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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 75756, 2284]*) (*NotebookOutlinePosition[ 76609, 2313]*) (* CellTagsIndexPosition[ 76565, 2309]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[StyleBox["Influence of age polyethism on longevity of workers \ in social insects\nAppendix", Evaluatable->False, AspectRatioFixed->True]], "Title", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times", FontSize->14, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell["Adam Tofilski", "Text", CellDingbat->None], Cell["\<\ Bee Research Department Agricultural University 29 Listopada 52 31-425 Krakow, Poland rotofils@cyf-kr.edu.pl\ \>", "Text", CellDingbat->None], Cell[TextData[{ StyleBox["This is a ", FontVariations->{"CompatibilityType"->0}], StyleBox["Mathematica", FontSlant->"Italic"], " 4.0 notebook." }], "Text", CellDingbat->None], Cell[CellGroupData[{ Cell["1 Simplified model", "Section", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times", FontSize->12, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[TextData[{ "It is assumed that there are two sets of tasks, A and B, which are \ associated with aging-independent mortality rates ", StyleBox["mA", "Input"], " and ", StyleBox["mB", "Input", FontVariations->{"CompatibilityType"->0}], " respectively. A fixed proportion of time ", StyleBox["f", "Input"], " is devoted to A-type tasks. A worker cannot spend more than the maximum \ resource ", StyleBox["k", "Input"], " available for the whole life. The rates of resource expenditure during \ tasks A and B are ", StyleBox["cA", "Input"], " and ", StyleBox["cB", "Input"], " respectively. In the simplified model aging does not affect the mortality \ of workers until a certain age is reached, when resources become exhausted. \ At this time all workers die. In the model the expected longevity of workers \ in colonies with and without age polyethism is compared. If there is no \ polyethism the workers perform tasks A and B in turn. If age polyethism is \ present the workers perform A-type tasks first and after time ", StyleBox["ts", "Input"], " they switch to B-type tasks. 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" }], "Text", AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[{ \(Solve[aAs\/\(aAs + aBs\) == f, ts]\), "\[IndentingNewLine]", \(Simplify[%]\)}], "Input", CellDingbat->None], Cell[BoxData[ \({{ts \[Rule] \(k\ mA + cA\ Log[\(f\ mA + mB - f\ mB\)\/\(f\ mA + \ \[ExponentialE]\^\(\(k\ mA\)\/cA\)\ mB - \[ExponentialE]\^\(\(k\ mA\)\/cA\)\ \ f\ mB\)]\)\/\(cA\ mA\)}}\)], "Output"], Cell[BoxData[ \({{ts \[Rule] k\/cA + Log[\(f\ mA + mB - f\ mB\)\/\(f\ mA + \ \[ExponentialE]\^\(\(k\ mA\)\/cA\)\ mB - \[ExponentialE]\^\(\(k\ mA\)\/cA\)\ \ f\ mB\)]\/mA}}\)], "Output"] }, Open ]], Cell[TextData[{ "It can be demonstrated that the expected longevity of workers in colonies \ with age polyethism is the same as in colonies without polyethism when", StyleBox[" ", "Input"], StyleBox[Cell[BoxData[ \(cB = \(mB\ cA\)\/mA\)], "Input", CellDingbat->None], "Input"], "." }], "Text", AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[{ \(pas = aAs + aBs /. %[\([1]\)]\), "\n", \(pas = Simplify[%]\)}], "Input", CellDingbat->None], Cell[BoxData[ \(\(1 - \[ExponentialE]\^\(\(-mA\)\ \((k\/cA + Log[\(f\ mA + mB - f\ mB\)\ \/\(f\ mA + \[ExponentialE]\^\(\(k\ mA\)\/cA\)\ mB - \[ExponentialE]\^\(\(k\ \ mA\)\/cA\)\ f\ mB\)]\/mA)\)\)\)\/mA - \(\[ExponentialE]\^\(-\(\(k\ mA\)\/cA\)\ \) - \[ExponentialE]\^\(\(-mA\)\ \((k\/cA + Log[\(f\ mA + mB - f\ mB\)\/\(f\ \ mA + \[ExponentialE]\^\(\(k\ mA\)\/cA\)\ mB - \[ExponentialE]\^\(\(k\ \ mA\)\/cA\)\ f\ mB\)]\/mA)\)\)\)\/mB\)], "Output"], Cell[BoxData[ \(\(1 - \[ExponentialE]\^\(-\(\(k\ mA\)\/cA\)\)\)\/\(f\ mA + mB - f\ mB\)\ \)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(p0s \[Equal] pas\)], "Input", CellDingbat->None], Cell[BoxData[ \(True\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["1.3 Numerical solution", "Subsection", Evaluatable->False, AspectRatioFixed->True, FontSize->12, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell["\<\ The way in which the problem was solved does not imply that there \ is only one solution. 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Cell["\<\ The expected longevity of workers during B-type tasks is given by\ \ \>", "Text", AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[{ \(aB0 = Exp[\(-mA\)\ ts] \(\[Integral]\_0\%\[Infinity] Exp[\(-mB\)\ t] \[DifferentialD]t\)\), "\[IndentingNewLine]", \(aB0 = % /. {Re[mB] > 0 \[Rule] True}\)}], "Input", CellDingbat->None], Cell[BoxData[ RowBox[{\(\[ExponentialE]\^\(\(-mA\)\ ts\)\), " ", RowBox[{"If", "[", RowBox[{\(Re[mB] > 0\), ",", \(1\/mB\), ",", RowBox[{ SubsuperscriptBox["\[Integral]", "0", InterpretationBox["\[Infinity]", DirectedInfinity[ 1]]], \(\(\[ExponentialE]\^\(\(-mB\)\ t\)\) \[DifferentialD]t\ \)}]}], "]"}]}]], "Output"], Cell[BoxData[ \(\[ExponentialE]\^\(\(-mA\)\ ts\)\/mB\)], "Output"] }, Open ]], Cell[TextData[{ "When there is no aging the switching time ", StyleBox["ts", "Input"], " can be found analytically. " }], "Text", AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[{ \(Solve[aA0\/\(aA0 + aB0\) == f, ts]\), "\[IndentingNewLine]", \(Simplify[%]\)}], "Input", CellDingbat->None], Cell[BoxData[ \({{ts \[Rule] Log[\(\(-f\)\ mA - mB + f\ mB\)\/\(\((\(-1\) + f)\)\ \ mB\)]\/mA}}\)], "Output"], Cell[BoxData[ \({{ts \[Rule] Log[\(f\ mA + mB - f\ mB\)\/\(mB - f\ mB\)]\/mA}}\)], "Output"] }, Open ]], Cell["\<\ When there is no aging the expected longevity of workers in \ colonies with age polyethism is the same as in colonies without polyethism.\ \ \>", "Text", AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[{ \(pa0 = aA0 + aB0 /. %[\([1]\)]\), "\[IndentingNewLine]", \(pa0 = Simplify[%]\)}], "Input", CellDingbat->None], Cell[BoxData[ \(\(mB - f\ mB\)\/\(mB\ \((f\ mA + mB - f\ mB)\)\) + \(1 - \(mB - f\ mB\)\ \/\(f\ mA + mB - f\ mB\)\)\/mA\)], "Output"], Cell[BoxData[ \(1\/\(f\ mA + mB - f\ mB\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(p00 == pa0\)], "Input", CellDingbat->None], Cell[BoxData[ \(True\)], "Output"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["2 More realistic model", "Section", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times", FontSize->12, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[TextData[{ StyleBox["It is assumed that mortality rate ", "Text", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["r", "Input", FontVariations->{"CompatibilityType"->0}], StyleBox[" increases exponentially with age ", "Text", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["t", "Input", FontVariations->{"CompatibilityType"->0}], StyleBox[": ", "Text", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox[Cell[BoxData[ FormBox[ RowBox[{ StyleBox["r", FontFamily->"Courier", FontSlant->"Plain"], StyleBox["=", "Input", FontFamily->"Courier", FontWeight->"Plain"], RowBox[{ StyleBox["m", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], StyleBox["+", "Input", FontFamily->"Courier", FontWeight->"Plain"], StyleBox[" ", "Input", FontFamily->"Courier", FontWeight->"Plain"], StyleBox[\(\[Alpha]t\^\[Beta]\), "Input", FontFamily->"Courier"]}]}], TraditionalForm]], "Input", FontVariations->{"CompatibilityType"->0}], "Input"], StyleBox[" where ", "Text", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["m", "Input", FontVariations->{"CompatibilityType"->0}], StyleBox[" is aging-independent mortality. Aging-related mortality is \ described by two parameters: ", "Text", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["\[Alpha]", "Input", FontVariations->{"CompatibilityType"->0}], StyleBox[" controlling the magnitude of this mortality, and ", "Text", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["\[Beta]", "Input", FontVariations->{"CompatibilityType"->0}], StyleBox[" controlling the shape of the curve depicting changes of \ aging-related mortality with age. ", "Text", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["\[Alpha]", "Input", FontVariations->{"CompatibilityType"->0}], StyleBox[" associated with tasks A and B is ", "Text", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["\[Alpha]A", "Input", FontVariations->{"CompatibilityType"->0}], StyleBox[" and ", "Text", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["\[Alpha]B", "Input", FontVariations->{"CompatibilityType"->0}], StyleBox[" respectively. Tasks A and B are associated with the same ", "Text", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["\[Beta]", "Input", FontVariations->{"CompatibilityType"->0}], StyleBox[".\n", "Text", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}] }], "Text", CellDingbat->None], Cell["\<\ The expected longevity of workers during A-type tasks is given by\ \ \>", "Text", CellDingbat->None], Cell[BoxData[ \(\(aAx = \[Integral]\_0\%ts Exp[\(-mA\)\ t - \(\[Alpha]A\ t\^\(\[Beta] + 1\)\)\/\(\[Beta] + \ 1\)] \[DifferentialD]t;\)\)], "Input", CellDingbat->None], Cell["\<\ The expected longevity of workers during B-type tasks is given by\ \ \>", "Text", CellDingbat->None], Cell[BoxData[ \(\(aBx = \[Integral]\_0\%\[Infinity] Exp[\(-mB\) \((t + tm)\) - \(\[Alpha]B\ \((t + t\[Alpha])\)\^\(\[Beta] + 1\ \)\)\/\(\[Beta] + 1\)] \[DifferentialD]t;\)\)], "Input", CellDingbat->None], Cell[TextData[{ "The correction terms ", StyleBox["tm", "Input"], " and ", StyleBox["t\[Alpha]", "Input"], " adjust the survivorship at the beginning of B-type tasks period to that \ the end of A-type tasks period." }], "Text", CellDingbat->None], Cell[CellGroupData[{ Cell[BoxData[{ \(Solve[Exp[\(-mB\)\ tm] \[Equal] Exp[\(-mA\)\ ts], tm]\), "\[IndentingNewLine]", \(tm = \(tm /. %[\([1]\)]\) /. 