(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 9.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 157, 7] NotebookDataLength[ 8117858, 167925] NotebookOptionsPosition[ 8012247, 164755] NotebookOutlinePosition[ 8013002, 164784] CellTagsIndexPosition[ 8012866, 164778] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Gametic selection, sex-ratio bias, and transitions between sex-determining \ systems\ \>", "Title", CellChangeTimes->{{3.6756461584547977`*^9, 3.675646190672323*^9}, { 3.675691343813216*^9, 3.675691344643425*^9}, {3.675718324413807*^9, 3.6757183349582043`*^9}, {3.675719587684537*^9, 3.675719616196001*^9}, 3.6759056672964783`*^9, 3.676676533093017*^9, 3.678542642367774*^9, { 3.678560549515456*^9, 3.678560568313982*^9}, 3.6820500476906357`*^9, { 3.6854453306884403`*^9, 3.685445347162414*^9}, 3.7180443827487497`*^9, { 3.7189766786378527`*^9, 3.718976679730826*^9}, {3.725136836370063*^9, 3.7251368503374023`*^9}},ExpressionUUID->"97d8766c-5237-4a0d-9ded-\ 3d8ba8ed1570"], Cell["Michael F Scott, Matthew M Osmond, Sarah P Otto", "Subtitle", CellChangeTimes->{{3.6756909919076443`*^9, 3.675690999875959*^9}, { 3.679939506599341*^9, 3.679939536485776*^9}, {3.683815470889019*^9, 3.683815474656125*^9}},ExpressionUUID->"cb3fbe32-2a9a-46fa-96fa-\ cde66bae3ef7"], Cell["\<\ Notes: 1) The notation in this notebook differs slightly from the text: here we do \ not use male and female symbols or sub- and super-scripts. 2) Throughout we mainly refer to an ancestral XY sex-determination system. By \ consistently flipping male and female labels, this is equivalent to an \ ancestral ZW system (noting that the heterogametic sex described in this \ notebook as XY would refer to heterogametic ZW females).\ \>", "Text", CellChangeTimes->{{3.718043951830217*^9, 3.718044092947372*^9}, { 3.718044123893067*^9, 3.718044151022295*^9}, {3.718044640505638*^9, 3.718044640705206*^9}, {3.71804793069186*^9, 3.7180479540503283`*^9}, { 3.71900369484517*^9, 3.719003697468656*^9}, {3.719954533103353*^9, 3.7199545338224573`*^9}, {3.724255556013454*^9, 3.724255694446392*^9}},ExpressionUUID->"cf73440c-40b2-4f1f-a3b2-\ 4bc3ad075914"], Cell[CellGroupData[{ Cell["General", "Section", CellChangeTimes->{{3.7190780194919767`*^9, 3.719078020713764*^9}, { 3.7190784512351217`*^9, 3.719078453507393*^9}, 3.720455564831493*^9},ExpressionUUID->"1f66c879-c56e-427e-af67-\ cd149a80361c"], Cell["Assumptions used to simplify the results:", "Text", CellChangeTimes->{{3.545561882571726*^9, 3.545561907371326*^9}, { 3.6967203158567533`*^9, 3.696720321197453*^9}},ExpressionUUID->"c8328382-ad67-462a-aac6-\ c4d4fa3382ee"], Cell[BoxData[ RowBox[{ RowBox[{"simpcond", "=", RowBox[{"Reduce", "[", RowBox[{"{", RowBox[{ RowBox[{"0", "<", "MAA"}], ",", RowBox[{"0", "<", "MAa"}], ",", RowBox[{"0", "<", "Maa"}], ",", RowBox[{"0", "<", "FAA"}], ",", RowBox[{"0", "<", "FAa"}], ",", RowBox[{"0", "<", "Faa"}], ",", RowBox[{"0", "<", "wAf"}], ",", RowBox[{"0", "<", "waf"}], ",", RowBox[{"0", "<", "wAm"}], ",", RowBox[{"0", "<", "wam"}], ",", RowBox[{"0", "<", "\[Alpha]m", "<", "1"}], ",", RowBox[{"0", "<", "\[Alpha]f", "<", "1"}], ",", RowBox[{"0", "\[LessEqual]", "Rm", "\[LessEqual]", FractionBox["1", "2"]}], ",", RowBox[{"0", "\[LessEqual]", "Rf", "\[LessEqual]", FractionBox["1", "2"]}]}], "}"}], "]"}]}], ";"}]], "Input", CellChangeTimes->{{3.5455618637051*^9, 3.545561881106247*^9}, { 3.545561911254034*^9, 3.545561916894547*^9}, {3.5455619842291718`*^9, 3.545562007676374*^9}, {3.545562291836935*^9, 3.545562292812436*^9}, 3.68522500095753*^9, {3.688401012923118*^9, 3.688401034851068*^9}, { 3.696720334925744*^9, 3.696720394634172*^9}}, ExpressionUUID -> "d0ab1aab-d032-4e43-9465-685b196e888a"] }, Closed]], Cell[CellGroupData[{ Cell["Recursion equations", "Section", CellChangeTimes->{{3.6756910197792797`*^9, 3.675691024113359*^9}, { 3.67569412935745*^9, 3.6756941314240303`*^9}, {3.675707349839982*^9, 3.675707351884247*^9}, {3.675709974013658*^9, 3.675709982915987*^9}, { 3.6757923469668207`*^9, 3.675792350124728*^9}, 3.676676690696854*^9, { 3.677350035605269*^9, 3.67735004069977*^9}, 3.677358409575016*^9, { 3.678845319012044*^9, 3.678845320057397*^9}, 3.692607135315508*^9, { 3.6926071764577847`*^9, 3.692607187122102*^9}, {3.693550212113646*^9, 3.693550235766447*^9}, {3.693554320708564*^9, 3.6935543247324944`*^9}, 3.693559930483079*^9, {3.718039868602145*^9, 3.718039880681121*^9}, { 3.719078448234187*^9, 3.719078450306834*^9}, {3.720455566676992*^9, 3.720455570340613*^9}},ExpressionUUID->"6975c5b3-4a2d-49cd-b453-\ d1cd06314bec"], Cell[CellGroupData[{ Cell["Haploid competition", "Subsection", CellChangeTimes->{{3.676680390552682*^9, 3.676680393224043*^9}, { 3.718041441611265*^9, 3.718041442986106*^9}},ExpressionUUID->"a5040726-719f-4c5b-a70c-\ 8fa26911e067"], Cell[TextData[{ "We start with haploid competition in each sex separately (e.g., pollen \ competition for eggs). We assume that the allele at an autosomal locus ", StyleBox["A", FontWeight->"Bold"], " determines competitive ability, with relative fitnesses wAf and waf in \ females and wAm and wam in males (in the text this is, e.g., ", Cell[BoxData[ FormBox[ SuperscriptBox[ SubscriptBox["w", "A"], "female"], TraditionalForm]],ExpressionUUID-> "6d56d0dc-3f92-4af3-9eca-ac286ada62aa"], "). Let the frequency of each gamete from each sex before selection be \ denoted by four letters, the first being the allele at the ancestral SDR (X \ or Y), the second being the allele at the selected locus (A or a), the third \ being the allele at the novel SDR (M or m), and the fourth indicating which \ sex the gamete came from (m or f). E.g., XAMf is the frequency of the X-A-M \ genotype among eggs (in the text this is ", Cell[BoxData[ FormBox[ SuperscriptBox[ SubscriptBox["x", "1"], "female"], TraditionalForm]],ExpressionUUID-> "fed18914-d4ef-4470-97bd-27d6050677bc"], "). Let the frequencies after haploid selection end in an s, e.g., XAMfs. 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3.676680492557345*^9, 3.676680499967092*^9}},ExpressionUUID->"b66bf2cb-36ca-4ee6-b847-\ d993896f4537"], Cell[TextData[{ "Let the probability that a zygote develops into a female be denoted with k \ followed by ", StyleBox["four", Background->RGBColor[1, 1, 0]], " letters: the first two indicating the ancestral SDR genotype, and the \ second two denoting the novel SDR genotype. For example, kXYMm is the \ probabiliy that a zygote with XY at the ancestral SDR and Mm at the novel SDR \ develops into a female. Let the frequency of zygotes be denoted by the two \ gamete genotypes followed by the sex, e.g., XAMYamfemale is the frequency of \ zygotes that are female and composed of X-A-M and Y-a-M gametes (it does not \ matter which came from which parent, i.e., XAM could come from the mother or \ father). Then, after random mating the frequencies of the diploid genotypes \ are" }], "Text", CellChangeTimes->{{3.675646094888542*^9, 3.675646099390658*^9}, { 3.675646217785323*^9, 3.675646250704771*^9}, {3.67570268816877*^9, 3.6757027045136023`*^9}, {3.676678026615859*^9, 3.676678026799192*^9}, { 3.7180401684994783`*^9, 3.718040181027934*^9}, {3.718040219709977*^9, 3.718040227525466*^9}, {3.71804156102256*^9, 3.718041609297316*^9}, { 3.7180418129470377`*^9, 3.718041906140842*^9}, {3.718041954949628*^9, 3.718042112052557*^9}, {3.718042144558155*^9, 3.718042185415447*^9}, { 3.7180428865312653`*^9, 3.71804289044309*^9}, 3.72425576107549*^9},ExpressionUUID->"ef6ac1e3-4fcd-4990-81ef-\ 9ee6eab28c55"], Cell[BoxData[ RowBox[{ RowBox[{"(*", RowBox[{"MM", " ", "Homozygotes"}], "*)"}], "\[IndentingNewLine]", RowBox[{"(*", RowBox[{"XM", "-", RowBox[{"XM", " ", "females"}]}], "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"XAMXAMfemale", "=", RowBox[{"kXXMM", " ", RowBox[{"(", RowBox[{"XAMfs", 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Thus, either the numerators and denominators must be both \ positive or both negative, the equilibrium will not be valid otherwise. 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impossible to \ satisfy the intercept condition for stability." }], "Text", CellChangeTimes->{{3.697040178274156*^9, 3.6970402312832403`*^9}, { 3.697041302189094*^9, 3.697041357107964*^9}},ExpressionUUID->"5500e276-2c37-4c2b-9fb0-\ 2a78c8e12d09"], Cell["\<\ We conclude that only under the first validity conditions (equivalent to \ overdominance) is stability and validity of the equilibrium possible. \ Furthermore, under the first validity conditions, we are guaranteed that the \ intercept is positive at \[Lambda]=1.\ \>", "Text", CellChangeTimes->{{3.6884006558966312`*^9, 3.688400823724429*^9}, 3.697040255881069*^9},ExpressionUUID->"bae79930-edb5-492c-a656-\ a8e0e42ab352"], Cell["\<\ To complete the proof, we show below that the slope at \[Lambda]=1 provides \ no further conditions on stability:\ \>", "Text", CellChangeTimes->{{3.688394493292367*^9, 3.68839450118009*^9}, { 3.6970413637162867`*^9, 3.6970413962664757`*^9}},ExpressionUUID->"954dcb44-ca7c-422d-b3fe-\ 56e62f082a30"], 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We conclude that in no case can equilC0 represent a stable equilibrium.\ \>", "Text", CellChangeTimes->{{3.697043057134006*^9, 3.697043084507807*^9}, { 3.719064154990081*^9, 3.719064157958358*^9}},ExpressionUUID->"15d68ddc-0dbc-4064-8578-\ fd9f32b9ce5e"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Weak selection", "Subsection", CellChangeTimes->{{3.718980550060027*^9, 3.718980551890119*^9}, { 3.7189807630587387`*^9, 3.7189807644377193`*^9}},ExpressionUUID->"d74a74bf-cfa9-4cc1-80bd-\ c104fb27fa21"], Cell[CellGroupData[{ Cell["Assumptions", "Subsubsection", CellChangeTimes->{{3.719517732168395*^9, 3.719517733632373*^9}},ExpressionUUID->"29f8666b-218c-47bc-96df-\ 1220c80fb139"], Cell["\<\ In this section we will assume weak selection in both haploids and diploids\ \>", "Text", CellChangeTimes->{{3.675694205010009*^9, 3.675694211923259*^9}, 3.67570298472752*^9, {3.6757030454432507`*^9, 3.6757030568360033`*^9}, { 3.71897823052678*^9, 3.7189782329437923`*^9}, {3.718978635166725*^9, 3.718978643531952*^9}},ExpressionUUID->"08174312-9ce0-4234-88ec-\ 8e7845f79079"], Cell[BoxData[ RowBox[{ RowBox[{"weaksel", "=", RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{"MAA", "\[Rule]", RowBox[{"1", "+", RowBox[{"sAm", " ", "\[Epsilon]"}]}]}], ",", "\[IndentingNewLine]", RowBox[{"MAa", "\[Rule]", RowBox[{"1", "+", RowBox[{"hAm", " ", "sAm", " ", "\[Epsilon]"}]}]}], ",", "\[IndentingNewLine]", RowBox[{"Maa", "\[Rule]", "1"}], ",", "\[IndentingNewLine]", RowBox[{"FAA", "\[Rule]", RowBox[{"1", "+", RowBox[{"sAf", " ", "\[Epsilon]"}]}]}], ",", "\[IndentingNewLine]", RowBox[{"FAa", "\[Rule]", RowBox[{"1", "+", RowBox[{"hAf", " ", "sAf", " ", "\[Epsilon]"}]}]}], ",", "\[IndentingNewLine]", RowBox[{"Faa", "\[Rule]", "1"}], ",", "\[IndentingNewLine]", RowBox[{"wAf", "\[Rule]", RowBox[{"1", "+", RowBox[{"tf", " ", "\[Epsilon]"}]}]}], ",", "\[IndentingNewLine]", RowBox[{"waf", "\[Rule]", "1"}], ",", "\[IndentingNewLine]", RowBox[{"wAm", "\[Rule]", RowBox[{"1", "+", RowBox[{"tm", " ", "\[Epsilon]"}]}]}], ",", "\[IndentingNewLine]", RowBox[{"wam", "\[Rule]", "1"}], ",", "\[IndentingNewLine]", RowBox[{"\[Alpha]m", "\[Rule]", RowBox[{ RowBox[{"1", "/", "2"}], "+", RowBox[{"\[Alpha]1m", " ", "\[Epsilon]"}]}]}], ",", "\[IndentingNewLine]", RowBox[{"\[Alpha]f", "\[Rule]", RowBox[{ RowBox[{"1", "/", "2"}], "+", RowBox[{"\[Alpha]1f", " ", "\[Epsilon]"}]}]}]}], "\[IndentingNewLine]", "}"}]}], ";"}]], "Input", CellChangeTimes->{{3.67564806523287*^9, 3.675648221558539*^9}, { 3.675694367971285*^9, 3.6756943845803022`*^9}, 3.6757029847280283`*^9, { 3.693574530059822*^9, 3.693574549458498*^9}, {3.6935749735257874`*^9, 3.693574996145659*^9}}, ExpressionUUID -> "0c8d9ff3-9d46-4025-a242-9135459b9b8c"], Cell["\<\ All selection terms are of order \[Epsilon]. We will solve for genotype \ frequencies using a Taylor Series. 