(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "Macintosh Mathematica Notebook Front End Version 2.2"; MacintoshStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e8, 24, "Times"; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e6, 18, "Times"; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, e6, 14, "Times"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, a20, 18, "Times"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, a15, 14, "Times"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, a12, 12, "Times"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times"; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L-5, 12, "Courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R65535, L-5, 12, "Courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B65535, L-5, 12, "Courier"; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, 12, "Courier"; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, 10, "Geneva"; fontset = header, inactive, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = leftheader, inactive, L2, 12, "Times"; fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7, 12, "Times"; fontset = leftfooter, inactive, L2, 12, "Times"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times"; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; paletteColors = 128; automaticGrouping; currentKernel; ] :[font = section; inactive; preserveAspect; startGroup] CHAPTER 9 :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] 1 :[font = text; inactive; preserveAspect; endGroup] Use pencil and paper. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] 2 :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] 2a :[font = text; inactive; preserveAspect; endGroup] Use pencil and paper. :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] 2b :[font = input; preserveAspect; endGroup] Eigenvalues[{{c,2c},{c,0}}] :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] 2c :[font = text; inactive; preserveAspect] Find the optimal clutch size under Godfray's model: :[font = input; preserveAspect] s=S Exp[-h(n-1)]; a=D[n s,n]; Solve[Numerator[Together[a]]==0,n] :[font = text; inactive; preserveAspect] Define c as a function of n; plot 2c, the dominant eigenvalue, against n, and observe that, approximately, 2c < 1 when n > 4; and then find the critical value of n above which 2c < 1 exactly. :[font = input; preserveAspect] c=(1-.5^n)S/(n s)/.h->1/n; :[font = input; preserveAspect] Plot[2c,{n,1,10}] :[font = input; preserveAspect; endGroup; endGroup] FindRoot[2c==1,{n,4}] :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] 3, 4, 5 :[font = text; inactive; preserveAspect; endGroup] Use pencil and paper. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] 6 :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] 6a :[font = text; inactive; preserveAspect] Evaluate f[n] for different values of n: ;[s] 5:0,0;9,1;13,0;38,1;39,0;41,-1; 2:3,13,9,Times,0,12,0,0,0;2,13,9,Times,1,12,0,0,0; :[font = input; preserveAspect] A={{1/2,1/2},{1,0}}; f[n_]:=MatrixPower[N[A],n]//MatrixForm :[font = text; inactive; preserveAspect] Find the eigenvalues and left eigenvectors of the matrix and hence identify the dominant left eigenvector: :[font = input; preserveAspect; endGroup] Eigensystem[Transpose[A]] :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] 6b :[font = text; inactive; preserveAspect; endGroup; endGroup] Repeat this exercise for the matrix in Equation 20. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] 7, 8 :[font = text; inactive; preserveAspect; endGroup; endGroup] Use pencil and paper. ^*)