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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[ 177320, 5175]*) (*NotebookOutlinePosition[ 179781, 5258]*) (* CellTagsIndexPosition[ 179425, 5242]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["\<\ H3 Help File for Discrete Dynamical Systems in Several Variables\ \ \>", "Subtitle", FontSize->16, FontWeight->"Bold"], Cell[CellGroupData[{ Cell["How to Use This Help File", "Subsection", Background->RGBColor[0, 1, 1]], Cell[TextData[{ "When evaluating a cell for the first time, a window with the following \ text will be displayed:\n", StyleBox[ "Do you want to automatically evaluate all the initialization cells in the \ notebook 'DDSnH.nb?", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["\n", FontWeight->"Bold"], "Click on ", StyleBox["Yes", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontWeight->"Bold"], " to insure that all the functions are available for this help file to \ function properly." }], "Text", CellFrame->True, Background->GrayLevel[0.849989]], Cell[BoxData[ \(<< DDSn`\)], "Input", InitializationCell->True] }, Open ]], Cell[CellGroupData[{ Cell["Common Features of the Functions", "Subsection"], Cell[TextData[{ "The palette functions described in this help file are designed for use \ with a first-order discrete dynamical system (DDS) of several equations of \ the form\n\n\t\t\t\t", Cell[BoxData[ \(TraditionalForm\`x\_1\)], FontWeight->"Bold"], "(n+1) = ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["f", FontWeight->"Bold"], "1"], TraditionalForm]]], "(", Cell[BoxData[ \(TraditionalForm\`x\_1\)], FontWeight->"Bold"], "(n), ", Cell[BoxData[ \(TraditionalForm\`x\_2\)], FontWeight->"Bold"], "(n),....,", Cell[BoxData[ \(TraditionalForm\`x\_k\)], FontWeight->"Bold"], "(n))", StyleBox["\n\t\t\t\t", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`x\_2\)], FontWeight->"Bold"], "(n+1) = ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["f", FontWeight->"Bold"], "2"], TraditionalForm]]], "(", Cell[BoxData[ \(TraditionalForm\`x\_1\)], FontWeight->"Bold"], "(n), ", Cell[BoxData[ \(TraditionalForm\`x\_2\)], FontWeight->"Bold"], "(n),....,", Cell[BoxData[ \(TraditionalForm\`x\_k\)], FontWeight->"Bold"], "(n))", StyleBox["\t\n\t\t\t\t\t\[VerticalEllipsis]\n\t\t\t\t", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`x\_k\)], FontWeight->"Bold"], "(n+1) = ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["f", FontWeight->"Bold"], "k"], TraditionalForm]]], "(", Cell[BoxData[ \(TraditionalForm\`x\_1\)], FontWeight->"Bold"], "(n), ", Cell[BoxData[ \(TraditionalForm\`x\_2\)], FontWeight->"Bold"], "(n),....,", Cell[BoxData[ \(TraditionalForm\`x\_k\)], FontWeight->"Bold"], "(n))\n\nThe use of the palette functions applicable to this general DDS \ are illustrated based on Example 1." }], "Text"], Cell[TextData[{ StyleBox["Example 1: ", CellFrame->True, FontWeight->"Bold", Background->None], StyleBox["The", CellFrame->True, Background->None], StyleBox[" ", CellFrame->True, FontSize->10, Background->None], StyleBox["Island Car Rental", CellFrame->True, FontSlant->"Italic", Background->None], StyleBox[ " company has two rental locations on a Caribbean island. Cars can be \ rented and returned at either of the two locations, but cannot be taken off \ the island. From past records, the following patterns have emerged for the \ return of rental cars on the same day:\n\t\t\t\n\t\t\t ", CellFrame->True, Background->None], StyleBox["rented at A\t rented at B", CellFrame->True, FontWeight->"Bold", Background->None], StyleBox["\n\t", CellFrame->True, Background->None], StyleBox["returned