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Cell[CellGroupData[{ Cell["The relevant quadratic", "Subsubsection", CellChangeTimes->{{3.7266032144292192`*^9, 3.7266032251665697`*^9}},ExpressionUUID->"0637f7eb-1f30-4d8b-88a9-\ df4afe47dc68"], Cell["\<\ Let us instead track the sum of the mutant-selected allele haplotypes across \ X and Y backgrounds in each sex, XAm+YAm (call this S), as well as the \ difference, Xam-Yam (call this D): \ \>", "Text", CellChangeTimes->{{3.726114043787024*^9, 3.726114150846752*^9}, { 3.7261173847562027`*^9, 3.726117397859269*^9}, {3.726117546771161*^9, 3.7261175851839314`*^9}, 3.726118175384413*^9, {3.726603762128442*^9, 3.726603877730934*^9}},ExpressionUUID->"4f11c9f2-d82e-4c8d-b3c5-\ 0a658f0aa06b"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"subsumdif", "=", RowBox[{"Flatten", "[", RowBox[{"Solve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"SAmf", "==", RowBox[{"YAmf", "+", "XAmf"}]}], ",", RowBox[{"Samf", "==", RowBox[{"Yamf", "+", "Xamf"}]}], ",", RowBox[{"DAmf", "\[Equal]", RowBox[{"XAmf", 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RowBox[{ RowBox[{"-", FractionBox["DAmf", "2"]}], "+", FractionBox["SAmf", "2"]}]}], ",", RowBox[{"Yamf", "\[Rule]", RowBox[{ RowBox[{"-", FractionBox["Damf", "2"]}], "+", FractionBox["Samf", "2"]}]}], ",", RowBox[{"XAmm", "\[Rule]", FractionBox[ RowBox[{"DAmm", "+", "SAmm"}], "2"]}], ",", RowBox[{"Xamm", "\[Rule]", FractionBox[ RowBox[{"Damm", "+", "Samm"}], "2"]}], ",", RowBox[{"YAmm", "\[Rule]", RowBox[{ RowBox[{"-", FractionBox["DAmm", "2"]}], "+", FractionBox["SAmm", "2"]}]}], ",", RowBox[{"Yamm", "\[Rule]", RowBox[{ RowBox[{"-", FractionBox["Damm", "2"]}], "+", FractionBox["Samm", "2"]}]}]}], "}"}]], "Output", CellChangeTimes->{{3.726603882436138*^9, 3.7266039113638973`*^9}, 3.726690847908771*^9, 3.726919684264328*^9, 3.726919919415851*^9, 3.7269469107950497`*^9, 3.7329767428692503`*^9, 3.7355849550675497`*^9}, ExpressionUUID -> "93ce139a-b320-4e0e-bfc1-a839d5d74054"] }, Closed]], Cell["\<\ This is just a translation of variables with no loss of information.\ \>", "Text", CellChangeTimes->{{3.726604148760791*^9, 3.7266041635243196`*^9}, 3.7266048791829853`*^9},ExpressionUUID->"b9ee7e83-9954-44f5-9235-\ 015a6e42999a"], Cell["\<\ Calculating the Jacobian matrix for this new set of variables: \ \>", "Text", CellChangeTimes->{{3.726117588023155*^9, 3.726117608638603*^9}, 3.7261181753879957`*^9},ExpressionUUID->"a0681dfe-56ed-4f09-8720-\ aaa1cdbe4ff3"], Cell[BoxData[{ RowBox[{ RowBox[{"eqsmutNEW", "=", RowBox[{ RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"nextXAmf", "+", "nextYAmf"}], ",", RowBox[{"nextXamf", "+", "nextYamf"}], ",", RowBox[{"nextXAmf", "-", "nextYAmf"}], ",", RowBox[{"nextXamf", "-", "nextYamf"}], ",", RowBox[{"nextXAmm", "+", "nextYAmm"}], ",", RowBox[{"nextXamm", "+", "nextYamm"}], ",", RowBox[{"nextXAmm", "-", "nextYAmm"}], ",", RowBox[{"nextXamm", "-", "nextYamm"}]}], "}"}], "/.", "SUBS"}], "/.", "subsumdif"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"matExtFullNEW", "=", RowBox[{ RowBox[{ RowBox[{ RowBox[{"Transpose", "[", 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CellChangeTimes->{{3.694892752137855*^9, 3.69489297225054*^9}, 3.6948931884875803`*^9, {3.69704338443299*^9, 3.697043416822537*^9}, 3.697043451334461*^9, {3.710776017690818*^9, 3.71077601791486*^9}, { 3.710776081032503*^9, 3.7107760848717413`*^9}, {3.710776240800933*^9, 3.7107762562940483`*^9}, 3.719079279679315*^9, {3.726113354740488*^9, 3.726113445429648*^9}, {3.7261135062445993`*^9, 3.726113539105657*^9}, { 3.72611415831678*^9, 3.72611418764834*^9}, {3.726114433851915*^9, 3.7261144438497887`*^9}, {3.726116641069388*^9, 3.726116676551949*^9}, { 3.7261167636496077`*^9, 3.726116768512115*^9}, {3.72611693091066*^9, 3.7261169966289873`*^9}, {3.726117078992219*^9, 3.7261170827939672`*^9}, { 3.726117119609941*^9, 3.7261171271936417`*^9}, {3.7261174856175327`*^9, 3.7261174999753447`*^9}, {3.726117614161941*^9, 3.7261176264595747`*^9}, { 3.7261176832076483`*^9, 3.726117693812958*^9}, {3.726117800000971*^9, 3.726117804855088*^9}, {3.726117987308525*^9, 3.7261179913322678`*^9}, 3.726118175388443*^9, 3.7266039355864763`*^9}, ExpressionUUID -> "8231e29e-4b57-4fd0-af8f-bd7d9f936d2d"], Cell["\<\ The advantage of this new variable system is that it now has a block (2x2) by \ (2x2) by (4x4) form for a neo-W into a neo-Y system (a \ \[OpenCurlyDoubleQuote]1\[CloseCurlyDoubleQuote] in the following means there \ is a potential non-zero entry):\ \>", "Text", CellChangeTimes->{{3.7261142065949497`*^9, 3.726114278856094*^9}, { 3.726114527827333*^9, 3.726114531546603*^9}, {3.72611656587186*^9, 3.726116573006241*^9}, {3.726117644297414*^9, 3.726117665893461*^9}, 3.726118175389727*^9, {3.7266039865808077`*^9, 3.7266040001842527`*^9}, { 3.7266041913095207`*^9, 3.7266042157225847`*^9}},ExpressionUUID->"c70f8714-028e-403c-af98-\ 012676d41706"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"PossibleZeroQ", "[", RowBox[{"matExtFullNEW", "/.", RowBox[{"k", "\[Rule]", "1"}]}], "]"}], "/.", RowBox[{"True", "\[Rule]", "0"}]}], "/.", RowBox[{"False", "\[Rule]", "1"}]}], "//", "MatrixForm"}]], "Input", CellChangeTimes->{{3.726113797634996*^9, 3.7261138343050547`*^9}, { 3.726116695925877*^9, 3.726116698491214*^9}, {3.726116934533935*^9, 3.726116934644672*^9}, 3.7261176329420443`*^9, 3.726118175389933*^9}, ExpressionUUID -> "84c2f892-4377-4d4e-8901-866a0bd5b15f"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "1", "0", "0", "1", "1", "0", "0"}, {"1", "1", "0", "0", "1", "1", "0", "0"}, {"1", "1", "1", "1", "1", "1", "1", "1"}, {"1", "1", "1", "1", "1", "1", "1", "1"}, {"0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", "0"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{3.72660400236172*^9, 3.726690891689684*^9, 3.726919684415346*^9, 3.726919971626739*^9, 3.726946946644556*^9, 3.732976777957658*^9, 3.735585001890407*^9}, ExpressionUUID -> "90bb06ef-79e8-4d9a-a5da-cd1f730e8dc7"] }, Closed]], Cell["\<\ The 2x2 in the top left describes the dynamics of the sum total frequencies \ of the neo-W-A and neo-W-a in females, summed over the old-X and old-Y \ backgrounds. The 2x2 below and right of it describes the dynamics of the \ difference between old-X and old-Y backgrounds in the frequencies of the \ neo-W-A and neo-W-a types in females.\ \>", "Text", CellChangeTimes->{{3.726115746523552*^9, 3.7261157660659037`*^9}, { 3.726116275076853*^9, 3.7261162991725073`*^9}, {3.7261177797880497`*^9, 3.726117782232609*^9}, 3.726118175393482*^9, {3.7266040395672483`*^9, 3.7266040664783297`*^9}, {3.7266049115197268`*^9, 3.726604986402114*^9}, { 3.726605124606518*^9, 3.726605124686276*^9}},ExpressionUUID->"d2e2c7c4-06ce-431a-802a-\ c1685e0515c2"], Cell["The same thing goes for a neo-Y invading an XY system.", "Text", CellChangeTimes->{{3.726604185339794*^9, 3.72660418624485*^9}, { 3.726604221146559*^9, 3.726604224218453*^9}, {3.726604262566516*^9, 3.7266042728846073`*^9}},ExpressionUUID->"e3d5108c-5219-432a-9015-\ 2dc3000bb8c5"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"PossibleZeroQ", "[", RowBox[{"matExtFullNEW", "/.", RowBox[{"k", "\[Rule]", "0"}]}], "]"}], "/.", RowBox[{"True", "\[Rule]", "0"}]}], "/.", RowBox[{"False", "\[Rule]", "1"}]}], "//", "MatrixForm"}]], "Input", CellChangeTimes->{{3.726604179054022*^9, 3.7266041791650553`*^9}}, ExpressionUUID -> "a4ae9714-342a-4506-8038-2f307e3d27a4"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", "0"}, {"1", "1", "0", "0", "1", "1", "0", "0"}, {"1", "1", "0", "0", "1", "1", "0", "0"}, {"1", "1", "1", "1", "1", "1", "1", "1"}, {"1", "1", "1", "1", "1", "1", "1", "1"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{3.726604179592861*^9, 3.726690891932214*^9, 3.726919684496687*^9, 3.7269199716884212`*^9, 3.726946946825272*^9, 3.73297677800738*^9, 3.735585002116489*^9}, ExpressionUUID -> "4d80cea6-3460-493a-b500-a7f7624c4cde"] }, Closed]], Cell["\<\ Now the bottom right 2x2 describes the dynamics of the difference between \ old-X and old-Y backgrounds in the frequencies of the neo-Y-A and neo-Y-a \ types in males. The 2x2 above and to the left it describes the sum total \ frequencies of the neo-Y-A and neo-Y-a in males, summed over the old-X and \ old-Y backgrounds.\ \>", "Text", CellChangeTimes->{{3.726605028036524*^9, 3.726605146442978*^9}},ExpressionUUID->"7f2bd2a3-ac53-495f-99eb-\ ff9a9e7e3714"], Cell["\<\ Now, the difference cannot grow faster than the sum because the new mutant \ type is assumed rare and cannot become negative in frequency. So at the very \ worst, the difference might increase until one of the old-X or old-Y types \ becomes exceedingly rare, but at that point the difference and the sum would \ have to grow at the same rate. Consequently, the behaviour of the system is \ determined by the behaviour of the sum. We can therefore look only at the \ eigenvalues emerging from the 2x2 Jacobian describing the changes in the \ summed frequencies over X and Y backgrounds of the m-A and m-a haplotypes in \ the sex that m makes.\ \>", "Text", CellChangeTimes->{{3.726604288024499*^9, 3.7266043570231028`*^9}, { 3.726604396793913*^9, 3.726604459531609*^9}, {3.726604759989481*^9, 3.726604761211133*^9}, {3.726604803364719*^9, 3.7266048104526987`*^9}, { 3.7266051529267197`*^9, 3.726605169484251*^9}},ExpressionUUID->"ec89b3aa-39e4-426a-b27d-\ f33b8272214d"], Cell["\<\ The relevant part of the characteristic polynomial for a neo-W is thus\ \>", "Text", CellChangeTimes->{{3.7266051845343018`*^9, 3.7266052042715483`*^9}},ExpressionUUID->"4bc2519e-2cfe-4532-b419-\ 8e09a1540f54"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"nextXAmf", "+", "nextYAmf"}], ",", RowBox[{"nextXamf", "+", "nextYamf"}]}], "}"}], "/.", "SUBS"}], "/.", "subsumdif"}], ";"}], 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or \[OpenCurlyDoubleQuote]a\[CloseCurlyDoubleQuote] from the father (first \ row) or \[OpenCurlyDoubleQuote]a\[CloseCurlyDoubleQuote] from the mother and \ either \[OpenCurlyDoubleQuote]A\[CloseCurlyDoubleQuote] or \ \[OpenCurlyDoubleQuote]a\[CloseCurlyDoubleQuote] from the father (second \ row), conditioning on the fact that an X must have been inherited from the \ dad (i.e., dividing by the frequency of female zygotes):\ \>", "Text", CellChangeTimes->{{3.697135349044276*^9, 3.697135353715836*^9}, { 3.697135991414695*^9, 3.69713600071249*^9}, {3.697136079571793*^9, 3.6971361031227503`*^9}, {3.697158803516185*^9, 3.697158833675169*^9}, { 3.724293018616994*^9, 3.7242930196471033`*^9}, {3.724293111758217*^9, 3.724293231633936*^9}},ExpressionUUID->"da88ff91-e0c0-4e75-80e7-\ 100b11775c62"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"wbarDipFemale", "=", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"pXf", RowBox[{"(", RowBox[{ RowBox[{"FAA", " ", FractionBox["wAf", "wbarHapFemale"], " ", RowBox[{"(", FractionBox[ RowBox[{"pXm", RowBox[{"(", RowBox[{"1", "-", "q"}], ")"}], " ", FractionBox["wAm", "wbarHapMale"]}], "freqFemale"], ")"}]}], "+", RowBox[{"FAa", " ", FractionBox["wAf", "wbarHapFemale"], " ", RowBox[{"(", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"1", "-", "pXm"}], ")"}], RowBox[{"(", RowBox[{"1", "-", "q"}], ")"}], " ", FractionBox["wam", "wbarHapMale"]}], "freqFemale"], ")"}]}]}], ")"}]}], "+", "\[IndentingNewLine]", RowBox[{ RowBox[{"(", RowBox[{"1", "-", "pXf"}], ")"}], RowBox[{"(", RowBox[{ RowBox[{"FAa", " ", FractionBox["waf", "wbarHapFemale"], " ", RowBox[{"(", FractionBox[ RowBox[{"pXm", RowBox[{"(", RowBox[{"1", "-", "q"}], ")"}], " ", FractionBox["wAm", "wbarHapMale"]}], "freqFemale"], ")"}]}], "+", RowBox[{"Faa", " ", FractionBox["waf", "wbarHapFemale"], " ", RowBox[{"(", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"1", "-", "pXm"}], ")"}], RowBox[{"(", RowBox[{"1", "-", "q"}], ")"}], " ", FractionBox["wam", "wbarHapMale"]}], "freqFemale"], ")"}]}]}], ")"}]}]}], "/.", "SUBS"}], "/.", "subequil"}], "//", "Simplify"}]}]], "Input", CellChangeTimes->{{3.697134772385618*^9, 3.697134812070602*^9}, { 3.697135080111429*^9, 3.6971351315874557`*^9}, {3.697135239834922*^9, 3.6971352704525843`*^9}, {3.697135313134502*^9, 3.697135341637377*^9}, { 3.697136032963991*^9, 3.697136037774029*^9}, {3.697136099803734*^9, 3.697136114355753*^9}, {3.6971527940895443`*^9, 3.697152796899786*^9}, { 3.6971528296409693`*^9, 3.697152860281275*^9}, {3.697152905278425*^9, 3.697152905656858*^9}, 3.6971576805668993`*^9, {3.6971577786769447`*^9, 3.6971577808951063`*^9}, {3.6971578259228983`*^9, 3.697157834381138*^9}, { 3.697158773602851*^9, 3.6971587971546926`*^9}, {3.697158837743108*^9, 3.6971588506223183`*^9}, {3.724292216621266*^9, 3.724292220523419*^9}, { 3.724292387408206*^9, 3.724292388783979*^9}, {3.724293027981629*^9, 3.7242930284365807`*^9}, 3.724293147041274*^9, {3.724293524397935*^9, 3.72429352674609*^9}, {3.724293662191028*^9, 3.724293662255575*^9}}, ExpressionUUID -> "78e6117b-90fe-4a3e-8799-3bc82988c6ce"], Cell[BoxData[ FractionBox[ RowBox[{ RowBox[{"Faa", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "pXf"}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "pXm"}], ")"}], " ", "waf", " ", "wam"}], "+", RowBox[{"FAA", " ", "pXf", " ", "pXm", " ", "wAf", " ", "wAm"}], "+", RowBox[{"FAa", " ", RowBox[{"(", RowBox[{ RowBox[{"pXm", " ", "waf", " ", "wAm"}], "-", RowBox[{"pXf", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "pXm"}], ")"}], " ", "wAf", " ", "wam"}], "+", RowBox[{"pXm", " ", "waf", " ", "wAm"}]}], ")"}]}]}], ")"}]}]}], RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "pXf"}], ")"}], " ", "waf"}], "-", RowBox[{"pXf", " ", "wAf"}]}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "pXm"}], ")"}], " ", "wam"}], "-", RowBox[{"pXm", " ", "wAm"}]}], ")"}]}]]], "Output", CellChangeTimes->{3.724292389339031*^9, 3.724293029482667*^9, 3.724293527457349*^9, 3.72429368116224*^9, 3.725116344185381*^9, 3.7251359781097937`*^9, 3.725385438393711*^9, 3.725631722530265*^9, 3.7257168235902433`*^9, 3.726605529449945*^9, 3.7266909011968307`*^9, 3.7269196851454487`*^9, 3.7269199824909973`*^9, 3.726946954588846*^9, 3.7329767854059267`*^9, 3.7355850119502897`*^9}, ExpressionUUID -> "fee12b14-bcab-4c52-b616-27cc2d495caf"] }, Closed]], Cell["\<\ Similarly, the mean fitness of a diploid male can be written as:\ \>", "Text", CellChangeTimes->{{3.697135349044276*^9, 3.697135353715836*^9}, { 3.697135991414695*^9, 3.69713600071249*^9}, {3.697136079571793*^9, 3.6971361031227503`*^9}, {3.697158803516185*^9, 3.697158833675169*^9}, { 3.724293018616994*^9, 3.7242930196471033`*^9}, {3.72429323430059*^9, 3.7242932410490026`*^9}},ExpressionUUID->"da88ff91-e0c0-4e75-80e7-\ 100b11775c62"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"wbarDipMale", "=", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"pXf", RowBox[{"(", RowBox[{ RowBox[{"MAA", " ", FractionBox["wAf", "wbarHapFemale"], " ", RowBox[{"(", FractionBox[ RowBox[{"pYm", " ", "q", " ", FractionBox["wAm", "wbarHapMale"]}], "freqMale"], ")"}]}], "+", RowBox[{"MAa", " ", FractionBox["wAf", "wbarHapFemale"], " ", RowBox[{"(", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"1", "-", "pYm"}], ")"}], "q", " ", FractionBox["wam", "wbarHapMale"]}], "freqMale"], ")"}]}]}], ")"}]}], "+", RowBox[{ RowBox[{"(", RowBox[{"1", "-", "pXf"}], ")"}], RowBox[{"(", RowBox[{ RowBox[{"MAa", " ", FractionBox["waf", "wbarHapFemale"], " ", RowBox[{"(", FractionBox[ RowBox[{"pYm", " ", "q", " ", FractionBox["wAm", "wbarHapMale"]}], "freqMale"], ")"}]}], "+", RowBox[{"Maa", " ", FractionBox["waf", "wbarHapFemale"], " ", RowBox[{"(", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"1", "-", "pYm"}], ")"}], "q", " ", FractionBox["wam", "wbarHapMale"]}], "freqMale"], ")"}]}]}], ")"}]}]}], "/.", "SUBS"}], "/.", "subequil"}], "//", "Simplify"}]}]], "Input", CellChangeTimes->{{3.697134772385618*^9, 3.697134812070602*^9}, { 3.697135080111429*^9, 3.6971351315874557`*^9}, {3.697135239834922*^9, 3.6971352704525843`*^9}, {3.697135313134502*^9, 3.697135341637377*^9}, { 3.697136032963991*^9, 3.697136037774029*^9}, {3.697136099803734*^9, 3.697136114355753*^9}, {3.6971527940895443`*^9, 3.697152796899786*^9}, { 3.6971528296409693`*^9, 3.697152860281275*^9}, {3.697152905278425*^9, 3.697152905656858*^9}, 3.6971576805668993`*^9, {3.6971577786769447`*^9, 3.6971577808951063`*^9}, {3.6971578259228983`*^9, 3.697157834381138*^9}, { 3.697158773602851*^9, 3.6971587971546926`*^9}, {3.697158837743108*^9, 3.6971588506223183`*^9}, {3.724292216621266*^9, 3.724292220523419*^9}, { 3.724292387408206*^9, 3.724292388783979*^9}, {3.724293027981629*^9, 3.7242930284365807`*^9}, {3.724293244216419*^9, 3.724293294060156*^9}, { 3.724293530315403*^9, 3.724293532936624*^9}, {3.724293669597776*^9, 3.7242936698011093`*^9}}, ExpressionUUID -> "78e6117b-90fe-4a3e-8799-3bc82988c6ce"], Cell[BoxData[ FractionBox[ RowBox[{ RowBox[{"Maa", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "pXf"}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "pYm"}], ")"}], " ", "waf", " ", "wam"}], "+", RowBox[{"MAA", " ", "pXf", " ", "pYm", " ", "wAf", " ", "wAm"}], "+", RowBox[{"MAa", " ", RowBox[{"(", RowBox[{ RowBox[{"pYm", " ", "waf", " ", "wAm"}], "-", RowBox[{"pXf", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "pYm"}], ")"}], " ", "wAf", " ", "wam"}], "+", RowBox[{"pYm", " ", "waf", " ", "wAm"}]}], ")"}]}]}], ")"}]}]}], RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "pXf"}], ")"}], " ", "waf"}], "-", RowBox[{"pXf", " ", "wAf"}]}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "pYm"}], ")"}], " ", "wam"}], "-", RowBox[{"pYm", " ", "wAm"}]}], ")"}]}]]], "Output", CellChangeTimes->{ 3.724292389339031*^9, 3.724293029482667*^9, {3.724293284538796*^9, 3.724293295872778*^9}, 3.724293533456543*^9, 3.725116347168006*^9, 3.725135978160911*^9, 3.72538543847866*^9, 3.725631722589016*^9, 3.725716823663583*^9, 3.726605529518544*^9, 3.7266909012359467`*^9, 3.7269196852467947`*^9, 3.7269199825408173`*^9, 3.726946954642653*^9, 3.7329767854559917`*^9, 3.735585012006866*^9}, ExpressionUUID -> "6d30f208-4f83-45d6-8daf-67a5983600f1"] }, Closed]], Cell["\<\ Note that if we define mean haploid fitnesses of X and Y bearing male gametes\ \ \>", "Text", CellChangeTimes->{{3.7256395266666*^9, 3.7256395707225018`*^9}},ExpressionUUID->"7aeaab2d-d674-4528-90e6-\ 4dc376de3fa1"], Cell[BoxData[{ RowBox[{ RowBox[{"wbarHapMaleX", "=", RowBox[{ RowBox[{"(", RowBox[{"1", "-", "q"}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"1", "-", "pXm"}], ")"}], " ", "wam"}], "+", RowBox[{"pXm", " ", "wAm"}]}], ")"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"wbarHapMaleY", "=", RowBox[{"q", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"1", "-", "pYm"}], ")"}], " ", "wam"}], "+", RowBox[{"pYm", " ", "wAm"}]}], ")"}]}]}], ";"}]}], "Input", CellChangeTimes->{{3.725639531765478*^9, 3.725639577975878*^9}}, ExpressionUUID -> "fdbfb4d0-d1da-49d6-b5d1-09a08195f454"], Cell["then ", "Text", CellChangeTimes->{{3.725639581924101*^9, 3.725639584801519*^9}},ExpressionUUID->"08441281-45de-4d98-9f1b-\ 9101df35c162"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"freqFemale", " ", "wbarHapMale"}], "==", "wbarHapMaleX"}], "/.", "SUBS"}], "/.", "subequil"}], "//", "Simplify"}]], "Input", CellChangeTimes->{{3.725639305567787*^9, 3.7256393390789337`*^9}, { 3.725639600317526*^9, 3.7256396006462383`*^9}}, ExpressionUUID -> "b60ffb81-6287-412f-a88b-0f69b5873e79"], Cell[BoxData["True"], "Output", CellChangeTimes->{{3.725639317037269*^9, 3.725639339697888*^9}, 3.7256396019008493`*^9, 3.725716823761684*^9, 3.726605529643736*^9, 3.7266909013124933`*^9, 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their formulae: \ \>", "Text", CellChangeTimes->{{3.724372496132658*^9, 3.7243725264104223`*^9}},ExpressionUUID->"bd416093-b0f7-4551-9c97-\ fcf719c4b7c0"], Cell[BoxData[ RowBox[{ RowBox[{"reverse", "=", RowBox[{ RowBox[{ RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"meandipM", "->", "wbarDipMale"}], ",", RowBox[{"meandipF", "->", "wbarDipFemale"}], ",", RowBox[{"\[Zeta]", "->", "freqMale"}], ",", RowBox[{"meanhapM", "->", "wbarHapMale"}], ",", RowBox[{"meanhapF", "->", "wbarHapFemale"}]}], "}"}], "/.", "SUBS"}], "/.", "subequil"}], "//", "Simplify"}]}], ";"}]], "Input", CellChangeTimes->{{3.724357868188628*^9, 3.724357937580203*^9}, { 3.72435809838582*^9, 3.7243581026786537`*^9}, 3.724358305194646*^9, 3.724358345481083*^9, {3.724358672896387*^9, 3.724358702551667*^9}}, ExpressionUUID -> "721ea217-d37a-4eaa-bf2b-1e03039c1386"], Cell["The following removes haploid selection:", "Text", CellChangeTimes->{{3.724372496132658*^9, 3.724372538889409*^9}},ExpressionUUID->"222c5c1a-8211-498f-9305-\ ee1d2a155960"], Cell[BoxData[ RowBox[{ RowBox[{"nohap", "=", RowBox[{"{", RowBox[{ RowBox[{"wam", "\[Rule]", "1"}], ",", RowBox[{"wAm", "\[Rule]", "1"}], ",", RowBox[{"waf", "\[Rule]", "1"}], ",", RowBox[{"wAf", "\[Rule]", "1"}], ",", RowBox[{"\[Alpha]m", "\[Rule]", RowBox[{"1", "/", "2"}]}], ",", RowBox[{"\[Alpha]f", "\[Rule]", RowBox[{"1", "/", "2"}]}]}], "}"}]}], ";"}]], "Input", CellChangeTimes->{{3.724372148848508*^9, 3.724372168102141*^9}}, ExpressionUUID -> "cdc4ac7c-82b3-4f2b-b58e-ea55848f9c76"] }, Closed]], Cell[CellGroupData[{ Cell["neo-Y characteristic polynomial", "Subsubsection", CellChangeTimes->{{3.719079895296085*^9, 3.719079912889949*^9}, { 3.7190807923459806`*^9, 3.719080794206753*^9}, {3.719261707039863*^9, 3.719261744473093*^9}, {3.724294131874865*^9, 3.724294141481683*^9}, { 3.724295896569313*^9, 3.7242958998099623`*^9}, {3.725116808187736*^9, 3.725116808617956*^9}, {3.726603229439431*^9, 3.726603231335416*^9}, 3.726610444976877*^9},ExpressionUUID->"1ed66246-49e8-4672-bbe2-\ 030d88979ab0"], Cell["\<\ As shown above, the relevant part of the characteristic polynomial for a \ masculinizing mutant is\ \>", "Text", CellChangeTimes->{{3.72660531781271*^9, 3.7266053420761747`*^9}},ExpressionUUID->"30ac6d36-f892-48a6-9170-\ cdc90613df7e"], Cell[CellGroupData[{ Cell[BoxData["charpoly\[Alpha]0"], "Input", CellChangeTimes->{{3.726605365055249*^9, 3.726605378305461*^9}, { 3.7266074360000362`*^9, 3.726607438881092*^9}}, ExpressionUUID -> "af13f416-021a-474d-a886-3aef03d59036"], Cell[BoxData[ RowBox[{ FractionBox[ RowBox[{"wam", " ", "wAm", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "Maa"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "pXf"}], ")"}], " ", "waf", " ", RowBox[{"(", RowBox[{ RowBox[{"MAA", " ", "pXf", " ", "wAf"}], "+", RowBox[{"2", " ", "MAa", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "pXf"}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "R"}], ")"}], " 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We call these the \ \[OpenCurlyDoubleQuote]haplotype-only growth rates\[CloseCurlyDoubleQuote], \ as they indicate how rapidly the haplotypes would spread if recombination \ never altered the background of the neo-SD.