(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 5190597, 86006] NotebookOptionsPosition[ 5184418, 85811] NotebookOutlinePosition[ 5185800, 85853] CellTagsIndexPosition[ 5185757, 85850] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Exercises lesson 2", "Title", CellChangeTimes->{{3.4323368555625*^9, 3.4323368600625*^9}, { 3.432915856640625*^9, 3.43291586221875*^9}}], Cell[CellGroupData[{ Cell["1. DISCRETE GENERATION MODELS (RECURRENCE EQUATION MODELS)", "Section", CellChangeTimes->{{3.432925002453125*^9, 3.432925019578125*^9}, { 3.4329309591875*^9, 3.432930976859375*^9}, {3.43293521625*^9, 3.43293521775*^9}, {3.4329531121875*^9, 3.43295311253125*^9}, 3.432954155984375*^9}], Cell[CellGroupData[{ Cell["\<\ 1.1 The melanic form of the British peppered moth was observed to increase \ from a frequency of 0.00001 to 0.8 in 50 generations. The melanic form is dominant and we will call the fitnesses of the three \ genotypes: \tWaa = 1 \t(wild type) \tWAa = 1+s\t(melanic form, heterozygote) \tWAA = 1+s\t(melanic form, homozygote) \t The recurrence equation for diploid selection developed in class is : \tp[t+1] = ((1+s)*p[t]^2+(1+s)*p[t]*(1-p[t])) / \ ((1+s)*p[t]^2+(1+s)*2*p[t]*(1-p[t])+(1-p[t])^2) Evaluate this equation numerically for various values of s to estimate how strong selection must have been to take p from 0.00001 to 0.8 in 50 \ generations.\ \>", "Subsection", CellChangeTimes->{{3.432915853421875*^9, 3.432915919265625*^9}, { 3.43291600403125*^9, 3.432916004546875*^9}, {3.432916133125*^9, 3.43291613521875*^9}, {3.432925063828125*^9, 3.4329250735625*^9}, { 3.432953114453125*^9, 3.432953122984375*^9}, {3.432954159359375*^9, 3.432954159609375*^9}, 3.432971627*^9, {3.43299095840625*^9, 3.43299096078125*^9}}, ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[{ RowBox[{ RowBox[{"p", "[", RowBox[{"t_", ",", "s_"}], "]"}], 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RowBox[{"t", "-", "1"}], ",", "s"}], "]"}]}], ")"}], "^", "2"}]}]]}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"p", "[", RowBox[{"0", ",", "s_"}], "]"}], ":=", "0.00001"}]}], "Input", CellChangeTimes->{ 3.432915977984375*^9, {3.432916026375*^9, 3.4329160681875*^9}, { 3.432916118953125*^9, 3.432916119078125*^9}, {3.432916177765625*^9, 3.432916205484375*^9}, {3.432916972109375*^9, 3.432916977859375*^9}, { 3.432917735703125*^9, 3.43291773596875*^9}, {3.432917773625*^9, 3.432917773890625*^9}, 3.432918121421875*^9, {3.432918205953125*^9, 3.432918206796875*^9}, {3.432918319765625*^9, 3.432918319984375*^9}, { 3.43292120234375*^9, 3.4329212446875*^9}}], Cell["\<\ You can see that the frequency of the melanic type after 50 generations is \ 0.8 for s around 0.4 :\ \>", "Text", CellChangeTimes->{{3.43297492065625*^9, 3.43297496246875*^9}}], Cell[CellGroupData[{ Cell["\<\ ListPlot[Table[{s,p[50,s]},{s,0,0.5,0.01}], \tJoined->True,Frame->True,FrameLabel->{\"Selective advantage s of melanic \ type\",\"Frequency of melanic type after 50 gens\"}]\ \>", "Input", CellChangeTimes->{ 3.432916260484375*^9, {3.432917907546875*^9, 3.43291791353125*^9}, { 3.432918004046875*^9, 3.43291803165625*^9}, {3.43291821659375*^9, 