at A", CellFrame->True, FontWeight->"Bold", Background->None], StyleBox[" 0.8\t 0.1\n\t", CellFrame->True, Background->None], StyleBox["returned at B", CellFrame->True, FontWeight->"Bold", Background->None], StyleBox[" 0.2\t 0.9\n\t\t\nIf ", CellFrame->True, Background->None], StyleBox["A(n) ", CellFrame->True, FontWeight->"Bold", Background->None], StyleBox["and ", CellFrame->True, Background->None], StyleBox["B(n) ", CellFrame->True, FontWeight->"Bold", Background->None], StyleBox["denote the number of cars after ", CellFrame->True, Background->None], StyleBox["n", CellFrame->True, FontWeight->"Bold", Background->None], StyleBox[ " days at location A and B, respectively, then the table above translates \ into the following system of equations:\n\n\t\t", CellFrame->True, Background->None], StyleBox[ "A(n+1) = 0.8 A(n) + 0.1 B(n)\n\t\tB(n+1) = 0.2 A(n) + 0.9 B(n) \n\n", CellFrame->True, FontWeight->"Bold", Background->None], StyleBox[ "This system can be written in matrix form as follows:\n\n \t \t ", CellFrame->True, Background->None], Cell[BoxData[ RowBox[{ RowBox[{"(", StyleBox[GridBox[{ { StyleBox[\(A \((n + 1)\)\), FontWeight->"Bold"]}, {\(B \((n + 1)\)\)} }], FontWeight->"Bold"], ")"}], "="}]], Background->None], Cell[BoxData[ RowBox[{ RowBox[{"(", GridBox[{ {"0.8", "0.1"}, {"0.2", "0.9"} }], ")"}], "\[CenterDot]"}]], Background->None], Cell[BoxData[ RowBox[{"(", StyleBox[GridBox[{ { StyleBox[\(A \((n)\)\), FontWeight->"Bold"]}, {\(B \((n)\)\)} }], FontWeight->"Bold"], ")"}]], Background->None] }], "Text", CellFrame->True, Background->GrayLevel[0.849989]], Cell[TextData[{ "\n\nSeveral of the palette functions (those whose name starts with ", StyleBox["Sys", FontWeight->"Bold"], ")", StyleBox[" ", FontWeight->"Bold"], "are designed for a special case, namely a DDS in two variables of the form \ \n\t\t\t\t", StyleBox["x(n+1) = ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\(c\_1\ \)\)]], StyleBox["x(n) + ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`c\_2\)]], StyleBox["x(n) y(n)\n\t\t\t\ty(n+1) = ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\(c\_3\ \)\)]], "y", StyleBox["(n) + ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`c\_4\)]], StyleBox["x(n) y(n).\n\n", FontWeight->"Bold"], "These palette functions require as input the values {", Cell[BoxData[ \(TraditionalForm\`\(c\_1\ \)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`c\_2\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\(c\_3\ \)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`c\_4\)]], "} and are illustrated based on Example 2.", StyleBox["\t", FontWeight->"Bold"] }], "Text"], Cell[TextData[{ StyleBox["Example 2: ", CellFrame->True, FontWeight->"Bold"], StyleBox[ "Blue Whales and Fin Whales live in the same environment and have \ overlapping food sources. This competition for food has a detrimental effect \ on each of the two populations and can be modeled as proportional to the \ product of the two species. Research on each species indicates that their \ intrinsic growth rates are 5% and 8% ,respectively. The size of the effect of \ the food competition is not well established; we will assume that the two \ rates resulting from the competition are equal to 0.00001. If we let ", CellFrame->True], StyleBox["B(n)", CellFrame->True, FontWeight->"Bold"], StyleBox[" and ", CellFrame->True], StyleBox["F(n) ", CellFrame->True, FontWeight->"Bold"], StyleBox["denote the number of Blue Whales and Fin Whales in year ", CellFrame->True], StyleBox["n, ", CellFrame->True, FontWeight->"Bold"], StyleBox[ "respectively, then we have the