\n\nIn this section, we show for \ R>0 that:\n\t* ", StyleBox["Lemma 1:", FontWeight->"Bold"], " invasion always becomes more difficult as R rises.\n\t* ", StyleBox["Lemma 2:", FontWeight->"Bold"], " invasion is always prohibited if both \[Lambda]mA<1 and \[Lambda]ma<1.\n\t\ * ", StyleBox["Lemma 3:", FontWeight->"Bold"], " invasion is guaranteed for any R if both \[Lambda]mA>1 and \[Lambda]ma>1. \ In particular, invasion of a neo-SD on a different chromosome (R = 1/2) will \ occur in this case.\n\t* ", StyleBox["Lemma 4:", FontWeight->"Bold"], " invasion becomes more difficult as R rises if only one of \[Lambda]mA and \ \[Lambda]ma is greater than one, requiring that equation (1) in the paper be \ satisfied.\n" }], "Text", CellChangeTimes->{{3.7242707389341784`*^9, 3.7242708076290607`*^9}, { 3.7242721835170097`*^9, 3.724272192907447*^9}, {3.724273872229392*^9, 3.724273909404004*^9}, {3.724273996894698*^9, 3.724274013219116*^9}, { 3.7242741326054564`*^9, 3.724274412537304*^9}, {3.724274928597633*^9, 3.7242749928836193`*^9}, {3.724275696029413*^9, 3.724275703880847*^9}, { 3.724282095085225*^9, 3.7242821327780733`*^9}, {3.724286167208143*^9, 3.724286215114114*^9}, {3.724286749584754*^9, 3.724286790320301*^9}, { 3.7242888761765413`*^9, 3.724288888082183*^9}, {3.724289201603519*^9, 3.7242892149767437`*^9}, {3.724289547256996*^9, 3.724289559247699*^9}, { 3.7243714264705057`*^9, 3.724371436674306*^9}},ExpressionUUID->"73b07879-040b-49fb-9e78-\ 45a9b4c8fa97"], Cell[TextData[{ StyleBox["Preliminaries: ", FontWeight->"Bold"], "We are guaranteed to have both roots of f(\[Lambda]) be greater than one if \ the slope at \[Lambda] = 1 is negative, i.e., if ", Cell[BoxData[ FormBox[ SuperscriptBox["f", "l"], TraditionalForm]],ExpressionUUID-> "d55be27d-25d0-4dc9-a8a4-42c5b8a55fdc"], "(1) = 2 + b < 0, or equivalently when ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"-", FractionBox["b", "2"]}], "=", RowBox[{ FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"\[Lambda]mA", " ", "-", " ", "\[Chi]mA"}], ")"}], " ", "+", " ", RowBox[{"(", RowBox[{"\[Lambda]ma", " ", "-", " ", "\[Chi]ma"}], ")"}]}], "2"], " ", ">", "1"}]}], TraditionalForm]],ExpressionUUID-> "b544ab17-3d82-4b41-96d3-b5851291ade2"], ". This is because the Perron-Frobenius theorem for a stability matrix \ without negative entries guarantees that the leading eigenvalue is positive \ and real and we know f(\[Lambda]) is concave up. Thus, if the quadratic has a \ negative slope at \[Lambda]=1 then it will cross zero and cross back at some \ \[Lambda]>1. In words, the neo-SD is guaranteed to spread when the \ haplotype-only growth rate minus its loss due to recombination averaged for \ the mA and ma haplotypes is greater than 1. We will see that this condition \ is a bit stronger than we really need, because it does not consider the fact \ that recombination also creates mutant haplotypes.\n\nWhen - ", Cell[BoxData[ FormBox[ FractionBox["b", "2"], TraditionalForm]],ExpressionUUID-> "b544ab17-3d82-4b41-96d3-b5851291ade2"], " < 1, then we will still have one \[Lambda]>1 when f(1) = 1 + b + c < 0. \ Again, the leading eigenvalue must be real and postitive and f(\[Lambda]) is \ concave up, so if it is negative at \[Lambda]=1 it must cross zero at some \ \[Lambda]>1. This condition can be rearranged to give equation (1)\n\n\t(\ \[Lambda]mA-1) (\[Lambda]ma-1) < \[Chi]ma (\[Lambda]mA-1)+\[Chi]mA \ (\[Lambda]ma-1)" }], "Text", CellChangeTimes->{{3.7157187626221848`*^9, 3.715719007333971*^9}, { 3.7157191529753113`*^9, 3.715719153590147*^9}, {3.715719184967559*^9, 3.715719288555563*^9}, {3.715719352480856*^9, 3.71571938180089*^9}, { 3.715720168785968*^9, 3.7157201698100853`*^9}, {3.715721417711936*^9, 3.715721418447949*^9}, {3.715723665626091*^9, 3.715723820751107*^9}, { 3.715723862369293*^9, 3.7157238646815357`*^9}, {3.715723919387598*^9, 3.7157239739257936`*^9}, {3.715724011943788*^9, 3.715724019856412*^9}, { 3.7157240985172663`*^9, 3.715724264090788*^9}, {3.7191481190905724`*^9, 3.719148146130077*^9}, 3.724270831843523*^9, {3.724270972105464*^9, 3.7242709762149973`*^9}, {3.724271033668889*^9, 3.724271085384211*^9}, { 3.724271167387083*^9, 3.724271175066794*^9}, {3.724271317841717*^9, 3.724271327744176*^9}, {3.724272247727338*^9, 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"\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"%", "-", RowBox[{"(", RowBox[{"1", "+", "b", "+", "c"}], ")"}]}], "/.", RowBox[{"b", "->", RowBox[{"-", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"\[Lambda]mA", " ", "-", " ", "\[Chi]mA"}], ")"}], " ", "+", " ", RowBox[{"(", RowBox[{"\[Lambda]ma", " ", "-", " ", "\[Chi]ma"}], ")"}]}], ")"}]}]}]}], " ", "/.", RowBox[{"c", "->", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"\[Lambda]mA", "-", "\[Chi]mA"}], ")"}], " ", RowBox[{"(", RowBox[{"\[Lambda]ma", "-", "\[Chi]ma"}], ")"}]}], "-", RowBox[{"\[Chi]mA", " ", "\[Chi]ma"}]}]}]}], "//", "Factor"}]}], "Input",\ CellChangeTimes->{{3.724271104777646*^9, 3.724271159160001*^9}, { 3.7242712953448553`*^9, 3.724271298853526*^9}}, ExpressionUUID -> "44418cb4-5d16-47df-bf31-bef2c8750beb"], Cell[BoxData["0"], "Output", CellChangeTimes->{{3.724271144430477*^9, 3.724271160186349*^9}, 3.7242713000327597`*^9, 3.7242822700590982`*^9, 3.7251176697455273`*^9, 3.7251361984752073`*^9, 3.725385668035912*^9, 3.7256319563654947`*^9, 3.725717040023752*^9, 3.726691120435381*^9, 3.72694713533506*^9, 3.732976953271509*^9, 3.735585235350482*^9}, ExpressionUUID -> "f78a6b9c-9651-4eeb-a45f-ed2ad6827f42"] }, Closed]], Cell[TextData[{ "Furthermore, the leading eigenvalue, ", Cell[BoxData[ FormBox[ SubscriptBox["\[Lambda]", "L"], TraditionalForm]],ExpressionUUID-> "cc49410b-9e48-480c-8f2a-a923bbf7f83c"], ", is bounded between \[Lambda]mA and \[Lambda]ma, given that the quadratic \ has opposite signs when evaluated at these two values:" }], "Text", CellChangeTimes->{{3.724330874270041*^9, 3.724330944415247*^9}},ExpressionUUID->"9329f782-da4a-4b8a-8059-\ 607bf691a6ad"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["\[Lambda]", "2"], "+", RowBox[{"b", " ", "\[Lambda]"}], " ", "+", "c"}], "/.", RowBox[{"b", "->", RowBox[{"-", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"\[Lambda]mA", " ", "-", " ", "\[Chi]mA"}], ")"}], " ", "+", " ", 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CellChangeTimes->{3.724288737997121*^9, 3.724330920250079*^9, 3.725117669820827*^9, 3.725136198525324*^9, 3.7253856681182337`*^9, 3.725631956423357*^9, 3.725717040072221*^9, 3.726691120472374*^9, 3.7269471353854637`*^9, 3.7329769533221607`*^9, 3.735585235384362*^9}, ExpressionUUID -> "daeafc40-c54d-41e8-acb8-beec5b23dc36"], Cell[BoxData[ RowBox[{ RowBox[{"-", RowBox[{"(", RowBox[{"\[Lambda]ma", "-", "\[Lambda]mA"}], ")"}]}], " ", "\[Chi]mA"}]], "Output", CellChangeTimes->{3.724288737997121*^9, 3.724330920250079*^9, 3.725117669820827*^9, 3.725136198525324*^9, 3.7253856681182337`*^9, 3.725631956423357*^9, 3.725717040072221*^9, 3.726691120472374*^9, 3.7269471353854637`*^9, 3.7329769533221607`*^9, 3.73558523538771*^9}, ExpressionUUID -> "daeafc40-c54d-41e8-acb8-beec5b23dc36"] }, Closed]], Cell["\<\ when R = 0, the \[Chi] terms are zero and the larger of \[Lambda]ma and \ \[Lambda]mA is the leading eigenvalue.\ \>", "Text", CellChangeTimes->{{3.724330946695024*^9, 3.724330975910329*^9}},ExpressionUUID->"43c41836-49b7-44ed-98e2-\ f4d6d809bc1a"], Cell[TextData[{ StyleBox["Proof of Lemma 1: ", FontWeight->"Bold"], "We next prove that the leading eigenvalue always declines as R increases. \ First we note that both \[Chi]ma and \[Chi]mA are proportional to R and we \ make this explicit in the quadratic f(\[Lambda]):" }], "Text", CellChangeTimes->{{3.724289570615861*^9, 3.724289586166252*^9}},ExpressionUUID->"f7fabfde-a6ad-4bff-a745-\ be742b828895"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox[ RowBox[{"\[Lambda]", "[", "R", "]"}], "2"], "+", RowBox[{"b", " ", RowBox[{"\[Lambda]", "[", "R", "]"}]}], "+", "c"}], "/.", RowBox[{"b", "->", RowBox[{"-", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"\[Lambda]mA", " ", "-", " ", "\[Chi]mA"}], ")"}], " ", "+", " ", RowBox[{"(", RowBox[{"\[Lambda]ma", " ", "-", " ", "\[Chi]ma"}], ")"}]}], ")"}]}]}]}], " ", "/.", RowBox[{"c", "->", RowBox[{ RowBox[{ RowBox[{"(", 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Cell[BoxData[ FormBox[ FractionBox["d\[Lambda]", "dR"], TraditionalForm]],ExpressionUUID-> "137fd7d9-f20c-4053-8712-f7b9adb47b61"], " and returning to the \[Chi] terms:" }], "Text", CellChangeTimes->{{3.7242869486177483`*^9, 3.724287018735746*^9}},ExpressionUUID->"c4821998-2d4a-40e2-9a9d-\ a9ccb31f1b70"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{"D", "[", RowBox[{"%", ",", "R"}], "]"}], "\[Equal]", "0"}], ",", RowBox[{ SuperscriptBox["\[Lambda]", "\[Prime]", MultilineFunction->None], "[", "R", "]"}]}], " ", "]"}], "/.", RowBox[{ SubscriptBox["f", "A"], "->", RowBox[{"\[Chi]mA", "/", "R"}]}]}], "/.", RowBox[{ SubscriptBox["f", "a"], "->", RowBox[{"\[Chi]ma", "/", "R"}]}]}], "/.", RowBox[{ RowBox[{"\[Lambda]", "[", "R", "]"}], "->", "\[Lambda]"}]}], "//", "Flatten"}], "//", "Simplify"}]], "Input", CellChangeTimes->{{3.724271475836294*^9, 3.724271498235908*^9}, 3.724284482786578*^9, {3.7242845243599243`*^9, 3.724284532821599*^9}, { 3.724286986133524*^9, 3.724286999061125*^9}, {3.724287049484766*^9, 3.724287052090775*^9}}, ExpressionUUID -> "b07d2d86-59f7-4008-b952-d01bc23dda62"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox["\[Lambda]", "\[Prime]", MultilineFunction->None], "[", "R", "]"}], "\[Rule]", FractionBox[ RowBox[{ RowBox[{"\[Lambda]mA", " ", "\[Chi]ma"}], "+", RowBox[{"\[Lambda]ma", " ", "\[Chi]mA"}], "-", RowBox[{"\[Lambda]", " ", RowBox[{"(", RowBox[{"\[Chi]ma", "+", "\[Chi]mA"}], ")"}]}]}], RowBox[{"R", " ", RowBox[{"(", RowBox[{ RowBox[{"2", " ", "\[Lambda]"}], "-", "\[Lambda]ma", "-", "\[Lambda]mA", "+", "\[Chi]ma", "+", "\[Chi]mA"}], ")"}]}]]}], "}"}]], "Output", CellChangeTimes->{{3.724271485451877*^9, 3.7242715036554937`*^9}, 3.7242842301726522`*^9, 3.724284261893035*^9, 3.724284365571692*^9, 3.7242844868271103`*^9, {3.724284518567246*^9, 3.7242845348212748`*^9}, 3.724285313713846*^9, 3.7242868950487337`*^9, 3.724286999481278*^9, { 3.724287052665565*^9, 3.724287057093011*^9}, 3.7251176699358053`*^9, 3.7251361986251373`*^9, 3.725385668268549*^9, 3.7256319565555887`*^9, 3.725717040173925*^9, 3.726691120579616*^9, 3.7269471355031843`*^9, 3.732976953423621*^9, 3.73558523548875*^9}, ExpressionUUID -> "cd687e2a-ea99-4622-938e-aaafca778ce7"] }, Closed]], Cell[TextData[{ "This can be written as:\n\n", Cell[BoxData[ RowBox[{ RowBox[{"-", " ", FractionBox[ RowBox[{"\[Chi]ma", "+", "\[Chi]mA"}], RowBox[{"R", " ", "slope"}]]}], RowBox[{"(", RowBox[{ SubscriptBox["\[Lambda]", "L"], "-", FractionBox[ RowBox[{ RowBox[{"\[Lambda]mA", " ", "\[Chi]ma"}], "+", RowBox[{"\[Lambda]ma", " ", "\[Chi]mA"}]}], RowBox[{"\[Chi]ma", "+", "\[Chi]mA"}]]}], ")"}]}]], CellChangeTimes->{{3.724288575750654*^9, 3.724288601200779*^9}}, ExpressionUUID->"6d121827-f36d-42ef-8376-0c6d872206c2"], "\n\nwhere \[OpenCurlyDoubleQuote]slope\[CloseCurlyDoubleQuote] = ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ RowBox[{"2", SubscriptBox["\[Lambda]", "L"]}], "+", "b"}], ")"}], TraditionalForm]], ExpressionUUID->"80f6b458-8d3a-4e81-919c-96bc41b53911"], " is the slope of the quadratic, ", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"df", "(", "\[Lambda]", ")"}], "d\[Lambda]"], TraditionalForm]], ExpressionUUID->"410f2845-a008-46be-95e2-e63c140afba1"], ", evaluated at the leading eigenvalue ", Cell[BoxData[ FormBox[ SubscriptBox["\[Lambda]", "L"], TraditionalForm]],ExpressionUUID-> "74e63800-b9f7-42cc-b1e3-63643c2397f0"], ", which must be positive (because f(\[Lambda]) is concave up and we\ \[CloseCurlyQuote]re asking about the right-most root). Hence the first \ fraction is positive.