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Log[Exp[\(-\(\(ts\^\(1 + \[Beta]\)\ \[Alpha]A\)\/\(1 + \ \[Beta]\)\)\)]] -> \(-\(\(ts\^\(1 + \[Beta]\)\ \[Alpha]A\)\/\(1 + \[Beta]\)\)\ \)\)}], "Input", CellDingbat->None], Cell[BoxData[ \({{t\[Alpha] \[Rule] \((\(-\(\(\((1 + \[Beta])\)\ \ Log[\[ExponentialE]\^\(-\(\(ts\^\(1 + \[Beta]\)\ \[Alpha]A\)\/\(1 + \[Beta]\)\ \)\)]\)\/\[Alpha]B\)\))\)\^\(1\/\(1 + \[Beta]\)\)}}\)], "Output"], Cell[BoxData[ \(\((\(ts\^\(1 + \[Beta]\)\ \[Alpha]A\)\/\[Alpha]B)\)\^\(1\/\(1 + \[Beta]\ \)\)\)], "Output"] }, Open ]], Cell[TextData[{ "The expected longevity of workers in colonies without polyethism ", StyleBox["P0", "Input"], " depends on probability of survival ", StyleBox["sm", "Input"], " associated with aging-independent mortality and probability of survival \ ", StyleBox["s\[Alpha]", "Input"], " associated with aging-related mortality. The probability of survival to \ age ", StyleBox["t", "Input"], " associated with aging-independent mortality is given by" }], "Text", CellDingbat->None], Cell[BoxData[ \(\(sm = Exp[\(-\((mA\ f + mB \((1 - f)\))\)\) t];\)\)], "Input", CellDingbat->None], Cell[TextData[{ "The probability of survival to age ", StyleBox["t", "Input"], " associated with aging-related mortality equals the probability of \ surviving B-type tasks during proportion (1-", StyleBox["f", "Input"], ") of time ", StyleBox["t", "Input"], ", taking into account the correction term ", StyleBox["t\[Alpha]", "Input"], ", where ", StyleBox["ts", "Input"], " is replaced by the proportion ", StyleBox["f", "Input"], " of time ", StyleBox["t", "Input"], " spent performing the A-type tasks." }], "Text", CellDingbat->None], Cell[BoxData[ \(\(s\[Alpha] = Exp[\(-\(1\/\(\[Beta] + 1\)\)\) \[Alpha]B \((\((1 - f)\) t + \(\@\(\[Alpha]A\/\ \[Alpha]B\)\%\(\[Beta] + 1\)\) f\ t)\)\^\(\[Beta] + 1\)];\)\)], "Input", CellDingbat->None], Cell["\<\ The expected longevity of workers in colonies without polyethism is \ given by\ \>", "Text", CellDingbat->None], Cell[BoxData[ \(\(p0x = \[Integral]\_0\%\[Infinity] sm\ s\[Alpha] \[DifferentialD]t;\)\)], "Input", CellDingbat->None], Cell["Numerical values were assigned to the parameters. ", "Text", AspectRatioFixed->True], Cell[BoxData[{ \(\(\[Alpha]A = 1.*^-7;\)\), "\n", \(\(\[Alpha]B = 1.*^-6;\)\), "\n", \(\(\[Beta] = 3;\)\), "\n", \(\(f = 0.5;\)\), "\n", \(\(mA = 0.01;\)\)}], "Input"], Cell[TextData[{ "Approximate function ", StyleBox["tsi", "Input"], " represents the switching time ", StyleBox["ts", "Input"], "." }], "Text", AspectRatioFixed->True], Cell[BoxData[{ \(tsv[\((mB_)\)?NumberQ]\ := \ ts\ /. FindRoot[aAx\/\(aAx + aBx\) == f, \ {ts, \ 20}]\), "\n", \(\(tsi = \ FunctionInterpolation[tsv[mB], {mB, \ 0.001, \ 0.06}];\)\)}], "Input"], Cell["\<\ To simplify computations the expected longevity of workers in \ colonies with age polyethism is expressed as\ \>", "Text", AspectRatioFixed->True], Cell[BoxData[ \(\(pax = aAx\/f;\)\)], "Input", CellDingbat->None], Cell[TextData[{ " Switching time ", StyleBox["ts", "Input"], " is replaced with approximate function ", StyleBox["tsi", "Input"], "." }], "Text", AspectRatioFixed->True], Cell[BoxData[ \(\(pax = pax /. ts \[Rule] tsi[mB];\)\)], "Input", CellDingbat->None], Cell["\<\ The difference in expected longevity of workers between colonies \ with and without age polyethism is plotted.\ \>", "Text", AspectRatioFixed->True], Cell[CellGroupData[{ Cell["\<\ Plot[pax-p0x,{mB, 0.001, \ 0.06},AxesLabel\[Rule]{\"mB\",\"pax-p0x\"}];\ \>", "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 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