following iterative model equations:\n", CellFrame->True], StyleBox["\n\t\t\t", CellFrame->True, Background->GrayLevel[0.849989]], StyleBox[ "B(n+1) = 1.05 B(n) - 0.00001 B(n) F(n)\n\t\t\tF(n+1) = 1.08 F(n) - 0.00001 \ B(n) F(n)", CellFrame->True, FontWeight->"Bold", Background->GrayLevel[0.849989]] }], "Text", CellFrame->True, Background->GrayLevel[0.849989]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ How to Locate Specific Palette Functions within the Help File\ \>", "Subsection"], Cell["\<\ When you click on any of the function buttons below, you will be \ linked to a description of the selected function. A highlighted cell bracket \ will indicate the cell containing the relevant explanation. The buttons are \ grouped vertically according to their usage. Palette functions in the first \ column are applicable to a general DDS of several variables, and compute and \ graph system values. Functions in the second column are geared to the special \ type of systems in two variables described above (before Example 2). The last \ column contains functions related to finding the explicit solution of a \ DDS.\ \>", "Text"], Cell[BoxData[GridBox[{ { ButtonBox["InteratedValueSeq", ButtonData:>"ValueSeq", ButtonStyle->"Hyperlink"], ButtonBox["SysEquil", ButtonData:>"SysEquil", ButtonStyle->"Hyperlink"], ButtonBox["LiveVecs", ButtonData:>"LiveVecs", ButtonStyle->"Hyperlink"]}, { ButtonBox["SeqsGraph", ButtonData:>"GraphSet", ButtonStyle->"Hyperlink"], ButtonBox["SysDyn", ButtonData:>"SysDyn", ButtonStyle->"Hyperlink"], ButtonBox["GenSol", ButtonData:>"GenSol", ButtonStyle->"Hyperlink"]}, {"\[Placeholder]", ButtonBox["SysGraph", ButtonData:>"SysGraph", ButtonStyle->"Hyperlink"], ButtonBox["sol", ButtonData:>"sol", ButtonStyle->"Hyperlink"]}, {"\[Placeholder]", ButtonBox["range", ButtonData:>"range", ButtonStyle->"Hyperlink"], ButtonBox["Solve", ButtonData:>"Solve", ButtonStyle->"Hyperlink"]} }, ColumnLines->True]], "Input", Evaluatable->False, TextAlignment->-0.5, FontWeight->"Plain", CellTags->"top"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Description of Palette Functions Contained in the DDSnP \ Palette\ \>", "Subsection"], Cell[TextData[{ StyleBox["\nIteratedValueSeqs[fun, var, init, nmax, s] ", FontWeight->"Bold"], "gives the values of the DDS defined by ", StyleBox["var", FontSlant->"Italic"], "(", StyleBox["n+1", FontSlant->"Italic"], ")", StyleBox["=fun", FontSlant->"Italic"], "(", StyleBox["var", FontSlant->"Italic"], "(", StyleBox["n", FontSlant->"Italic"], ")) for n = 0 to n = ", StyleBox["nmax", FontWeight->"Bold"], ", in steps of size ", StyleBox["s", FontWeight->"Bold"], ", where ", StyleBox["var", FontWeight->"Bold"], " is the list of variables, ", StyleBox["fun", FontWeight->"Bold"], " is the list of functions defining the system, and", StyleBox[" init", FontWeight->"Bold"], " gives the list of initial values for the system. The step size ", StyleBox["s", FontWeight->"Bold"], " is optional; if no value for", StyleBox[" s", FontWeight->"Bold"], " is given, the step size will be 1. To compute the values of the system at \ single time ", StyleBox["n = nmax", FontWeight->"Bold"], ", set the step size ", StyleBox["s =", FontWeight->"Bold"], " ", StyleBox["nmax", FontWeight->"Bold"], ". Here is an illustration of the two different uses of ", StyleBox["IteratedValueSeqs", FontWeight->"Bold"], ", based on the car rental problem (Example 1). " }], "Text", CellTags->"ValueSeq"], Cell[TextData[{ "Suppose you are the manager of the car rental company and would like to \ use the palette function ", StyleBox["IteratedValueSeqs", FontWeight->"Bold"], " to see the availability of cars at each of the two rental locations for \ the next 30 days. You know that there are currently 300 cars at location A \ and 200 cars at location B. To use ", StyleBox["IteratedValueSeqs", FontWeight->"Bold"], ", you need to determine the relevant input values.