\n\nFurthermore, the second fraction must be positive, \ since the leading eigenvalue is strictly above ", Cell[BoxData[ FractionBox[ RowBox[{ RowBox[{"\[Lambda]mA", " ", "\[Chi]ma"}], "+", RowBox[{"\[Lambda]ma", " ", "\[Chi]mA"}]}], RowBox[{"\[Chi]ma", "+", "\[Chi]mA"}]]], CellChangeTimes->{{3.7242873545367126`*^9, 3.7242873857881927`*^9}, { 3.7242885024714127`*^9, 3.7242885182309923`*^9}},ExpressionUUID-> "1867b4e7-1dff-4ea7-b009-bd11b03a057b"], ". This is because ", Cell[BoxData[ RowBox[{"f", RowBox[{"(", FractionBox[ RowBox[{ RowBox[{"\[Lambda]mA", " ", "\[Chi]ma"}], "+", RowBox[{"\[Lambda]ma", " ", "\[Chi]mA"}]}], RowBox[{"\[Chi]ma", "+", "\[Chi]mA"}]], ")"}]}]], CellChangeTimes->{{3.7242873545367126`*^9, 3.7242873857881927`*^9}, { 3.7242885024714127`*^9, 3.7242885182309923`*^9}},ExpressionUUID-> "b617cea0-a461-403d-9ba0-fd1a08c9bffe"], "is strictly negative and so the leading eigenvalue must occur at a higher \ value of \[Lambda]:" }], "Text", CellChangeTimes->{{3.724287024176738*^9, 3.724287193017831*^9}, { 3.724287230551898*^9, 3.724287252406035*^9}, {3.7242874172361717`*^9, 3.724287417394834*^9}, {3.724287500241075*^9, 3.7242875027046223`*^9}, { 3.724287802331688*^9, 3.72428781019217*^9}, {3.724288538240334*^9, 3.7242885694379263`*^9}, 3.724288608386856*^9, 3.7242886506034613`*^9, { 3.7242886830933523`*^9, 3.724288711228965*^9}, {3.724288763997159*^9, 3.7242888611606483`*^9}, {3.72428895606783*^9, 3.724289121458405*^9}, { 3.724289305341631*^9, 3.724289446630069*^9}},ExpressionUUID->"0b9eba5d-8706-44c9-953b-\ 9adb4868f638"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["\[Lambda]", "2"], "+", RowBox[{"b", " ", "\[Lambda]"}], " ", "+", "c"}], "/.", RowBox[{"b", "->", RowBox[{"-", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"\[Lambda]mA", " ", "-", " ", "\[Chi]mA"}], ")"}], " ", "+", " ", RowBox[{"(", RowBox[{"\[Lambda]ma", " ", "-", " ", "\[Chi]ma"}], ")"}]}], ")"}]}]}]}], " ", "/.", RowBox[{"c", "->", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"\[Lambda]mA", "-", "\[Chi]mA"}], ")"}], " ", RowBox[{"(", RowBox[{"\[Lambda]ma", "-", "\[Chi]ma"}], ")"}]}], "-", RowBox[{"\[Chi]mA", " ", "\[Chi]ma"}]}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"%", "/.", RowBox[{"\[Lambda]", "\[Rule]", FractionBox[ RowBox[{ RowBox[{"\[Lambda]mA", " ", "\[Chi]ma"}], "+", RowBox[{"\[Lambda]ma", " ", "\[Chi]mA"}]}], RowBox[{"\[Chi]ma", "+", "\[Chi]mA"}]]}]}], "//", "Factor"}]}], "Input", CellChangeTimes->{{3.7242873545367126`*^9, 3.7242873857881927`*^9}, { 3.7242885024714127`*^9, 3.7242885182309923`*^9}, {3.72428872152035*^9, 3.724288737212182*^9}}, ExpressionUUID -> "ad6b8ab9-0b0e-4e7c-94fc-591d23f83b5e"], Cell[BoxData[ RowBox[{"-", FractionBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"\[Lambda]ma", "-", "\[Lambda]mA"}], ")"}], "2"], " ", "\[Chi]ma", " ", "\[Chi]mA"}], SuperscriptBox[ RowBox[{"(", RowBox[{"\[Chi]ma", "+", "\[Chi]mA"}], ")"}], "2"]]}]], "Output", CellChangeTimes->{3.724288737997121*^9, 3.7251176699917316`*^9, 3.7251361986754293`*^9, 3.7253856683546133`*^9, 3.7256319566245623`*^9, 3.725717040221609*^9, 3.726691120622862*^9, 3.726947135551794*^9, 3.732976953475439*^9, 3.735585235537033*^9}, ExpressionUUID -> "b26a8707-7a5e-4d8f-b0bc-bd5af71a6f36"] }, Closed]], Cell["\<\ We conclude that the leading eigenvalue must fall as R increases. \ \>", "Text", CellChangeTimes->{{3.724289136911166*^9, 3.724289149014613*^9}},ExpressionUUID->"c65af33f-1552-4e34-bfed-\ 9bb074e442fc"], Cell[TextData[{ StyleBox["Corollary of Lemma 1: ", FontWeight->"Bold"], " The non-leading eigenvalue will also decline as R increase, because \ \[OpenCurlyDoubleQuote]slope\[CloseCurlyDoubleQuote] is then negative and the \ term, ", Cell[BoxData[ RowBox[{"(", RowBox[{ SubscriptBox["\[Lambda]", "N"], "-", FractionBox[ RowBox[{ RowBox[{"\[Lambda]mA", " ", "\[Chi]ma"}], "+", RowBox[{"\[Lambda]ma", " ", "\[Chi]mA"}]}], RowBox[{"\[Chi]ma", "+", "\[Chi]mA"}]]}], ")"}]], CellChangeTimes->{{3.724288575750654*^9, 3.724288601200779*^9}}, ExpressionUUID->"7e202986-d698-4918-9f4e-d649bdc3455e"], ", will also be negative (because the non-leading eigenvalue must fall below \ ", Cell[BoxData[ FractionBox[ RowBox[{ RowBox[{"\[Lambda]mA", " ", "\[Chi]ma"}], "+", RowBox[{"\[Lambda]ma", " ", "\[Chi]mA"}]}], RowBox[{"\[Chi]ma", "+", "\[Chi]mA"}]]], CellChangeTimes->{{3.724288575750654*^9, 3.724288601200779*^9}}, ExpressionUUID->"8e2c0e96-f7e3-4e12-a86b-76c644ca3c92"], " is the quadratic is negative at that point)." }], "Text", CellChangeTimes->{{3.7242897756934557`*^9, 3.724289779074728*^9}, { 3.724289811747401*^9, 3.72428991688706*^9}},ExpressionUUID->"48898b02-012a-4a2b-b5b3-\ 3196c59dedee"], Cell[TextData[{ StyleBox["Proof of Lemma 2: ", FontWeight->"Bold"], "When both the haplotype-only growth rates (\[Lambda]mA and \[Lambda]ma) are \ less than 1, then ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"-", FractionBox["b", "2"]}], "=", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"\[Lambda]mA", " ", "-", " ", "\[Chi]mA"}], ")"}], " ", "+", " ", RowBox[{"(", RowBox[{"\[Lambda]ma", " ", "-", " ", "\[Chi]ma"}], ")"}]}], "2"]}], TraditionalForm]],ExpressionUUID->"1d9c5b67-8bf9-42eb-97d9-dfd5785a9e34"], " must be less than one (recall that the \[Chi] terms are positive) and 1 + \ b + c cannot be negative because (\[Lambda]mA - 1) (\[Lambda]ma - 1) < \ \[Chi]ma (\[Lambda]mA - 1) + \[Chi]mA (\[Lambda]ma - 1) is false (left hand \ side is positive while right hand side is negative). Thus, the neo-SD never \ invades when neither haplotype can grow in the absence of recombination \ (R=0). \n\nCheck: Can invasion occur?" }], "Text", CellChangeTimes->{{3.715721276545784*^9, 3.7157213181472893`*^9}, { 3.715721472826126*^9, 3.715721531156227*^9}, {3.715721592127339*^9, 3.715721661260517*^9}, {3.7157216967084513`*^9, 3.7157217960389*^9}, { 3.715724297532874*^9, 3.715724301532427*^9}, 3.715724333757893*^9, { 3.7157854982095003`*^9, 3.715785510867275*^9}, {3.7157884944670753`*^9, 3.715788626787586*^9}, {3.715788658944414*^9, 3.715788679563257*^9}, { 3.715788758869928*^9, 3.715788783291356*^9}, {3.715790374011229*^9, 3.715790381064714*^9}, {3.715867609224828*^9, 3.715867674751651*^9}, { 3.715867760904635*^9, 3.715867765250544*^9}, {3.715867830629575*^9, 3.715867867218644*^9}, {3.71586789899417*^9, 3.715867924978119*^9}, { 3.715873465458592*^9, 3.715873468731282*^9}, {3.715876863206894*^9, 3.715877153969789*^9}, {3.715877282548334*^9, 3.715877291376307*^9}, { 3.715949150179731*^9, 3.715949162628196*^9}, {3.7159509058254023`*^9, 3.715950910700091*^9}, 3.715951057659639*^9, 3.715951117177115*^9, { 3.715951211993087*^9, 3.715951236960308*^9}, {3.7191484370123262`*^9, 3.7191484406348124`*^9}, {3.719148471019236*^9, 3.71914852851727*^9}, { 3.724273775242256*^9, 3.724273863335085*^9}, {3.7242748816462803`*^9, 3.724274916443844*^9}, {3.7242749544810343`*^9, 3.724274958987906*^9}, { 3.724275016007628*^9, 3.724275032778013*^9}, 3.7242752929265957`*^9, { 3.724275326623315*^9, 3.724275326623405*^9}, {3.72427631020888*^9, 3.7242763289533052`*^9}, {3.724276361751498*^9, 3.724276398700664*^9}, { 3.724281387951131*^9, 3.724281391347183*^9}, 3.724289623028285*^9},ExpressionUUID->"e4ee7ead-2571-4330-94b1-\ 71cdd7231520"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{ RowBox[{"Reduce", "[", "\[IndentingNewLine]", RowBox[{"{", RowBox[{ RowBox[{ FractionBox[ RowBox[{ "\[Lambda]mA", "-", "\[Chi]mA", "+", "\[Lambda]ma", "-", "\[Chi]ma"}], "2"], ">", "1"}], RowBox[{"(*", " ", RowBox[{"Slope", " ", "condition", " ", RowBox[{"met", ":", " ", RowBox[{ RowBox[{ RowBox[{"f", "'"}], RowBox[{"(", "1", ")"}]}], "<", " ", "0"}]}]}], " ", "*)"}], "\[IndentingNewLine]", "||", RowBox[{ RowBox[{ RowBox[{"\[Chi]ma", " ", RowBox[{"(", RowBox[{"\[Lambda]mA", "-", "1"}], ")"}]}], "+", RowBox[{"\[Chi]mA", " ", RowBox[{"(", RowBox[{"\[Lambda]ma", "-", "1"}], ")"}]}]}], ">", RowBox[{ RowBox[{"(", RowBox[{"\[Lambda]mA", "-", "1"}], ")"}], " ", RowBox[{"(", RowBox[{"\[Lambda]ma", "-", "1"}], ")"}]}]}]}], RowBox[{"(*", " ", RowBox[{"or", " ", "intercept", " ", "condition", " ", RowBox[{"met", ":", " ", RowBox[{ RowBox[{"f", RowBox[{"(", "1", ")"}]}], ">", "0"}]}]}], "*)"}], " ", "}"}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{ RowBox[{"\[Lambda]mA", "<", "1"}], ",", RowBox[{"\[Lambda]ma", "<", "1"}], ",", RowBox[{"0", "<", "\[Chi]mA"}], ",", RowBox[{"0", "<", "\[Chi]ma"}]}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.7242761526849833`*^9, 3.7242761788479156`*^9}, { 3.724276215555994*^9, 3.72427621801637*^9}, {3.724276286609914*^9, 3.724276295530012*^9}, {3.72427633140018*^9, 3.7242763920928917`*^9}, { 3.724277077846962*^9, 3.724277081956224*^9}, {3.724278149338642*^9, 3.724278150912915*^9}}, ExpressionUUID -> "9f0dbd49-67d1-47d7-8db7-25dcb31eba05"], Cell[BoxData["False"], "Output", CellChangeTimes->{{3.724276159727729*^9, 3.724276180282143*^9}, 3.7242762204901648`*^9, 3.724276297667194*^9, 3.7242770824921427`*^9, 3.724278151557322*^9, 3.725117671154912*^9, 3.725136199837441*^9, 3.725385669504407*^9, 3.725631957787365*^9, 3.725717041371804*^9, 3.726691121960864*^9, 3.726947136654544*^9, 3.73297695452747*^9, 3.735585236988085*^9}, ExpressionUUID -> "fd84073f-6d9d-4ae6-9d28-0bc5843567b8"] }, Closed]], Cell[TextData[{ StyleBox["Proof of Lemma 3: ", FontWeight->"Bold"], "When both the haplotype-only growth rates (\[Lambda]mA and \[Lambda]ma) are \ greater than 1, we consider two sub-cases:\n\n", StyleBox["Case 1: ", FontSlant->"Italic"], "Both haplotype-only growth rates are so large that the average of \ (\[Lambda]mA - \[Chi]mA) and (\[Lambda]ma - \[Chi]ma) is also greater than \ one. Then we are guaranteed that both roots of f(\[Lambda]) are greater than \ one regardless of R since ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"-", FormBox[ RowBox[{ FractionBox["b", "2"], "=", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"\[Lambda]mA", " ", "-", " ", "\[Chi]mA"}], ")"}], " ", "+", " ", RowBox[{"(", RowBox[{"\[Lambda]ma", " ", "-", " ", "\[Chi]ma"}], ")"}]}], "2"]}], TraditionalForm]}], " ", ">", "1"}], TraditionalForm]],ExpressionUUID-> "1d9c5b67-8bf9-42eb-97d9-dfd5785a9e34"], ".