3.43291821671875*^9}, {3.432925138359375*^9, 3.432925188046875*^9}, { 3.432971662359375*^9, 3.432971700109375*^9}, {3.432975425125*^9, 3.43297543140625*^9}}, ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ GraphicsBox[{{}, {}, {Hue[0.67, 0.6, 0.6], LineBox[{{0., 0.00001}, {0.01, 0.000016446106186165866`}, {0.02, 0.000026914969718244895`}, {0.03, 0.00004383609348199948}, {0.04, 0.00007105815504216794}, {0.05, 0.00011464999684024686`}, {0.06, 0.00018413739173779928`}, {0.07, 0.0002944027064491058}, {0.08, 0.0004685859975798154}, {0.09, 0.0007424860582448591}, {0.1, 0.0011711829964105257`}, {0.11, 0.0018389000817273264`}, {0.12, 0.002873481797206464}, {0.13, 0.004467220234333245}, {0.14, 0.006905909953722324}, {0.15, 0.01060746121542113}, {0.16, 0.016169132349738047`}, 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0.8582580124790286}}]}}, AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948], Axes->True, AxesOrigin->{0, 0}, Frame->True, FrameLabel->{ FormBox["\"Selective advantage s of melanic type\"", TraditionalForm], FormBox["\"Frequency of melanic type after 50 gens\"", TraditionalForm]}, PlotRange->{{0., 0.5}, {0., 0.8582580124790286}}, PlotRangeClipping->True, PlotRangePadding->{ Scaled[0.02], Scaled[0.02]}]], "Output", CellChangeTimes->{3.432971701125*^9, 3.43297260465625*^9, 3.432975431953125*^9, 3.43297798621875*^9}] }, Open ]], Cell["\<\ Unfortunately the standard way of Solving for a value, e.g. using \ FindRoot[p[50,s]==0.8,{s,0.4}], does not appear to work because the function \ is non differentiable:\ \>", "Text", CellChangeTimes->{{3.43297492065625*^9, 3.432975028671875*^9}, { 3.432975060078125*^9, 3.43297506453125*^9}, {3.432975170140625*^9, 3.4329751795*^9}}], Cell[BoxData[ RowBox[{"FindRoot", "[", RowBox[{ RowBox[{ RowBox[{"p", "[", RowBox[{"50", ",", "s"}], "]"}], "==", "0.8"}], ",", RowBox[{"{", RowBox[{"s", ",", "0.4"}], "}"}]}], "]"}]], "Input"], Cell["However you can find the value of s by trial and error:", "Text", CellChangeTimes->{{3.43297492065625*^9, 3.432975028671875*^9}, { 3.432975060078125*^9, 3.43297506453125*^9}, {3.4329751040625*^9, 3.432975112953125*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"p", "[", RowBox[{"50", ",", "0.415"}], "]"}]], "Input", CellChangeTimes->{{3.432918062171875*^9, 3.432918081453125*^9}, { 3.432974504328125*^9, 3.43297451234375*^9}, 3.4329748180625*^9, 3.432974887703125*^9, {3.432975119046875*^9, 3.4329751193125*^9}}], Cell[BoxData["0.8003097856494822`"], "Output", CellChangeTimes->{{3.432918063484375*^9, 3.432918082109375*^9}, 3.43292124921875*^9, 3.432925165703125*^9, 3.432971638515625*^9, 3.43297204925*^9, 3.432972740828125*^9, {3.432974505015625*^9, 3.4329745125625*^9}, 3.4329748185*^9, 3.432974888625*^9, 3.432975153921875*^9}] }, Open ]], Cell["\<\ Or you can calculate a large set of values, do a polynomial fit up to a large \ order (130), and then use FindRoot on this function:\ \>", "Text", CellChangeTimes->{{3.43297492065625*^9, 3.432975028671875*^9}, { 3.432975060078125*^9, 3.43297506453125*^9}, {3.4329751040625*^9, 3.432975112953125*^9}, {3.432975162734375*^9, 3.432975230640625*^9}, { 3.432978167703125*^9, 