\n\nYou may choose ", StyleBox["A", FontWeight->"Bold"], " and ", StyleBox["B ", FontWeight->"Bold"], "as the names of the two variables. (Do not put A(n) or B(n) as those would \ indicate specific values at time n !!!) With this choice, the corresponding \ iterative model functions are ", StyleBox["0.8 A + 0.1 B", FontWeight->"Bold"], " (for variable ", StyleBox["A", FontWeight->"Bold"], ") and ", StyleBox["0.2 A + 0.9 B ", FontWeight->"Bold"], "(for variable ", StyleBox["B", FontWeight->"Bold"], "). The initial values are ", StyleBox["300", FontWeight->"Bold"], " for ", StyleBox["A", FontWeight->"Bold"], " and ", StyleBox["200", FontWeight->"Bold"], " for ", StyleBox["B", FontWeight->"Bold"], ". This gives the following assignments for ", StyleBox["fun", FontWeight->"Bold"], ", ", StyleBox["var", FontWeight->"Bold"], ", and ", StyleBox["init:", FontWeight->"Bold"], "\n\t\n\t", StyleBox["fun = { ", FontWeight->"Bold"], " ", StyleBox[ "0.8 A + 0.1 B, 0.2 A + 0.9 B }\n\tvar = { A, B }\n\tinit = { 300, 200}\n\t\ \n", FontWeight->"Bold"], "Be careful to put the entries in each list in the correct order (the first \ element relates to variable ", StyleBox["A", FontWeight->"Bold"], " and the second one relates to variable ", StyleBox["B", FontWeight->"Bold"], " in each case). Finally, since we want to see the values for every day up \ to 30 days, we set ", StyleBox["nmax = 30", FontWeight->"Bold"], " and ", StyleBox["s = 1", FontWeight->"Bold"], ". Altogether, we use" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"IteratedValueSeqs", "[", StyleBox[ \({\ 0.8\ A\ + \ 0.1\ B, \ 0.2\ A\ + \ 0.9\ B\ }, {A, B}, {300, 200}, 30, 1\), FontWeight->"Bold"], "]"}]], "Input"], Cell[BoxData[ TagBox[GridBox[{ {\("n"\), TagBox[\("A(n)"\), DisplayForm], TagBox[\("B(n)"\), DisplayForm]}, {"0", "300", "200"}, {"1", "260.000000000000008`", "239.999999999999991`"}, {"2", "231.999999999999984`", "268.000000000000015`"}, {"3", "212.400000000000011`", "287.600000000000033`"}, {"4", "198.680000000000056`", "301.320000000000032`"}, {"5", "189.076000000000092`", "310.924000000000066`"}, {"6", "182.353200000000086`", "317.646800000000073`"}, {"7", "177.647240000000072`", "322.352760000000149`"}, {"8", "174.353068000000055`", "325.646932000000077`"}, {"9", "172.047147600000087`", "327.952852400000116`"}, {"10", "170.433003320000082`", "329.566996680000112`"}, {"11", "169.303102324000072`", "330.696897676000123`"}, {"12", "168.512171626800082`", "331.487828373200121`"}, {"13", "167.95852013876007`", "332.041479861240151`"}, {"14", "167.570964097132097`", "332.429035902868141`"}, {"15", "167.299674867992509`", "332.700325132007757`"}, {"16", "167.109772407594796`", "332.89022759240554`"}, {"17", "166.976840685316396`", "333.023159314683914`"}, {"18", "166.883788479721539`", "333.116211520278815`"}, {"19", "166.818651935805135`", "333.181348064195237`"}, {"20", "166.773056355063653`", "333.226943644936701`"}, {"21", "166.741139448544606`", "333.258860551455793`"}, {"22", "166.718797613981273`", "333.28120238601917`"}, {"23", "166.703158329786944`", "333.296841670213472`"}, {"24", "166.69221083085091`", "333.307789169149515`"}, {"25", "166.684547581595694`", "333.315452418404811`"}, {"26", "166.679183307117035`", "333.320816692883425`"}, {"27", "166.675428314981992`", "333.324571685018478`"}, {"28", "166.67279982048746`", "333.327200179513027`"}, {"29", "166.670959874341306`", "333.329040125659181`"}, {"30", "166.669671912038985`", "333.330328087961547`"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], (TableForm[ #]&)]], "Output"] }, Open ]], Cell[TextData[{ "It looks as if the number of cars at each location stabilizes (~167 at \ location A and ~333 at location B).