\n", StyleBox["Case 2: ", FontSlant->"Italic"], "If, however, ", Cell[BoxData[ FormBox[ RowBox[{"-", FormBox[ RowBox[{ FractionBox["b", "2"], "<", "1"}], TraditionalForm]}], TraditionalForm]],ExpressionUUID-> "1d9c5b67-8bf9-42eb-97d9-dfd5785a9e34"], ", then invasion will still occur if both \[Lambda]mA and \[Lambda]ma are \ greater than one. This is because the intercept can be written as:\n\n", Cell[BoxData[ RowBox[{"-", RowBox[{"[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"\[Lambda]mA", "-", "1"}], ")"}], " ", RowBox[{"(", RowBox[{"2", "+", "b"}], ")"}]}], "+", SuperscriptBox[ RowBox[{"(", RowBox[{"\[Lambda]mA", "-", "1"}], ")"}], "2"], "+", RowBox[{ RowBox[{"(", RowBox[{"\[Lambda]ma", "-", "\[Lambda]mA"}], ")"}], " ", "\[Chi]mA"}]}], "]"}]}]], CellChangeTimes->{{3.724280708553141*^9, 3.724280774230484*^9}}, ExpressionUUID->"e73b38e0-1d44-4719-9d6c-ee3cbd927543"], "\n\nand equivalently as:\n\n", Cell[BoxData[ RowBox[{"-", RowBox[{"[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"\[Lambda]ma", "-", "1"}], ")"}], " ", RowBox[{"(", RowBox[{"2", "+", "b"}], ")"}]}], "+", SuperscriptBox[ RowBox[{"(", RowBox[{"\[Lambda]ma", "-", "1"}], ")"}], "2"], "+", RowBox[{ RowBox[{"(", RowBox[{"\[Lambda]mA", "-", "\[Lambda]ma"}], ")"}], " ", "\[Chi]ma"}]}], "]"}]}]], CellChangeTimes->{{3.724280708553141*^9, 3.724280774230484*^9}}, ExpressionUUID->"82a97dc8-280d-41cd-a663-a3662ac2e7de"], "\n\nThus, if \[Lambda]ma \[GreaterEqual] \[Lambda]mA, we choose the first \ way to write the intercept, which is clearly negative (all terms in square \ brackets are positive given that both \[Lambda]mA and \[Lambda]ma are greater \ than one and ", Cell[BoxData[ FormBox[ RowBox[{"-", FormBox[ RowBox[{ FractionBox["b", "2"], "<", "1"}], TraditionalForm]}], TraditionalForm]],ExpressionUUID-> "1d9c5b67-8bf9-42eb-97d9-dfd5785a9e34"], "). If if \[Lambda]mA \[GreaterEqual] \[Lambda]ma, we choose the second way \ to write the intercept, which is also clearly negative. Either way, invasion \ is guaranteed regardless of R." }], "Text", CellChangeTimes->{{3.715721276545784*^9, 3.7157213181472893`*^9}, { 3.715721472826126*^9, 3.715721531156227*^9}, {3.715721592127339*^9, 3.715721661260517*^9}, {3.7157216967084513`*^9, 3.7157217960389*^9}, { 3.715724297532874*^9, 3.715724301532427*^9}, 3.715724333757893*^9, { 3.7157854982095003`*^9, 3.715785510867275*^9}, {3.7157884944670753`*^9, 3.715788626787586*^9}, {3.715788658944414*^9, 3.715788679563257*^9}, { 3.715788758869928*^9, 3.715788783291356*^9}, {3.715790374011229*^9, 3.715790381064714*^9}, {3.715867609224828*^9, 3.715867674751651*^9}, { 3.715867760904635*^9, 3.715867765250544*^9}, {3.715867830629575*^9, 3.715867867218644*^9}, {3.71586789899417*^9, 3.715867924978119*^9}, { 3.715873465458592*^9, 3.715873468731282*^9}, {3.715876863206894*^9, 3.715877153969789*^9}, {3.715877282548334*^9, 3.715877291376307*^9}, { 3.715949150179731*^9, 3.715949162628196*^9}, {3.7159509058254023`*^9, 3.715950910700091*^9}, 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first note that stability is determined \ solely by the intercept of f(\[Lambda]) at \[Lambda] = 1, because the slope \ at \[Lambda] = 1 is immaterial. This is because Lemma 1 and its corollary \ imply that both roots of f(\[Lambda]) decline as R rises. Thus, if only one \ root is greater than 1 at R = 0 and if both roots decline with R, then we can \ never have both roots greater than one. With only one root possibly greater \ than 1, this root will be greater than one if and only if f(1) is negative. \n\ \nThus, when only one of \[Lambda]mA and \[Lambda]ma is greater than one, \ invasion will occur according to the intercept condition:\n\n\t\ (\[Lambda]mA-1) (\[Lambda]ma-1) <", Cell[BoxData[ FormBox[ RowBox[{" ", "\[Chi]ma"}], TraditionalForm]],ExpressionUUID-> "86c89d8e-5c85-435a-8435-847ce96abe89"], " (\[Lambda]mA-1)+\[Chi]mA (\[Lambda]ma-1) \t\n\t\n\t\t[dividing both sides \ by (\[Lambda]mA-1) (\[Lambda]ma-1) and switching the inequality because this \ is negative]:\n\t\n\t1 > ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox["\[Chi]ma", RowBox[{"\[Lambda]ma", "-", "1"}]], "+", FractionBox["\[Chi]mA", RowBox[{"\[Lambda]mA", "-", "1"}]]}], TraditionalForm]],ExpressionUUID-> "f0d53d0b-043a-4ae2-ac5d-e03e39163d3b"], "\n\nwhich is equation (1). One of the terms on the right is positive and \ one negative. Because the \[Chi] are both positive and proportional to R, \ the condition is always satisfied when R = 0 (right-hand side is zero). \n\n\ In words, we need the amount of recombination off the growing background, \ weighted by the growth rate of the growing haplotype, to be small enough \ relative to the amount of recombination off the shrinking haplotype, weighted \ by its rate of decline. \n\nCheck: Can invasion occur if equation (1) is \ satisfied?" }], "Text", CellChangeTimes->{{3.724281996251914*^9, 3.724281998481538*^9}, { 3.724282143470478*^9, 3.724282165924045*^9}, {3.7242823337445393`*^9, 3.72428235175231*^9}, {3.7242827381649647`*^9, 3.724282785341885*^9}, { 3.724282866378118*^9, 3.724282893993227*^9}, {3.724283166294426*^9, 3.7242831998222017`*^9}, {3.724283313152452*^9, 3.724283359110807*^9}, { 3.724283389367065*^9, 3.724283395334827*^9}, {3.724283443148415*^9, 3.724283471845456*^9}, {3.724283657135232*^9, 3.72428368741015*^9}, { 3.724283798812879*^9, 3.72428387156509*^9}, 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But if ", Cell[BoxData[ RowBox[{ FractionBox[ RowBox[{ RowBox[{"fitness", "[", RowBox[{"W", "-", "a"}], "]"}], "/", "meandipF"}], RowBox[{"2", " ", "freqfemale"}]], ">", "1"}]], CellChangeTimes->{{3.6971404582251177`*^9, 3.697140466798635*^9}}, ExpressionUUID->"5b596b1c-062e-42f0-9506-69b49f107479"], " or if ", Cell[BoxData[ RowBox[{ FractionBox[ RowBox[{ RowBox[{"fitness", "[", RowBox[{"W", "-", "A"}], "]"}], "/", "meandipF"}], RowBox[{"2", " ", "freqfemale"}]], ">", "1"}]], CellChangeTimes->{{3.6971404582251177`*^9, 3.697140466798635*^9}}, ExpressionUUID->"05ef90d2-0df4-4473-aa73-f1f3924be64b"], " then invasion is possible." }], "Text", CellChangeTimes->{{3.6971405725355787`*^9, 3.697140638816691*^9}, { 3.6971407700483503`*^9, 3.6971409192108107`*^9}, {3.697140995948196*^9, 3.697141016514312*^9}, {3.697141062707622*^9, 3.697141105224299*^9}, { 3.6971411980001707`*^9, 3.6971411987038717`*^9}, {3.697159565845908*^9, 3.697159579041697*^9}, {3.697159660287785*^9, 3.697159685976252*^9}, { 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At the end of this section, we \ prove:\n\t", StyleBox["Lemma 5: ", FontWeight->"Bold"], "An unlinked neo-W (R=1/2) cannot invade a system with sexually antagonistic \ selection if selection is directional in each sex and haploid selection is \ absent.\n\nUnlinked neo-Ws can invade in several other cases, however. We \ know that if both \[Lambda]mA and \[Lambda]ma are greater than one, then \ invasion will occur for all R (as the leading eigenvalue is bounded between \ these two values, see \[OpenCurlyDoubleQuote]Invasion Conditions\ \[CloseCurlyDoubleQuote]). Scanning the above requirements, we can thus \ conclude that it is possible for unlinked neo-W chromosomes to invade with:\n\ \t* heterozygote advantage in males with directional selection in females (no \ haploid selection)" }], "Text", CellChangeTimes->{{3.724364370001005*^9, 3.7243645858589907`*^9}, { 3.724364686585678*^9, 3.724364719230373*^9}, {3.7243649507911863`*^9, 3.724365000328431*^9}, 3.724365169373115*^9, {3.724365273851008*^9, 3.724365277953088*^9}, {3.7243658190224257`*^9, 3.724365865738779*^9}, { 3.724365934034816*^9, 3.7243660377434683`*^9}, {3.724367516951556*^9, 3.724367520244789*^9}, {3.724371022941655*^9, 3.724371048655724*^9}, { 3.724371098866084*^9, 3.7243712665625877`*^9}, {3.724371539207206*^9, 3.724371539309132*^9}, {3.724450422250174*^9, 3.724450426889781*^9}, { 3.724450821757176*^9, 3.724450822262328*^9}},ExpressionUUID->"bf3db60a-2bbe-4dab-8482-\ bf13afcbc0be"], Cell["\<\ Recalling that the eigenvalues must decrease with increasing R (see \ \[OpenCurlyDoubleQuote]Invasion Conditions\[CloseCurlyDoubleQuote]), we know \ that sexually antagonistic selection alone (with opposing directional \ selection in the two sexes) will not allow the spread of a neo-W at \ equilibrium B under any linkage conditions because both \[Lambda]mA and \ \[Lambda]ma are less than one. We also know that if both \[Lambda]mA and \[Lambda]ma are greater than one, \ then invasion will occur for all R (as the leading eigenvalue is bounded \ between these two values, see \[OpenCurlyDoubleQuote]Invasion Conditions\ \[CloseCurlyDoubleQuote]). Scanning the above requirements, we can thus \ conclude that it is possible for unlinked neo-W chromosomes to invade with: \t* heterozygote advantage in males with directional selection in females if \ haploid selection is absent. \t Indeed, overdominance in males is required for both \[Lambda]mA>1 and \ \[Lambda]ma>1 in the absence of haploid selection: \ \>", "Text", CellChangeTimes->{{3.724364370001005*^9, 3.7243645858589907`*^9}, { 3.724364686585678*^9, 3.724364719230373*^9}, {3.7243649507911863`*^9, 3.724365000328431*^9}, 3.724365169373115*^9, {3.724365273851008*^9, 3.724365277953088*^9}, {3.7243658190224257`*^9, 3.724365865738779*^9}, { 3.724365934034816*^9, 3.7243660377434683`*^9}, {3.7243674934635572`*^9, 3.724367506492231*^9}, {3.7243677439388323`*^9, 3.7243678396429234`*^9}, { 3.724449061425387*^9, 3.72444912621952*^9}, {3.724449414583281*^9, 3.724449415894415*^9}, {3.7244508541251497`*^9, 3.724450859474063*^9}, { 3.724450933724112*^9, 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(ensuring unlinked neo-W can invade). \ For example, meiotic drive for the Y linked allele in males skews the sex \ ratio towards males, allowing a neo-W chromosome to invade (which balances \ the sex ratio). Meiotic drive in females can also work, if balanced by the \ right type of diploid selection. Haploid selection coupled for example with sexual antagonism in diploids:\ \>", "Text", CellChangeTimes->{ 3.7244508256304417`*^9, {3.724450891742837*^9, 3.724450892707912*^9}},ExpressionUUID->"f7024d74-8be8-4314-9548-\ 48bb1dd220e4"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"(*", RowBox[{ "Including", " ", "meiotic", " ", "drive", " ", "in", " ", "males"}], "*)"}], "\[IndentingNewLine]", RowBox[{"Simplify", "[", RowBox[{"Reduce", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"{", RowBox[{"stabcondA", ",", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"\[Lambda]mA1", ">", "1"}], ")"}], "&&", RowBox[{"(", RowBox[{"\[Lambda]ma1", ">", "1"}], ")"}]}], ")"}], ",", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"FAA", ">", "FAa", ">", "Faa", ">", "0"}], ")"}], "&&", RowBox[{"(", RowBox[{"Maa", ">", "MAa", ">", "MAA", ">", "0"}], ")"}]}], ")"}], "||", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"0", 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In this case (see \ the \[OpenCurlyDoubleQuote]Invasion Condition\[CloseCurlyDoubleQuote] \ section), we showed that a neo-W will invade if and only if the intercept of \ the primary quadratic is negative:\ \>", "Text", CellChangeTimes->{{3.7244509611759787`*^9, 3.7244510314540873`*^9}},ExpressionUUID->"edf93190-96f6-46aa-8a9f-\ 82b74d829fc2"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"1", "+", "b", "+", "c"}], ")"}], "/.", RowBox[{"b", "->", RowBox[{"-", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"\[Lambda]mA", " ", "-", " ", "\[Chi]mA"}], ")"}], " ", "+", " ", RowBox[{"(", RowBox[{"\[Lambda]ma", " ", "-", " ", "\[Chi]ma"}], ")"}]}], ")"}]}]}]}], " ", "/.", RowBox[{"c", "->", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"\[Lambda]mA", "-", "\[Chi]mA"}], ")"}], " ", RowBox[{"(", RowBox[{"\[Lambda]ma", "-", "\[Chi]ma"}], ")"}]}], "-", RowBox[{"\[Chi]mA", " ", "\[Chi]ma"}]}]}]}], "//", "Factor"}]], "Input", CellChangeTimes->{{3.724271104777646*^9, 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\[OpenCurlyDoubleQuote]A\ \[CloseCurlyDoubleQuote] in females). 