3.432978168*^9}, {3.432978483765625*^9, 3.432978484375*^9}, {3.432992777625*^9, 3.432992782890625*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"listvals", "=", RowBox[{"Table", "[", RowBox[{ RowBox[{"{", RowBox[{"s", ",", RowBox[{"p", "[", RowBox[{"50", ",", "s"}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"s", ",", "0", ",", "0.5", ",", "0.0025"}], "}"}]}], "]"}]}], ";"}]], "Input", CellChangeTimes->{{3.43297425528125*^9, 3.432974276671875*^9}, 3.432977998703125*^9, 3.43297820340625*^9, {3.432978234984375*^9, 3.432978235359375*^9}, {3.4329783313125*^9, 3.432978332890625*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"fittedfunc", "[", "s_", "]"}], "=", RowBox[{"Fit", "[", 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CellChangeTimes->{ 3.432974392984375*^9, {3.432974435625*^9, 3.43297449846875*^9}, { 3.43297458434375*^9, 3.43297460196875*^9}, {3.432974793828125*^9, 3.432974809890625*^9}, {3.432974858078125*^9, 3.43297487415625*^9}, { 3.432975238109375*^9, 3.432975252046875*^9}, 3.43297810859375*^9, 3.43297818003125*^9, {3.432978248578125*^9, 3.4329782725625*^9}, 3.432978319609375*^9, {3.432978357546875*^9, 3.432978372390625*^9}, 3.432978407359375*^9}] }, Open ]], Cell["This is indeed a little bit more accurate:", "Text", CellChangeTimes->{{3.43297492065625*^9, 3.432975028671875*^9}, { 3.432975060078125*^9, 3.43297506453125*^9}, {3.4329751040625*^9, 3.432975112953125*^9}, {3.432975162734375*^9, 3.432975230640625*^9}, { 3.432975287953125*^9, 3.432975293515625*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"p", "[", RowBox[{"50", ",", "0.41470"}], "]"}]], "Input", CellChangeTimes->{{3.432918062171875*^9, 3.432918081453125*^9}, { 3.432974504328125*^9, 3.43297451234375*^9}, 3.4329748180625*^9, 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This shows the increase in frequency of the melanic form as a function of \ time for s=0.41470 :\ \>", "Text", CellChangeTimes->{{3.432925088796875*^9, 3.432925116890625*^9}, { 3.432975299375*^9, 3.432975328171875*^9}, {3.43297538528125*^9, 3.432975393296875*^9}, {3.4329754486875*^9, 3.4329754488125*^9}, 3.432978477875*^9}], Cell[CellGroupData[{ Cell["\<\ ListPlot[Table[p[t,0.41470],{t,0,50}], \tJoined->True,Frame->True,FrameLabel->{\"Generation t\",\"Frequency of \ melanic type p\"}]\ \>", "Input", CellChangeTimes->{ 3.432916260484375*^9, {3.432917907546875*^9, 3.43291791353125*^9}, { 3.432918004046875*^9, 3.43291803165625*^9}, {3.43291821659375*^9, 3.43291821671875*^9}, {3.432925138359375*^9, 3.432925188046875*^9}, { 3.432975398375*^9, 3.432975399765625*^9}, 3.432978423734375*^9, 3.43297847246875*^9}, ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ GraphicsBox[{{}, {}, {Hue[0.67, 0.6, 0.6], LineBox[{{1., 0.00001}, {2., 0.000014146882666341837`}, {3., 0.00002001336008448561}, {4., 0.00002831243055590056}, {5., 0.00004005265499108888}, {6., 0.00005666060880674523}, {7., 0.00008015399660393702}, {8., 0.00011338632140624449`}, {9., 0.00016039254601410912`}, {10., 0.00022687715589436348`}, {11., 0.0003209027344122781}, {12., 0.00045386031977760505`}, {13., 0.0006418346422801729}, {14., 0.0009075205161302969}, {15., 