\n\nTo compute the number of cars at each \ location after 15 days (without seeing the intermediate values for the other \ days), we set the step size to ", StyleBox["s", FontWeight->"Bold"], " = ", StyleBox["nmax", FontWeight->"Bold"], " =", StyleBox[" 15", FontWeight->"Bold"], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"IteratedValueSeqs", "[", StyleBox[ \({\ 0.8\ A\ + \ 0.1\ B, \ 0.2\ A\ + \ 0.9\ B\ }, {A, B}, {300, 200}, 15, 15\), FontWeight->"Bold"], "]"}]], "Input"], Cell[BoxData[ TagBox[GridBox[{ {\("n"\), TagBox[\("A(n)"\), DisplayForm], TagBox[\("B(n)"\), DisplayForm]}, {"0", "300", "200"}, {"15", "167.299674867992509`", "332.700325132007757`"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], (TableForm[ #]&)]], "Output"] }, Open ]], Cell[BoxData[ StyleBox[ ButtonBox[ StyleBox[\(Back\ to\ top\), "IndentedText", FontWeight->"Plain"], ButtonData:>"top", ButtonStyle->"Hyperlink"], "IndentedText", FontWeight->"Plain"]], "Input", Evaluatable->False, FontSize->9], Cell[TextData[{ "\n", StyleBox["SeqsGraph[A, vars, joint]", FontWeight->"Bold"], " is used to display the values of several variables together in one graph. \ The values to be graphed are stored in list ", StyleBox["A", FontWeight->"Bold"], ", which has either been generated by the function ", StyleBox["IteratedValueSeqs ", FontWeight->"Bold"], "or is of the same form. The list ", StyleBox["vars", FontWeight->"Bold"], " indicates which of the variables are to be graphed. To graph the \ variables as functions of time (time is on the horizontal axis, the values of \ the variables are on the vertical axis), you need to include the value ", StyleBox["0 ", FontWeight->"Bold"], "(= time) in the list ", StyleBox["vars", FontWeight->"Bold"], ". The other variables are listed as ", StyleBox["1, 2, ...", FontWeight->"Bold"], ". referring to the order in which they are listed in the table A. \ Finally, the optional value ", StyleBox["joint", FontWeight->"Bold"], " determines how the graph is displayed. If ", StyleBox["joint", FontWeight->"Bold"], " is set to ", StyleBox["True", FontWeight->"Bold"], ", then the data points will be connected by line segments. If no value is \ given, then the points will not be connected.\n\nAs an example, we will graph \ the number of cars at the two rental locations over time. This graph has time \ on the horizontal axis and the values of A(n) and B(n) on the vertical axis. \ First, we compute the values and give the resulting list the name ", StyleBox["rents", FontWeight->"Bold"], ". Then we apply ", StyleBox["SeqsGraph", FontWeight->"Bold"], " to the list ", StyleBox["rents", FontWeight->"Bold"], "; the variables to be displayed are time (", StyleBox["0", FontWeight->"Bold"], "), A (", StyleBox["1", FontWeight->"Bold"], "), and B (", StyleBox["2", FontWeight->"Bold"], "). 