(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "To determine the criterion for annuals to invade a population of \ perennials, first determine the equilibrium number of perennials, then P", Evaluatable->False, AspectRatioFixed->True], StyleBox["1", Evaluatable->False, AspectRatioFixed->True, FontSize->10, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" and then b*/(1\[Dash]P", Evaluatable->False, AspectRatioFixed->True], StyleBox["1", Evaluatable->False, AspectRatioFixed->True, FontSize->10, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[ "). This is a little more complicated to calculate. 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Define the objective function as j=q[22], \ and then evaluate it in terms of values in week 21:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["j=q[22];\nj=Expand[j/.func[21]]"], "Input", AspectRatioFixed->True], Cell[TextData[ "j is maximized by setting u[21]=0. We do this and evaluate j in terms of \ values in week 20:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["u[21]=0;\nj=Expand[j/.func[20]]"], "Input", AspectRatioFixed->True], Cell[TextData[ "This is maximized by setting u[20]=0. In general it is clear that j, \ expressed in terms of values of week t, depends on u[t] through a single term \ which is proportional to u[t] w[t] and is maximized by equating u[t] to 0 or \ 1 according as the coefficient of this term is negative or positive. 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The discrete-time \ analogue of Equation 41 with time measured in days is\n\t\t\t\t\tw(t+1) = \ 0.05u(t)w(t) + 0.977w(t)\n\t\t\t\t\tq(t+1) = 0.05[1 - u(t)]w(t) + q(t)\nWe \ now repeat the previous calculations with the season ending on day 154:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Clear[u]\nfunc[t_]:={w[t+1]->.05 u[t] w[t] + .977 w[t],\nq[t+1]->.05(1-u[t]) \ w[t] + q[t]}"], "Input", AspectRatioFixed->True], Cell[TextData[ "j=q[154];\nDo[j=Expand[j/.func[t-1]];\n\t\ u[t-1]=If[Negative[Coefficient[j,u[t-1] w[t-1]]],0,1],\n\t{t,184,1,-1}]\n\ Table[{t,u[t]},{t,0,153}]//MatrixForm"], "Input", AspectRatioFixed->True], Cell[TextData[StyleBox[ "The optimal strategy is to make workers only until the last 27 days and then \ to make queens only.", AspectRatioFixed->True, FontFamily->"Times", FontWeight->"Plain"]], "Input", AspectRatioFixed->True]}, Open]]}, Open]], Cell[CellGroupData[{Cell[TextData["5"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "As before we define a function which replaces w[t] and q[t] in terms of \ their values in the previous week according to the recurrence relation: "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Clear[u]\nfunc[t_]:={w[t+1]->.35 u[t] w[t] + .84 w[t],\nq[t+1]->.35(1-u[t]) \ w[t] + q[t]}"], "Input", AspectRatioFixed->True], Cell[TextData["In week 24 the objective function is j=q[24]:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["j=q[24];"], "Input", AspectRatioFixed->True], Cell[TextData[ "In week 23 the objective function is q[23]*q[24], which can be expanded in \ terms of values in week 23:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["j=Expand[q[23] j/.func[23]]"], "Input", AspectRatioFixed->True], Cell[TextData[ "This is maximized by setting u[23]=0; we then repeat the calculation for \ week 22, after updating the objective function:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["u[23]=0;\nj=Expand[q[22] j/.func[22]]"], "Input", AspectRatioFixed->True], Cell[TextData[ "It is not possible to find the value of u which maximizes this expression \ without knowing the relative numbers of queens and workers in week 22, which \ cannot be determined without knowing the values of u in previous weeks. 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We first assume a trial \ solution for u[t] and calculate the numbers of queens and workers in \ different weeks: "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Do[u[t]=If[t<18,1,0],{t,0,23}]\nww[t_]:=ww[t]=.35 u[t-1] ww[t-1] + .84 \ ww[t-1];\nqq[t_]:=qq[t]=.35(1-u[t-1]) ww[t-1] + qq[t-1];\nww[0]=10;qq[0]=0;\n\ Do[ww[t];qq[t],{t,1,24}]"], "Input", AspectRatioFixed->True], Cell[TextData[ "Clear the trial values for u and determine the optimal values for u assuming \ the numbers of queens and workers just calculated:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Clear[u]\nj=q[24];\nj=Expand[q[23] j/.func[23]];\nu[23]=0;\nj=Expand[q[22] \ j/.func[22]];\njj=j/.{w[22]->ww[22],q[22]->qq[22],u[22]->u}"], "Input", AspectRatioFixed->True], Cell[TextData[ "The value of u which maximizes this function can be seen by plotting jj \ against u; it is at u=0:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Plot[jj,{u,0,1}]"], "Input", AspectRatioFixed->True], Cell[TextData["Equate u[22] to zero and go back to week 21:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "u[22]=0;\nj=Expand[q[21] j/.func[21]];\n\ jj=j/.{w[21]->ww[21],q[21]->qq[21],u[21]->u};\nPlot[jj,{u,0,1}]"], "Input", AspectRatioFixed->True], Cell[TextData[ "The optimal value of u[21] is also zero. Continue backwards in this way. \ Remember that if j is the objective function at time t+1, the objective \ function at time t is q[t] j when t>=20 but is j when t<20. At t=17 you will \ find that there is an internal maximum near u=0.4; its exact value can be \ found by defining d=D[jj,u] and then solving for d=0 near u=0.4 using \ FindRoot. For t>17 the optimal strategy is u=0 and for t<17 it is u=1. \n \ Now recalculate the numbers of queens and workers using the above values \ of u, and then recalculate the optimal values of u. Continue this process \ until the values of u have converged to 3 decimal places."], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["6"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Calculate the l[x] and m[x] functions as in Exercise 4.2a"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "temp={.25,.46,.77,.65,.67,.64,.88};\n\ l=Table[Product[temp[[i]],{i,x}],{x,7}];"], "Input", AspectRatioFixed->True], Cell[TextData["m={1.28,2.28,2.28,2.28,2.28,2.28,2.28};"], "Input", AspectRatioFixed->True], Cell[TextData["The age-specific sensitivities from Equations 50 are:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Table[Sum[l[[x]] m[[x]],{x,a+1,7}],{a,0,6}]"], "Input", AspectRatioFixed->True], Cell[TextData[StyleBox[ "The age-specific sensitivities from Equations 51 are:", AspectRatioFixed->True, FontFamily->"Times", FontWeight->"Plain"]], "Input", AspectRatioFixed->True], Cell[TextData["l"], "Input", AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["7"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "I first define a matrix of the fitnesses of the four genotypes in the two \ environments:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["w={{1.00,0.6,0.785,0.776},{0.58,1.0,0.785,0.815}};"], "Input", AspectRatioFixed->True], Cell[TextData[ "I now calculate the arithmetic and geometric mean fitnesses for the four \ genotypes. 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You may extract \ appropriate elements from the list ", Evaluatable->False, AspectRatioFixed->True], StyleBox["res", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[ ". The idea of using a random number generator to simulatre a stochastic \ process is a very useful tool.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["res=NestList[f,{25,25,25,25},1000];"], "Input", AspectRatioFixed->True], Cell[TextData[" "], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["8"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["8a"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "I define the log geometric mean fitness from Equation 54 and then find the \ germination rate that maximizes it. 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