0.0012829040736284142`}, {16., 0.0018129965284045292`}, {17., 0.002560998707616486}, {18., 0.003615375314152233}, {19., 0.005099408052717854}, {20., 0.007183826413716594}, {21., 0.010102979213349376`}, {22., 0.014174510629289675`}, {23., 0.019821305710302984`}, {24., 0.02759208857326542}, {25., 0.03817299536124031}, {26., 0.05237670511406256}, {27., 0.07108996439249526}, {28., 0.09515960622366154}, {29., 0.1252102267204998}, {30., 0.1614209258919006}, {31., 0.2033362140983139}, {32., 0.24981283979091962`}, {33., 0.29916684420018114`}, {34., 0.3494853771176968}, {35., 0.3989769966489365}, {36., 0.44622771447851}, {37., 0.4903031708272187}, {38., 0.5307196737177394}, {39., 0.567345009596925}, {40., 0.6002839679597182}, {41., 0.629779863558329}, {42., 0.656142442033956}, {43., 0.6797007772682145}, {44., 0.7007754245403341}, {45., 0.7196637665676191}, {46., 0.7366336837733027}, {47., 0.7519221176748652}, {48., 0.765736294348319}, {49., 0.7782562361123256}, {50., 0.7896377601445774}, {51., 0.8000155212110011}}]}}, AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948], Axes->True, AxesOrigin->{0, 0}, Frame->True, FrameLabel->{ FormBox["\"Generation t\"", TraditionalForm], FormBox["\"Frequency of melanic type p\"", TraditionalForm]}, PlotRange->{{0., 51.}, {0., 0.8000155212110011}}, PlotRangeClipping->True, PlotRangePadding->{ Scaled[0.02], Scaled[0.02]}]], "Output", CellChangeTimes->{ 3.4329162650625*^9, 3.432916350828125*^9, 3.43291785903125*^9, 3.43291791409375*^9, {3.432918025921875*^9, 3.432918032125*^9}, 3.4329181268125*^9, {3.43291818521875*^9, 3.4329182173125*^9}, 3.43292125171875*^9, {3.432925160359375*^9, 3.432925190625*^9}, 3.43297540128125*^9, 3.4329784298125*^9, 3.432978473890625*^9}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "1.2 In the previous class we discussed the hawk-dove model of contest \ evolution whereby the fitness of the hawk and of the dove type was given by\n\ \n\t\tWH = w0 + (1 - p)*v + p*(v - c)/2\nand\n\t\tWD = w0 + (1 - p)*(v/2);\n\t\ \t\nwhere\t\t", StyleBox["v", FontSlant->"Italic"], " = value of the resource\n\t\t", StyleBox["c", FontSlant->"Italic"], " = fighting cost that occurs when two hawks meet\n\t\t", StyleBox["p", FontSlant->"Italic"], " = frequency of the hawk genotype in the population\n\t\t", StyleBox["w0", FontSlant->"Italic"], " = baseline fitness\n\t\t\nUse the haploid selection model discussed in \ class to calculate the equilibrium (ESS) frequency of the hawk type in the \ population (assuming we are dealing with haploid organisms than interact \ randomly).\nDetermine the equilibrium using Solve[] and investigate \ graphically which equilibrium is reached when v < c and when v > c by \ plotting the frequency of the hawk type p as a function of time using \ ListPlot." }], "Subsection", CellChangeTimes->{ 3.432342989015625*^9, {3.43234313646875*^9, 3.432343148421875*^9}, { 3.4324553335625*^9, 3.43245533703125*^9}, {3.432925640484375*^9, 3.432925795515625*^9}, {3.43292590009375*^9, 3.4329259014375*^9}, { 3.432925959234375*^9, 3.432925970375*^9}, {3.432926791375*^9, 3.432926803203125*^9}, {3.432927575984375*^9, 3.432927633734375*^9}, 