangeTimes->{3.433540523984375*^9, 3.433541240515625*^9, 3.433542794203125*^9, 3.43354298996875*^9, 3.433543117328125*^9, 3.43354325009375*^9, 3.433544190984375*^9}] }, Open ]], Cell["For females: number of Aa females produced*0.5 ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"pfng", "=", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"(", RowBox[{"1", "/", "2"}], ")"}], "*", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"Pfreq", "*", "AafemalesP"}], "+", RowBox[{"Mfreq", "*", "AafemalesM"}]}], ")"}], "/", 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CellChangeTimes->{{3.433542594140625*^9, 3.43354262671875*^9}, { 3.43354339303125*^9, 3.433543413234375*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["1.2 Worker optimum sex ratio under single mating", "Subsubsection", CellChangeTimes->{{3.4335397096875*^9, 3.43353973996875*^9}, 3.433540849859375*^9}], Cell["\<\ aa = wild type, workers produce sex ratio (female investment) F Aa = rare mutant, workers produce sex ratio (female investment) F+d, with d \ small n = number of offspring produced We assume that the males are produced by the queen\ \>", "Text", CellChangeTimes->{{3.43354159096875*^9, 3.433541605703125*^9}, { 3.4335432805*^9, 3.433543280828125*^9}}], Cell["\<\ When A gene is rare we need to consider 3 types of colonies: wild type colonies : aa x a freq 1-2pf-pm M type colonies : Aa x a freq 2pf P type colonies : aa x A freq pm If the frequency of the A gene in females and males is pf and pm, then when \ rare its frequency in wild type, M type and P type colonies are approx. \ 1-2pf-pm, 2pf and pm :\ \>", "Text", CellChangeTimes->{{3.4335408096875*^9, 3.4335408195*^9}, { 3.433540991296875*^9, 3.433541003*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"Wfreq", "=", RowBox[{"1", "-", RowBox[{"2", "*", "pf"}], "-", "pm"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Mfreq", "=", RowBox[{"2", "*", "pf"}]}], ";"}], " "}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Pfreq", "=", "pm"}], ";"}], " ", "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"AmalesM", "=", RowBox[{ RowBox[{"(", RowBox[{"n", "/", "2"}], ")"}], RowBox[{"(", RowBox[{"1", "-", RowBox[{"(", RowBox[{"F", "+", RowBox[{"d", "/", "2"}]}], ")"}]}], ")"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"AmalesP", "=", "0"}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"AafemalesM", "=", RowBox[{ RowBox[{"(", RowBox[{"n", "/", "2"}], ")"}], RowBox[{"(", RowBox[{"F", "+", RowBox[{"d", "/", "2"}]}], ")"}]}]}], ";"}], 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For males: number of A males produced\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"pmng", "=", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"Mfreq", "*", "AmalesM"}], "+", RowBox[{"Pfreq", "*", "AmalesP"}]}], ")"}], "/", "totmalesW"}], "]"}]}]], "Input"], Cell[BoxData[ FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "2"}], "+", "d", "+", RowBox[{"2", " ", "F"}]}], ")"}], " ", "pf"}], RowBox[{"2", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "F"}], ")"}]}]]], "Output", CellChangeTimes->{3.433540732359375*^9, 3.433541388671875*^9, 3.43354330271875*^9, 3.433544191171875*^9}] }, Open ]], Cell["For females: number of Aa females produced*0.5 ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"pfng", "=", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"(", RowBox[{"1", "/", "2"}], ")"}], "*", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"Pfreq", "*", "AafemalesP"}], "+", RowBox[{"Mfreq", "*", "AafemalesM"}]}], ")"}], "/", "totfemalesW"}]}], 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