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In the simplified model aging does not affect the mortality \ of workers until a certain age is reached, when resources become exhausted. \ At that time all workers die. 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 model the expected longevity of workers in colonies \ with and without caste polyethism is compared. If there is no polyethism, the \ workers perform tasks A and B in turn. If caste polyethism is present the \ workers from caste A perform only A-type tasks during the whole life and the \ workers from caste B perform only B-type tasks during the whole life. It is \ assumed that a fixed proportion of time ", StyleBox["f", "Input"], " is devoted in the colony to A-type tasks, ", StyleBox[Cell[BoxData[ \(f = tA\/\(tA + tB\)\)], "Input", CellDingbat->None], "Input"], StyleBox[", ", "Input"], "where ", StyleBox["tA", "Input"], " and ", StyleBox["tB", "Input"], " are the time spent in the colony on tasks A and B respectively. To \ simplify further computations it is better to write it another way:" }], "Text", CellDingbat->None, FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ \(\(tB = \(tA \((1 - f)\)\)\/f;\)\)], "Input", CellDingbat->None], Cell["\<\ In colonies with caste polyethism the proportion of workers from \ caste A is given by\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(fw = \(tA\/wA\)\/\(tA\/wA + tB\/wB\)\), "\[IndentingNewLine]", \(fw = Simplify[%]\)}], "Input", CellDingbat->None], Cell[BoxData[ \(tA\/\(wA\ \((tA\/wA + \(\((1 - f)\)\ tA\)\/\(f\ wB\))\)\)\)], "Output"], Cell[BoxData[ \(\(f\ wB\)\/\(wA - f\ wA + f\ wB\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["1.1 General case", "Subsection", Evaluatable->False, AspectRatioFixed->True, FontSize->12, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell["Error messages are switched off.", "Text", AspectRatioFixed->True], Cell[BoxData[{ \(Off[General::spell1]\), "\[IndentingNewLine]", \(Off[General::inum]\)}], "Input", CellDingbat->None], Cell["\<\ The expected longevity of workers in colonies without polyethism \ (for details see Tofilski submitted) is given by\ \>", "Text", AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[{ \(p0 = \[Integral]\_0\%\(k\/\(cA\ f\ + \ cB\ \((1 - f)\)\)\)Exp[\(-\((mA\ \ f + mB\ \((1 - f)\))\)\) t] \[DifferentialD]t\), "\[IndentingNewLine]", \(p0 = Simplify[%]\)}], "Input", CellDingbat->None], Cell[BoxData[ \(1\/\(f\ mA + mB - f\ mB\) - \[ExponentialE]\^\(-\(\(k\ \((f\ mA + mB - \ f\ mB)\)\)\/\(cB\ \((1 - f)\) + cA\ f\)\)\)\/\(f\ mA + mB - f\ mB\)\)], \ "Output"], Cell[BoxData[ \(\(-\(\(\(-1\) + \[ExponentialE]\^\(\(k\ \((f\ \((mA - mB)\) + mB)\)\)\/\ \(cB\ \((\(-1\) + f)\) - cA\ f\)\)\)\/\(f\ \((mA - mB)\) + mB\)\)\)\)], "Output"] }, Open ]], Cell["The expected longevity of workers from caste A is given by", "Text", AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[{ \(wA = \[Integral]\_0\%\(k\/cA\)Exp[\(-mA\)\ t] \[DifferentialD]t\), "\n\ ", \(wA = Simplify[%]\)}], "Input", CellDingbat->None], Cell[BoxData[ \(1\/mA - \[ExponentialE]\^\(-\(\(k\ mA\)\/cA\)\)\/mA\)], "Output"], Cell[BoxData[ \(\(1 - \[ExponentialE]\^\(-\(\(k\ mA\)\/cA\)\)\)\/mA\)], "Output"] }, Open ]], Cell["The expected longevity of workers from caste B is given by", "Text", AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[{ \(wB = \[Integral]\_0\%\(k\/cB\)Exp[\(-mB\)\ t] \[DifferentialD]t\), "\n\ ", \(wB = Simplify[%]\)}], "Input", CellDingbat->None], Cell[BoxData[ \(1\/mB - \[ExponentialE]\^\(-\(\(k\ mB\)\/cB\)\)\/mB\)], "Output"], Cell[BoxData[ \(\(1 - \[ExponentialE]\^\(-\(\(k\ mB\)\/cB\)\)\)\/mB\)], "Output"] }, Open ]], Cell["The proportion of workers from caste A is given by", "Text", AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(fw = \(wB\ f\)\/\(wA \((1 - f)\) + \ wB\ f\)\)], "Input", CellDingbat->None], Cell[BoxData[ \(\(\((1 - \[ExponentialE]\^\(-\(\(k\ mB\)\/cB\)\))\)\ f\)\/\(\((\(\((1 - \ \[ExponentialE]\^\(-\(\(k\ mA\)\/cA\)\))\)\ \((1 - f)\)\)\/mA + \(\((1 - \ \[ExponentialE]\^\(-\(\(k\ mB\)\/cB\)\))\)\ f\)\/mB)\)\ mB\)\)], "Output"] }, Open ]], Cell["\<\ The expected longevity of workers in colonies with caste polyethism \ is given by\ \>", "Text", AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[{ \(pw = wA\ fw + wB\ \((1 - fw)\)\), "\[IndentingNewLine]", \(pw = Simplify[%]\)}], "Input", CellDingbat->None], Cell[BoxData[ \(\(\((1 - \[ExponentialE]\^\(-\(\(k\ mA\)\/cA\)\))\)\ \((1 - \ \[ExponentialE]\^\(-\(\(k\ mB\)\/cB\)\))\)\ f\)\/\(mA\ \((\(\((1 - \ \[ExponentialE]\^\(-\(\(k\ mA\)\/cA\)\))\)\ \((1 - f)\)\)\/mA + \(\((1 - \ \[ExponentialE]\^\(-\(\(k\ mB\)\/cB\)\))\)\ f\)\/mB)\)\ mB\) + \(\((1 - \ \[ExponentialE]\^\(-\(\(k\ mB\)\/cB\)\))\)\ \((1 - \(\((1 - \[ExponentialE]\^\ \(-\(\(k\ mB\)\/cB\)\))\)\ f\)\/\(\((\(\((1 - \[ExponentialE]\^\(-\(\(k\ mA\)\ \/cA\)\))\)\ \((1 - f)\)\)\/mA + \(\((1 - \[ExponentialE]\^\(-\(\(k\ mB\)\/cB\ \)\))\)\ f\)\/mB)\)\ mB\))\)\)\/mB\)], "Output"], Cell[BoxData[ \(\(-\(\(\((\(-1\) + \[ExponentialE]\^\(\(k\ mA\)\/cA\))\)\ \((\(-1\) + \ \[ExponentialE]\^\(\(k\ mB\)\/cB\))\)\)\/\(\[ExponentialE]\^\(\(k\ mA\)\/cA\)\ \ f\ mA - \[ExponentialE]\^\(\(k\ mB\)\/cB\)\ \((\(-1\) + f)\)\ mB + \[ExponentialE]\^\(\(k\ mA\)\/cA + \(k\ mB\)\/cB\ \)\ \((\(-f\)\ mA - mB + f\ mB)\)\)\)\)\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " is unable to calculate analytically under what ", StyleBox["mB", "Input"], " the expected longevity of workers in colonies with and without caste \ polyethism is the same." }], "Text", AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[pw\ == \ p0, mB]\)], "Input", CellDingbat->None], Cell[BoxData[ \(Solve::"tdep" \(\(:\)\(\ \)\) "The equations appear to involve the variables to be solved for in an \ essentially non-algebraic way."\)], "Message"], Cell[BoxData[ \(Solve[\(-\(\(\((\(-1\) + \[ExponentialE]\^\(\(k\ mA\)\/cA\))\)\ \((\(-1\ \) + \[ExponentialE]\^\(\(k\ mB\)\/cB\))\)\)\/\(\[ExponentialE]\^\(\(k\ \ mA\)\/cA\)\ f\ mA - 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\[ExponentialE]\^\(-\(\(k\ mA\)\/cA\)\))\)\ f\)\/\(\((cB + \ cA\ f - cB\ f)\)\ mA\) + \(\((cA - cA\ \[ExponentialE]\^\(-\(\(k\ \ mA\)\/cA\)\))\)\ \((1 - \(cA\ f\)\/\(cB + cA\ f - cB\ f\))\)\)\/\(cB\ \ mA\)\)], "Output"], Cell[BoxData[ \(\(cA - cA\ \[ExponentialE]\^\(-\(\(k\ mA\)\/cA\)\)\)\/\(cB\ mA + cA\ f\ \ mA - cB\ f\ mA\)\)], "Output"] }, Open ]], Cell[TextData[{ "It can be demonstrated that the expected longevity of workers in colonies \ with and without caste polyethism is the same when", StyleBox[" ", "Input"], StyleBox[Cell[BoxData[ \(mB = \(cB\ \ mA\)\/cA\)], "Input", CellDingbat->None], "Input"], "." }], "Text", AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(pws \[Equal] p0s\)], "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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