3.432953128046875*^9, {3.43295416415625*^9, 3.432954164390625*^9}, { 3.4329928720625*^9, 3.432992879078125*^9}}], Cell["The fitness of individuals of the hawk genotype is", "Text", CellChangeTimes->{{3.432455342671875*^9, 3.43245537703125*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"WH", "=", RowBox[{"w0", "+", RowBox[{ RowBox[{"(", RowBox[{"1", "-", "p"}], ")"}], "*", "v"}], "+", RowBox[{"p", "*", RowBox[{ RowBox[{"(", RowBox[{"v", "-", "c"}], ")"}], "/", "2"}]}]}]}], ";"}]], "Input", CellChangeTimes->{{3.4323430275625*^9, 3.4323430573125*^9}, { 3.43245539421875*^9, 3.43245544859375*^9}, {3.432503578002963*^9, 3.432503579440463*^9}, 3.432506156362338*^9, 3.432925856390625*^9, 3.43292590390625*^9, {3.432926806140625*^9, 3.432926808515625*^9}}], Cell["The fitness of individuals of the dove genotype is", "Text", CellChangeTimes->{{3.432455342671875*^9, 3.43245537703125*^9}, { 3.43245546490625*^9, 3.432455465375*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"WD", "=", RowBox[{"w0", "+", RowBox[{ RowBox[{"(", RowBox[{"1", "-", "p"}], ")"}], "*", RowBox[{"(", RowBox[{"v", "/", "2"}], ")"}]}]}]}], ";"}]], "Input", CellChangeTimes->{{3.4323430275625*^9, 3.4323430573125*^9}, { 3.43245539421875*^9, 3.43245549328125*^9}, {3.432503581846713*^9, 3.432503583221713*^9}, 3.432506160221713*^9, 3.432925874390625*^9, { 3.432925907015625*^9, 3.43292590728125*^9}, 3.432926812421875*^9}], Cell["\<\ Following the haploid selection model the frequency of hawk in the next \ generation is p(t+1)=p(t).WH / (p(t).WH + (1-p(t)).WD), and there are three candidate \ equilibria (ESS frequencies of the hawk type) :\ \>", "Text", CellChangeTimes->{{3.432455342671875*^9, 3.43245537703125*^9}, { 3.43245546490625*^9, 3.432455465375*^9}, {3.432455533265625*^9, 3.43245554965625*^9}, {3.432506206987338*^9, 3.432506238706088*^9}, { 3.4329259298125*^9, 3.432925947453125*^9}, {3.432925986859375*^9, 3.432926020328125*^9}, {3.43292620259375*^9, 3.432926209328125*^9}, { 3.432926855765625*^9, 3.43292686753125*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Solve", "[", RowBox[{ RowBox[{"p", "==", RowBox[{"p", "*", RowBox[{"WH", "/", RowBox[{"(", RowBox[{ RowBox[{"p", "*", "WH"}], "+", RowBox[{ RowBox[{"(", RowBox[{"1", "-", "p"}], ")"}], "*", "WD"}]}], ")"}]}]}]}], ",", "p"}], "]"}]], "Input", CellChangeTimes->{{3.432921912734375*^9, 3.4329219149375*^9}, { 3.432922001640625*^9, 3.432922029125*^9}, {3.43292589053125*^9, 3.432925896984375*^9}, {3.432926127421875*^9, 3.432926168265625*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"p", "\[Rule]", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"p", "\[Rule]", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"p", "\[Rule]", FractionBox["v", "c"]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.432922029671875*^9, 3.4329261829375*^9, 3.432926817921875*^9, 3.432926869421875*^9}] }, Open ]], Cell["\<\ More simply you could also just determine the equilibrium using the condition \ Solve[WH ==WD, p])\ \>", "Text", CellChangeTimes->{{3.43297552909375*^9, 3.43297560309375*^9}}], Cell["\<\ To get rid of the curly brackets you can use [[1]], whereby [[1]] means the \ first part of a list or equation, [[2]] means the second part of a list or \ equation, etc... So if we use [[1]][[1]][[2]] it means the second part of the equation of the \ first element in the first element of the nested list, and we end up with \ just V/C which is the part we need. With Clip[] we can also specify that the biologically relevant solution has \ to lie between 0 and 1.