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", CellFrame->True, Background->None], "The interpretation of this graph is as follows: The system starts in the \ rightmost point (300, 200), then moves to the left and top. Each point is a \ pair of values for the number of cars at location A and B, respectively. From \ the graph we can read off that the number of cars at location B increases (as \ B(n) is on the vertical axis, points moving upward relate to increasing \ values), while the number of cars at location A decreases (as A(n) is shown \ on the horizontal axis, points moving to the left indicate decreasing \ values). The points get closer and closer together, indicating that the \ number of cars at each location do not change very much in the long run, \ i.e., the system stabilizes." }], "Text"], Cell[BoxData[ StyleBox[ ButtonBox[ StyleBox[\(Back\ to\ top\), "IndentedText", FontWeight->"Plain"], ButtonData:>"top", ButtonStyle->"Hyperlink"], "IndentedText", FontWeight->"Plain"]], "Input", Evaluatable->False, FontSize->9], Cell[TextData[{ "\n", StyleBox["SysEquil[{", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\(c\_1\ \)\)], FontWeight->"Bold"], StyleBox[", ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`c\_2\)], FontWeight->"Bold"], StyleBox[", ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\(c\_3\ \)\)], FontWeight->"Bold"], StyleBox[", ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`c\_4\)], FontWeight->"Bold"], StyleBox["}]", FontWeight->"Bold"], " computes the equilibrium value for the discrete dynamical system of the \ form\n\t\t\t\t\n\t\t\t\t", StyleBox["x(n+1) = ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\(c\_1\ \)\)]], StyleBox["x(n) + ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`c\_2\)]], StyleBox["x(n) y(n)\n\t\t\t\ty(n+1) = ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\(c\_3\ \)\)]], "y", StyleBox["(n) + ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`c\_4\)]], StyleBox["x(n) y(n).\n\n", FontWeight->"Bold"], "Example 2, which describes a model for the interaction between Blue Whales \ and Fin Whales, is of this form:\n\n\t\t\t", StyleBox[ "B(n+1) = 1.05 B(n) - 0.00001 B(n) F(n)\n\t\t\tF(n+1) = 1.08 F(n) - 0.00001 \ B(n) F(n)\n\t\t\t", CellFrame->True, FontWeight->"Bold", Background->None], "\nWe can read off the values for the constants needed to apply ", StyleBox["SysEquil", FontWeight->"Bold"], ": ", Cell[BoxData[ \(TraditionalForm\`\(c\_1\ \)\)]], "= 1.05, ", Cell[BoxData[ \(TraditionalForm\`c\_2\)]], " = -0.00001, ", Cell[BoxData[ \(TraditionalForm\`\(\ \ \ c\_3\)\)]], " = 1.08, ", Cell[BoxData[ \(TraditionalForm\`c\_4\)]], "= -0.00001. The equilibrium for the whale ecosystem is given by" }], "Text", CellTags->"SysEquil"], Cell[CellGroupData[{ Cell[BoxData[ \(SysEquil[{1.05, \(-0.00001\), \ 1.08, \ \(-0.00001\)}]\)], "Input"], Cell[BoxData[ \({8000.0000000000071`, 5000.00000000000355`}\)], "Output"] }, Open ]], Cell[TextData[{ "This result indicates that if the initial population consists of 8000 Blue \ Whales and 5000 Fin Whales, then their population levels will remain constant \ over time. (This can be verified with ", StyleBox["IteratedValueSeqs", FontWeight->"Bold"], ".)" }], "Text"], Cell[BoxData[ StyleBox[ ButtonBox[ StyleBox[\(Back\ to\ top\), "IndentedText", FontWeight->"Plain"], ButtonData:>"top", ButtonStyle->"Hyperlink"], "IndentedText", FontWeight->"Plain"]], "Input", Evaluatable->False, FontSize->9], Cell[TextData[{ "\n", StyleBox["SysGraph[{", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\(c\_1\ \)\)], FontWeight->"Bold"], StyleBox[", ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`c\_2\)], FontWeight->"Bold"], StyleBox[", ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\(c\_3\ \)\)], FontWeight->"Bold"], StyleBox[", ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`c\_4\)], FontWeight->"Bold"], StyleBox["}, init, nmax, joint]", FontWeight->"Bold"], " is a special version of ", StyleBox["SeqsGraph ", FontWeight->"Bold"], "for the discrete dynamical system of the form\n\t\t\t\t", StyleBox["x(n+1) = ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\(c\_1\ \)\)]], StyleBox["x(n) + ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`c\_2\)]], StyleBox["x(n) y(n)\n\t\t\t\ty(n+1) = ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\(c\_3\ \)\)]], StyleBox["y(n) + ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`c\_4\)]], StyleBox["x(n) y(n).