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ExpressionUUID -> "c4d6ed0e-6dd6-420f-975e-8f5c937f8532"], Cell[BoxData[ FractionBox[ RowBox[{"wAm", " ", "\[Alpha]m"}], RowBox[{"wam", "-", RowBox[{"wam", " ", "\[Alpha]m"}], "+", RowBox[{"wAm", " ", "\[Alpha]m"}]}]]], "Output", CellChangeTimes->{ 3.697135930397572*^9, 3.697138969719345*^9, 3.697159074143735*^9, 3.697213871903405*^9, 3.697213922128715*^9, 3.697284229983076*^9, 3.697284368919626*^9, 3.697284922547516*^9, 3.697285124242982*^9, 3.697288565543076*^9, 3.69729028337392*^9, 3.6972973695650167`*^9, 3.699818048872463*^9, 3.710776888310074*^9, 3.712019371175259*^9, 3.724361606244821*^9, {3.726692965804451*^9, 3.726692991535873*^9}, 3.726947202647744*^9, 3.732977018467165*^9, 3.735585320886518*^9}, ExpressionUUID -> "35844e16-dee5-4e9f-8775-12422bc9d821"] }, Closed]], Cell["The mean fitness of diploid females:", "Text", CellChangeTimes->{{3.697135349044276*^9, 3.697135353715836*^9}, { 3.697135991414695*^9, 3.69713600071249*^9}, {3.697136079571793*^9, 3.6971361031227503`*^9}, 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Essentially, the X is already as \ specialized as possible for the female beneficial allele (X-A is fixed), and \ the neo-W often makes daughters with the Y-a haplotype, increasing the flow \ of \[OpenCurlyDoubleQuote]a\[CloseCurlyDoubleQuote] alleles into the \ daughters, which reduces fitness of those daughters. \n\nIf selection doesn\ \[CloseCurlyQuote]t uniformly favor \[OpenCurlyDoubleQuote]A\ \[CloseCurlyDoubleQuote] in females, however, the neo-W can spread at this \ equilibium if FAa>FAA (from \[Lambda]mA) or if ", Cell[BoxData[ RowBox[{ RowBox[{ FractionBox["Faa", "2"], "+", FractionBox["FAa", "2"]}], ">", "FAA"}]], CellChangeTimes->{{3.6971404582251177`*^9, 3.697140466798635*^9}}, ExpressionUUID->"8b6eb2dc-208e-4357-a7cd-7fa5417fe2f1"], " (from \[Lambda]ma). If FAa>FAA, the implication is that \ \[OpenCurlyDoubleQuote]a\[CloseCurlyDoubleQuote] is favored in females \ despite \[OpenCurlyDoubleQuote]A\[CloseCurlyDoubleQuote] being fixed on the \ X, which must require that \[OpenCurlyDoubleQuote]X-A\[CloseCurlyDoubleQuote] \ is sufficiently favored in males to drive the frequency of \ \[OpenCurlyDoubleQuote]X-A\[CloseCurlyDoubleQuote] to one (specifically, from \ the stability conditions, we must have ", Cell[BoxData[ RowBox[{ FractionBox["MAa", RowBox[{ RowBox[{"(", RowBox[{"Maa", "+", "MAa"}], ")"}], "/", "2"}]], ">", FractionBox["FAa", "FAA"]}]], CellChangeTimes->{{3.697140941036002*^9, 3.6971409611853733`*^9}}, ExpressionUUID->"3b211fb1-fbe2-4b68-9df8-78ec3a1d54db"], ").\n\nEven if FAaFAa, such that ", Cell[BoxData[ RowBox[{ RowBox[{ FractionBox["Faa", "2"], "+", FractionBox["FAa", "2"]}], ">", "FAA"}]], CellChangeTimes->{{3.6971404582251177`*^9, 3.697140466798635*^9}}, ExpressionUUID->"159575ac-b41b-414a-b910-f955362803c3"], ". In this case, \[OpenCurlyDoubleQuote]a\[CloseCurlyDoubleQuote] is not \ favored in females near the equilibrium where females are AA (comparing Aa to \ AA genotypes) and yet the neo-W with \[OpenCurlyDoubleQuote]a\ \[CloseCurlyDoubleQuote] can spread because it produces female aa individuals \ at a high rate by capturing Y-a haplotypes and Faa is high enough." }], "Text", CellChangeTimes->{{3.697165727540505*^9, 3.6971659350478783`*^9}, { 3.697165996048627*^9, 3.697166192312763*^9}, {3.7243640122214212`*^9, 3.724364047792406*^9}},ExpressionUUID->"74493587-1967-45d2-b76d-\ 8ccef9ad2fa9"], Cell["\<\ Confirming that sexually antagonistic selection alone does not work:\ \>", "Text", CellChangeTimes->{{3.724362555244467*^9, 3.724362596393092*^9}, { 3.7243629712298603`*^9, 3.724362974203187*^9}, {3.724363973421528*^9, 3.7243639823398447`*^9}},ExpressionUUID->"485f1ad1-eccb-48f4-b825-\ b751dd13d19b"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{"Reduce", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"{", RowBox[{"stabcondB", ",", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"\[Lambda]mA1", ">", "1"}], ")"}], "||", RowBox[{"(", RowBox[{"\[Lambda]ma1", ">", "1"}], ")"}]}], ")"}], ",", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"FAA", ">", "FAa", ">", "Faa", ">", "0"}], ")"}], "&&", RowBox[{"(", RowBox[{"Maa", ">", "MAa", ">", "MAA", ">", "0"}], ")"}]}], ")"}], "||", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"0", "<", "FAA", "<", "FAa", "<", "Faa"}], ")"}], "&&", RowBox[{"(", RowBox[{"0", "<", "Maa", "<", "MAa", "<", "MAA"}], ")"}]}], ")"}]}], ")"}]}], "}"}], "/.", "reverse"}], "/.", RowBox[{"pAveM", "\[Rule]", RowBox[{ RowBox[{"q", " ", "pYm"}], "+", RowBox[{ RowBox[{"(", RowBox[{"1", "-", "q"}], ")"}], "pXm"}]}]}]}], "/.", "equilB0"}], "/.", "nohap"}], "/.", RowBox[{"Rm", "\[Rule]", "0"}]}], "/.", RowBox[{"Rf", "\[Rule]", "0"}]}], "]"}], "]"}]], "Input", CellChangeTimes->{{3.724362941984918*^9, 3.7243629593616667`*^9}, { 3.724363162255517*^9, 3.7243631835267076`*^9}, {3.724372402856997*^9, 3.724372405635332*^9}}, ExpressionUUID -> "e78b298a-ce35-4ef8-b682-ad1201c8acd3"], Cell[BoxData["False"], "Output", CellChangeTimes->{3.724363184526938*^9, 3.72437240623696*^9, 3.7266929663203173`*^9, 3.7269472033483667`*^9, 3.732977018892531*^9, 3.735585321287098*^9}, ExpressionUUID -> "8ae9b3b3-6a5d-48f4-94f7-764023c15999"] }, Closed]], Cell["\<\ Other forms of selection allow the neo-W to invade an internally stable \ equilibrium, even without haploid selection:\ \>", "Text", CellChangeTimes->{{3.719172341641839*^9, 3.7191723811038647`*^9}, { 3.719172439739073*^9, 3.71917245764992*^9}, {3.724363360164721*^9, 3.72436337442564*^9}, {3.724364054503463*^9, 3.724364055380584*^9}, { 3.724367098177601*^9, 3.724367119421631*^9}},ExpressionUUID->"a6e6bfa4-b56d-4dae-9e86-\ d974224537e7"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{"Reduce", "[", RowBox[{ RowBox[{ RowBox[{ 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We also know that if both \[Lambda]mA and \[Lambda]ma are greater than one, \ then invasion will occur for all R (as the leading eigenvalue is bounded \ between these two values, see \[OpenCurlyDoubleQuote]Invasion Conditions\ \[CloseCurlyDoubleQuote]). Scanning the above requirements, we can thus \ conclude that it is possible for unlinked neo-W chromosomes to invade with: \t* heterozygote advantage in males with directional selection in females Indeed, in the absence of haploid selection, overdominance in males is \ required for both \[Lambda]mA>1 and \[Lambda]ma>1: \ \>", "Text", CellChangeTimes->{{3.724364370001005*^9, 3.7243645858589907`*^9}, { 3.724364686585678*^9, 3.724364719230373*^9}, {3.7243649507911863`*^9, 3.724365000328431*^9}, 3.724365169373115*^9, {3.724365273851008*^9, 3.724365277953088*^9}, {3.7243658190224257`*^9, 3.724365865738779*^9}, { 3.724365934034816*^9, 3.7243660377434683`*^9}, {3.7243674934635572`*^9, 3.724367506492231*^9}, {3.7243677439388323`*^9, 3.7243678396429234`*^9}, { 3.724449061425387*^9, 3.72444912621952*^9}, {3.724449414583281*^9, 3.724449415894415*^9}},ExpressionUUID->"384c5b6a-0228-4040-8f76-\ 3b14372689c0"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"(*", RowBox[{ "Other", " ", "forms", " ", "of", " ", "selection", " ", "in", " ", "males"}], "*)"}], RowBox[{ RowBox[{"Simplify", "[", RowBox[{"Reduce", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"{", RowBox[{"stabcondB", ",", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"\[Lambda]mA1", ">", "1"}], ")"}], "&&", RowBox[{"(", RowBox[{"\[Lambda]ma1", ">", "1"}], ")"}]}], ")"}], ",", RowBox[{"(", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"Maa", "<", "MAa", "<", "MAA"}], ")"}], "||", RowBox[{"(", RowBox[{"Maa", ">", "MAa", "<", "MAA"}], ")"}], "||", RowBox[{"(", RowBox[{"Maa", ">", "MAa", ">", "MAA"}], ")"}]}], ")"}], ")"}]}], "}"}], "/.", "reverse"}], "/.", RowBox[{"pAveM", "\[Rule]", RowBox[{ RowBox[{"q", " ", "pYm"}], "+", RowBox[{ RowBox[{"(", RowBox[{"1", "-", "q"}], ")"}], "pXm"}]}]}]}], "/.", "equilB0"}], "\[IndentingNewLine]", "/.", "nohap"}], "/.", RowBox[{"Rm", "\[Rule]", "0"}]}], "/.", RowBox[{"Rf", "\[Rule]", "0"}]}], "]"}], "]"}], "\[IndentingNewLine]", RowBox[{"(*", RowBox[{"Overdominance", " ", "in", " ", "males"}], "*)"}], RowBox[{"Simplify", "[", RowBox[{"Reduce", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"{", RowBox[{"stabcondB", ",", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"\[Lambda]mA1", ">", "1"}], ")"}], "&&", RowBox[{"(", RowBox[{"\[Lambda]ma1", ">", "1"}], ")"}]}], ")"}], ",", RowBox[{"(", RowBox[{"Maa", "<", "MAa", ">", "MAA"}], ")"}]}], "}"}], "/.", "reverse"}], "/.", RowBox[{"pAveM", "\[Rule]", RowBox[{ RowBox[{"q", " ", "pYm"}], "+", RowBox[{ RowBox[{"(", RowBox[{"1", "-", "q"}], ")"}], "pXm"}]}]}]}], "/.", "equilB0"}], "\[IndentingNewLine]", "/.", "nohap"}], "/.", RowBox[{"Rm", "\[Rule]", "0"}]}], "/.", RowBox[{"Rf", "\[Rule]", "0"}]}], "]"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.724448525757742*^9, 3.7244485767179527`*^9}, { 3.7244487944589376`*^9, 3.724448795565898*^9}, {3.7244488475935802`*^9, 3.724448851974215*^9}, {3.7244490681660852`*^9, 3.724449074521701*^9}}, ExpressionUUID -> "5b1ff3f6-f69a-486e-b421-311ecad25a0c"], Cell[BoxData["False"], "Output", CellChangeTimes->{ 3.7244485311831503`*^9, {3.724448566105516*^9, 3.724448581065548*^9}, 3.72444879923344*^9, 3.72444885548101*^9, 3.724449074883088*^9, 3.726692968251583*^9, 3.726947204313889*^9, 3.732977019736599*^9, 3.735585322318946*^9}, ExpressionUUID -> "53b64658-1cc3-4b9b-be1b-9aa34928279f"], Cell[BoxData[ RowBox[{ RowBox[{"FAA", ">", "0"}], "&&", RowBox[{"FAA", "<", "FAa", "<", RowBox[{"2", " ", "FAA"}]}], "&&", RowBox[{"MAa", ">", "0"}], "&&", RowBox[{"Maa", ">", "0"}], "&&", RowBox[{ FractionBox[ RowBox[{"2", " ", "FAA", " ", "MAa"}], "FAa"], ">", RowBox[{"Maa", "+", "MAa"}]}], "&&", RowBox[{"0", "<", "MAA", "<", "MAa"}], "&&", RowBox[{ RowBox[{"Faa", "+", "FAa"}], ">", RowBox[{"2", " ", "FAA"}]}]}]], "Output", CellChangeTimes->{ 3.7244485311831503`*^9, {3.724448566105516*^9, 3.724448581065548*^9}, 3.72444879923344*^9, 3.72444885548101*^9, 3.724449074883088*^9, 3.726692968251583*^9, 3.726947204313889*^9, 3.732977019736599*^9, 3.735585322323447*^9}, ExpressionUUID -> "53b64658-1cc3-4b9b-be1b-9aa34928279f"] }, Closed]], Cell["\<\ Various forms of haploid selection are also consistent with both \[Lambda]mA \ and \[Lambda]ma being greater than one (ensuring unlinked neo-W can invade). \ For example, meiotic drive for the Y linked allele in males skews the sex \ ratio towards males, allowing a neo-W chromosome to invade (which balances \ the sex ratio). Meiotic drive in females can also work, if balanced by the \ right type of diploid selection. Haploid selection coupled for example with sexual antagonism in diploids:\ \>", "Text", CellChangeTimes->{{3.724364370001005*^9, 3.7243645858589907`*^9}, { 3.724364686585678*^9, 3.724364719230373*^9}, {3.7243649507911863`*^9, 3.724365000328431*^9}, 3.724365169373115*^9, {3.724365273851008*^9, 3.724365277953088*^9}, {3.7243658190224257`*^9, 3.724365865738779*^9}, { 3.724365934034816*^9, 3.7243660377434683`*^9}, {3.7243674934635572`*^9, 3.724367506492231*^9}, {3.7243677439388323`*^9, 3.7243678396429234`*^9}, { 3.724449143340851*^9, 3.724449161313517*^9}, {3.72444943337956*^9, 3.724449456603344*^9}, {3.724450321832451*^9, 