\ \>", "Text", CellChangeTimes->{{3.432455342671875*^9, 3.43245537703125*^9}, { 3.43245546490625*^9, 3.432455465375*^9}, {3.432455533265625*^9, 3.43245560509375*^9}, {3.432455640609375*^9, 3.432455724921875*^9}, { 3.432543396768125*^9, 3.43254343003375*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"ESS", "[", RowBox[{"v_", ",", "c_"}], "]"}], "=", RowBox[{"Clip", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"Solve", "[", RowBox[{ RowBox[{"p", "==", RowBox[{"p", "*", RowBox[{"WH", "/", RowBox[{"(", RowBox[{ RowBox[{"p", "*", "WH"}], "+", RowBox[{ RowBox[{"(", RowBox[{"1", "-", "p"}], ")"}], "*", "WD"}]}], ")"}]}]}]}], ",", "p"}], "]"}], "[", RowBox[{"[", "3", "]"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}], "[", RowBox[{"[", "2", "]"}], "]"}], ",", RowBox[{"{", RowBox[{"0", ",", "1"}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.432455496015625*^9, 3.43245551915625*^9}, { 3.43245572909375*^9, 3.432455746671875*^9}, {3.432459759546875*^9, 3.43245976521875*^9}, {3.432506170659213*^9, 3.432506182409213*^9}, { 3.432506244440463*^9, 3.432506251081088*^9}, {3.43254335184625*^9, 3.43254336265875*^9}, {3.43292635409375*^9, 3.43292636021875*^9}, { 3.432926824546875*^9, 3.43292682603125*^9}}], Cell[BoxData[ RowBox[{"Clip", "[", RowBox[{ FractionBox["v", "c"], ",", RowBox[{"{", RowBox[{"0", ",", "1"}], "}"}]}], "]"}]], "Output", CellChangeTimes->{ 3.432539898424375*^9, {3.432543363393125*^9, 3.43254337459625*^9}, { 3.4329263576875*^9, 3.432926360421875*^9}, 3.432926832453125*^9, 3.43292687203125*^9}] }, Open ]], Cell["\<\ This is a plot of the ESS frequency of the hawk type p as a function of V and \ C. The options PlotStyle \[Rule] None, ClippingStyle \[Rule] None results in \ a black & white wire mesh plot. You can rotate the graph by dragging on it \ with the mouse.\ \>", "Text", CellChangeTimes->{{3.432455342671875*^9, 3.43245537703125*^9}, { 3.43245546490625*^9, 3.432455465375*^9}, {3.432455533265625*^9, 3.43245560509375*^9}, {3.432455640609375*^9, 3.432455724921875*^9}, { 3.43253851703375*^9, 3.432538560424375*^9}, {3.43292633628125*^9, 3.432926339296875*^9}, {3.432926838875*^9, 3.43292684703125*^9}, { 3.432975618046875*^9, 3.432975627578125*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Plot3D", "[", RowBox[{ RowBox[{"ESS", "[", RowBox[{"v", ",", "c"}], "]"}], ",", RowBox[{"{", RowBox[{"c", ",", "1", ",", "10"}], "}"}], ",", RowBox[{"{", RowBox[{"v", ",", "1", ",", "10"}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{"0", ",", "1"}], "}"}]}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"c", ",", "v", ",", RowBox[{"ESS", " ", "p"}]}], "}"}]}], ",", RowBox[{"PlotStyle", "\[Rule]", "None"}], ",", RowBox[{"ClippingStyle", "\[Rule]", "None"}]}], 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