3.724450413545866*^9}, { 3.724450526941691*^9, 3.7244505683392687`*^9}},ExpressionUUID->"8623538d-88c9-49f8-9b2c-\ e835a6a35eaf"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"(*", RowBox[{ "Including", " ", "meiotic", " ", "drive", " ", "in", " ", "males"}], "*)"}], "\[IndentingNewLine]", RowBox[{"Simplify", "[", RowBox[{"Reduce", "[", RowBox[{ RowBox[{ RowBox[{ 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This allows \ the invasion of unlinked neo-W chromosomes. Here we consider when an unlinked neo-W can spread with only one of \ \[Lambda]mA or \[Lambda]ma greater than one. In this case (see the \ \[OpenCurlyDoubleQuote]Invasion Condition\[CloseCurlyDoubleQuote] section), \ we showed that a neo-W will invade if and only if the intercept of the \ primary quadratic is negative:\ \>", "Text", CellChangeTimes->{{3.724450634549142*^9, 3.724450715784717*^9}},ExpressionUUID->"41a62566-258f-4fce-9697-\ 4326b49b79e0"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"1", "+", "b", "+", "c"}], ")"}], "/.", RowBox[{"b", "->", RowBox[{"-", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"\[Lambda]mA", " ", "-", " ", "\[Chi]mA"}], ")"}], " ", "+", " ", RowBox[{"(", RowBox[{"\[Lambda]ma", " ", "-", " ", "\[Chi]ma"}], ")"}]}], ")"}]}]}]}], " ", "/.", RowBox[{"c", "->", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"\[Lambda]mA", "-", "\[Chi]mA"}], ")"}], " ", RowBox[{"(", RowBox[{"\[Lambda]ma", "-", "\[Chi]ma"}], ")"}]}], "-", RowBox[{"\[Chi]mA", " ", "\[Chi]ma"}]}]}]}], "//", "Factor"}]], "Input", CellChangeTimes->{{3.724271104777646*^9, 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The two selection terms in parentheses in \ the numerator are positive when there is a stable polymorphism\ \>", "Text", CellChangeTimes->{{3.7195147762388773`*^9, 3.7195148319956713`*^9}},ExpressionUUID->"929d5473-ad71-4ae4-830b-\ 8c57edf3c3fb"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"stabconds", "/.", RowBox[{"hAf", "\[Rule]", "hAm"}]}], "/.", RowBox[{"hAm", "\[Rule]", "hA"}]}], "/.", RowBox[{"R", "\[Rule]", RowBox[{"1", "/", "2"}]}]}], "/.", RowBox[{"tf", "\[Rule]", "0"}]}], "/.", RowBox[{"tm", "\[Rule]", "0"}]}], "/.", RowBox[{"\[Alpha]1m", "\[Rule]", "0"}]}], "//", "Simplify"}]], "Input", CellChangeTimes->{{3.719514484376442*^9, 3.719514490862681*^9}, { 3.719514555586302*^9, 3.7195145614463663`*^9}, {3.719514725696336*^9, 3.719514726370994*^9}, {3.719514756480866*^9, 3.7195147575367603`*^9}, { 3.719515020310872*^9, 3.719515020840652*^9}}, ExpressionUUID -> "0e7b9035-4714-4580-867e-e64583486bc6"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "hA"}], ")"}], " ", RowBox[{"(", RowBox[{"sAf", "+", "sAm"}], ")"}]}], ">", RowBox[{"2", " ", "\[Alpha]1f"}]}], ",", RowBox[{ RowBox[{ RowBox[{"hA", " ", RowBox[{"(", RowBox[{"sAf", "+", "sAm"}], ")"}]}], "+", RowBox[{"2", " ", "\[Alpha]1f"}]}], ">", "0"}]}], "}"}]], "Output", CellChangeTimes->{{3.7195144848805113`*^9, 3.7195144911478157`*^9}, { 3.719514555933391*^9, 3.719514561866493*^9}, 3.71951472683701*^9, 3.71951475782166*^9, 3.71951502124568*^9, 3.71951825836548*^9, 3.719756932384733*^9, 3.7198445761775503`*^9, 3.7199342094677677`*^9, 3.7201915363622913`*^9, 3.720355807456831*^9, 3.720368399524125*^9, 3.7204461748380423`*^9, 3.720450730831497*^9, 3.720451590005844*^9, 3.720464764879394*^9, 3.725136586557104*^9, 3.725719836791959*^9, 3.7269472116834717`*^9, 3.732977026558984*^9, 3.735585330423719*^9}, ExpressionUUID -> "aad5ccf9-be35-4a15-9eab-7144dec09970"] }, Closed]], Cell["\<\ Thus we need sAf sAm > 0 for a neo-W to invade when the only haploid \ selection is female drive.\ \>", "Text", CellChangeTimes->{{3.7195148340136423`*^9, 3.719514865057172*^9}, { 3.719514901227985*^9, 3.719514916344853*^9}, {3.719515046686426*^9, 3.719515047240765*^9}, {3.7195151323258457`*^9, 3.719515132407991*^9}},ExpressionUUID->"df498b96-300a-454d-9aad-\ 871577fd9f8b"], Cell["\<\ When the only haploid selection is male gamete competition a neo-W can invade \ only when the following is positive\ \>", "Text", CellChangeTimes->{{3.719512507506158*^9, 3.719512549690648*^9}, { 3.719514884356893*^9, 3.71951488668377*^9}, {3.7195149232759457`*^9, 3.719514936462377*^9}, {3.7195150650110283`*^9, 3.719515070833517*^9}},ExpressionUUID->"3529053c-a746-43d6-be14-\ 3e29183f6e63"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ FractionBox["1", "4"], "VA", " ", SuperscriptBox["SA", "2"], FractionBox[ RowBox[{"r", "-", "R"}], 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The two selection terms in parentheses in \ the numerator are positive when there is a stable polymorphism\ \>", "Text", CellChangeTimes->{{3.7195147762388773`*^9, 3.7195148319956713`*^9}},ExpressionUUID->"abd8f17e-80d5-4077-bee4-\ b04eb9db0401"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"stabconds", "/.", RowBox[{"hAf", "\[Rule]", "hAm"}]}], "/.", RowBox[{"hAm", "\[Rule]", "hA"}]}], "/.", RowBox[{"R", "\[Rule]", RowBox[{"1", "/", "2"}]}]}], "/.", RowBox[{"tf", "\[Rule]", "0"}]}], "/.", RowBox[{"\[Alpha]1m", "\[Rule]", "0"}]}], "/.", RowBox[{"\[Alpha]1f", "\[Rule]", "0"}]}], "//", "Simplify"}]], "Input", CellChangeTimes->{{3.719514484376442*^9, 3.719514490862681*^9}, { 3.719514555586302*^9, 3.7195145614463663`*^9}, {3.719514725696336*^9, 3.719514726370994*^9}, {3.719514756480866*^9, 3.7195147575367603`*^9}, { 3.719515020310872*^9, 3.719515020840652*^9}, {3.719515090134296*^9, 3.71951509599741*^9}}, ExpressionUUID -> "fe4bdf82-8646-47b4-8802-8ce8ed522dd7"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "hA"}], ")"}], " ", RowBox[{"(", RowBox[{"sAf", "+", "sAm"}], ")"}]}], ">", "tm"}], ",", RowBox[{ RowBox[{ RowBox[{"hA", " ", RowBox[{"(", RowBox[{"sAf", "+", "sAm"}], ")"}]}], "+", "tm"}], ">", "0"}]}], "}"}]], "Output", CellChangeTimes->{{3.7195144848805113`*^9, 3.7195144911478157`*^9}, { 3.719514555933391*^9, 3.719514561866493*^9}, 3.71951472683701*^9, 3.71951475782166*^9, 3.71951502124568*^9, 3.719515096347609*^9, 3.71951825844742*^9, 3.719756932484593*^9, 3.71984457627901*^9, 3.719934209590769*^9, 3.720191536452339*^9, 3.720355807561907*^9, 3.720368399623972*^9, 3.720446174954845*^9, 3.720450730949932*^9, 3.720451590122764*^9, 3.720464765013495*^9, 3.7251365867248917`*^9, 3.725719836892557*^9, 3.726947211832821*^9, 3.732977026652116*^9, 3.735585330522397*^9}, ExpressionUUID -> "6f3c455a-50cb-42ac-a255-4bde31a8923f"] }, Closed]], Cell["\<\ Thus we need sAf (sAf - sAm) < 0 for a neo-W to invade when the only haploid \ selection is male gamete competition.\ \>", "Text", CellChangeTimes->{{3.7195148340136423`*^9, 3.719514865057172*^9}, { 3.719514901227985*^9, 3.719514916344853*^9}, {3.719515046686426*^9, 3.719515047240765*^9}, {3.7195151186856127`*^9, 3.719515137030348*^9}, { 3.72036051114159*^9, 3.72036051152356*^9}},ExpressionUUID->"06b873fb-423e-45dd-ad3b-\ d3bd671f4573"], Cell["\<\ When the only haploid selection is female gamete competition a neo-W can \ invade only when the following is positive\ \>", "Text", CellChangeTimes->{{3.719512507506158*^9, 3.719512549690648*^9}, { 3.719514884356893*^9, 3.71951488668377*^9}, {3.7195149232759457`*^9, 3.719514936462377*^9}, {3.7195150650110283`*^9, 3.719515070833517*^9}, { 3.719515145314369*^9, 3.719515145384625*^9}},ExpressionUUID->"855d36e3-bc57-4804-99a0-\ 91b4cd849c90"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ FractionBox["1", "4"], "VA", " ", SuperscriptBox["SA", "2"], FractionBox[ RowBox[{"r", "-", "R"}], RowBox[{"r", " ", "R"}]]}], "+", RowBox[{ FractionBox[ RowBox[{" ", "dpAYmXm", " "}], "2"], RowBox[{"(", RowBox[{ RowBox[{"2", "\[Alpha]Dm"}], "-", RowBox[{"2", "\[Alpha]Df"}], "+", "tm", "-", "tf"}], ")"}]}]}], "/.", RowBox[{"SA", "\[Rule]", RowBox[{ RowBox[{"(", RowBox[{"Dm", "+", "\[Alpha]Dm", "+", "tm"}], ")"}], "-", RowBox[{"(", RowBox[{"Df", "+", "\[Alpha]Df", "+", "tf"}], ")"}]}]}]}], "/.", "trysub3"}], "/.", RowBox[{"realsol2", "[", RowBox[{"[", "3", "]"}], "]"}]}], "/.", RowBox[{"hAf", "\[Rule]", "hAm"}]}], "/.", RowBox[{"hAm", "\[Rule]", "hA"}]}], "/.", RowBox[{"R", "\[Rule]", RowBox[{"1", "/", "2"}]}]}], "/.", RowBox[{"tm", "\[Rule]", "0"}]}], "/.", RowBox[{"\[Alpha]1m", "\[Rule]", "0"}]}], "/.", RowBox[{"\[Alpha]1f", "\[Rule]", "0"}]}], "//", "Simplify"}]], "Input", CellChangeTimes->{{3.719342572113749*^9, 3.719342574437006*^9}, { 3.719513203888503*^9, 3.719513215868059*^9}, {3.719513469037504*^9, 3.7195134966026773`*^9}, {3.7195135425237827`*^9, 3.719513548551619*^9}, { 3.7195137120655518`*^9, 3.719513868789322*^9}, {3.719513902423485*^9, 3.719513903853347*^9}, {3.7195143340711927`*^9, 3.7195143778832893`*^9}, { 3.7195148694873543`*^9, 3.719514871845229*^9}, {3.719514949892138*^9, 3.71951495210909*^9}, {3.719515074907977*^9, 3.719515080162184*^9}, { 3.719515148741364*^9, 3.719515149113656*^9}}, ExpressionUUID -> "5dcc90ac-5c31-4f96-a898-03174376174f"], Cell[BoxData[ RowBox[{"-", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{"2", " ", "r"}]}], ")"}], " ", RowBox[{"(", RowBox[{"sAf", "-", "sAm"}], ")"}], " ", "sAm", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "hA"}], ")"}], " ", "sAf"}], "+", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "hA"}], ")"}], " ", "sAm"}], "-", "tf"}], ")"}], " ", SuperscriptBox["tf", "2"], " ", RowBox[{"(", RowBox[{ RowBox[{"hA", " ", RowBox[{"(", RowBox[{"sAf", "+", "sAm"}], ")"}]}], "+", "tf"}], ")"}]}], RowBox[{"2", " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "-", RowBox[{"2", " ", "hA"}]}], ")"}], "2"], " ", "r", " ", SuperscriptBox[ RowBox[{"(", RowBox[{"sAf", "+", "sAm"}], ")"}], "4"]}]]}]], "Output", CellChangeTimes->{ 3.719342574989887*^9, 3.719511656375464*^9, {3.719513204412901*^9, 3.719513216294621*^9}, {3.719513469384255*^9, 3.719513497034894*^9}, { 3.719513544652604*^9, 3.719513549673626*^9}, {3.719513712566848*^9, 3.719513762387082*^9}, {3.719513798545582*^9, 3.7195138716426687`*^9}, 3.719513908101605*^9, {3.719514344410779*^9, 3.719514378210956*^9}, 3.719514952527379*^9, 3.71951508126272*^9, 3.719515149616847*^9, 3.719518258482698*^9, 3.719756932535348*^9, 3.719844576328149*^9, 3.719934209653963*^9, 3.720191536493169*^9, 3.720355807612732*^9, 3.7203683996733427`*^9, 3.720446175004052*^9, 3.720450730999123*^9, 3.720451590173078*^9, 3.7204647650806103`*^9, 3.725136586820299*^9, 3.725719836943623*^9, 3.7269472118991547`*^9, 3.732977026698101*^9, 3.7355853305700283`*^9}, ExpressionUUID -> "0f349b00-1ab1-407a-bf41-ec83e9882ace"] }, Closed]], Cell["\<\ The denominator is always positive. The two selection terms in parentheses in \ the numerator are positive when there is a stable polymorphism\ \>", "Text", CellChangeTimes->{{3.7195147762388773`*^9, 3.7195148319956713`*^9}},ExpressionUUID->"fd6c7366-8a1e-4021-86ea-\ 407edad24a84"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"stabconds", "/.", RowBox[{"hAf", "\[Rule]", "hAm"}]}], "/.", RowBox[{"hAm", "\[Rule]", "hA"}]}], "/.", RowBox[{"R", "\[Rule]", RowBox[{"1", "/", "2"}]}]}], "/.", RowBox[{"tm", "\[Rule]", "0"}]}], "/.", RowBox[{"\[Alpha]1m", "\[Rule]", "0"}]}], "/.", RowBox[{"\[Alpha]1f", "\[Rule]", "0"}]}], "//", "Simplify"}]], "Input", CellChangeTimes->{{3.719514484376442*^9, 3.719514490862681*^9}, { 3.719514555586302*^9, 3.7195145614463663`*^9}, {3.719514725696336*^9, 3.719514726370994*^9}, {3.719514756480866*^9, 3.7195147575367603`*^9}, { 3.719515020310872*^9, 3.719515020840652*^9}, {3.719515090134296*^9, 3.71951509599741*^9}, {3.719515156640312*^9, 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