\n\nSysGraph", FontWeight->"Bold"], " displays the values of the variables ", StyleBox["x", FontSlant->"Italic"], "(", StyleBox["n", FontSlant->"Italic"], ")", StyleBox[" ", FontSlant->"Italic"], "and ", StyleBox["y", FontSlant->"Italic"], "(", StyleBox["n", FontSlant->"Italic"], ") in the form of pairs, with the first variable displayed on the \ horizontal axis and the second variable on the vertical axis. (Time is not \ displayed on any axis.) The parameters needed for ", StyleBox["SysGraph", FontWeight->"Bold"], " are the values of the constants ", StyleBox["{", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\(c\_1\ \)\)], FontWeight->"Bold"], StyleBox[", ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`c\_2\)], FontWeight->"Bold"], StyleBox[", ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\(c\_3\ \)\)], FontWeight->"Bold"], StyleBox[", ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`c\_4\)], FontWeight->"Bold"], StyleBox["} ", FontWeight->"Bold"], "for this special DDS. We can display, simultaneously, the sequence of \ pairs for several starting conditions. The list ", StyleBox["init ", FontWeight->"Bold"], "contains the (pairs of) starting value(s) for which the sequences are to \ be displayed. (Note that if you want to see just a single sequence, i.e., \ only a single starting point is given, you still need to use a list - hence, \ there are double curly braces.) The values are computed up to time ", StyleBox["nmax", FontWeight->"Bold"], "; if ", StyleBox["joint", FontWeight->"Bold"], " is set to ", StyleBox["True", FontWeight->"Bold"], ", then the individual points are connected. " }], "Text", CellTags->"SysGraph"], Cell[TextData[{ "We will use Example 2, which models whale interaction, to illustrate the \ use of this function. From the set of equations\n\n\t\t\t", StyleBox[ "B(n+1) = 1.05 B(n) - 0.00001 B(n) F(n)\n\t\t\tF(n+1) = 1.08 F(n) - 0.00001 \ B(n) F(n)\n\t\t\t", CellFrame->True, FontWeight->"Bold", Background->None], "\nwe read off ", Cell[BoxData[ \(TraditionalForm\`\(c\_1\ \)\)]], "= 1.05, ", Cell[BoxData[ \(TraditionalForm\`c\_2\)]], " = -0.00001, ", Cell[BoxData[ \(TraditionalForm\`\(\ c\_3\)\)]], " = 1.08, ", Cell[BoxData[ \(TraditionalForm\`c\_4\)]], "= -0.00001. We will look at the evolution of the whale populations for the \ following intial populations: \n\n\t1) 7000 Blue Whales and 3000 Fin Whales ( \ initial point = {7000, 3000}),\n\t2) 4000 Blue Whales and 4000 Fin Whales ( \ initial point = {4000, 4000}),\n\t3) 6000 Blue Whales and 6000 Fin Whales ( \ initial point = {6000, 6000}),\n\t4) 10000 Blue Whales and 8000 Fin Whales ( \ initial point = {10000, 8000}), and\n\t5) 12000 Blue Whales and 6000 Fin \ Whales ( initial point = {12000, 6000})\n\t\nThus, ", StyleBox[ "init = {{7000, 3000}, {4000, 4000}, {6000, 6000}, {10000, 8000}, {12000, \ 6000}}", FontWeight->"Bold"], ". 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Recall that the horizontal axis displays \ the Blue Whale population size (first variable of the system), and the \ vertical axis displays the Fin Whale population size. We see that there are \ two main behaviors: ", "\n", "1) For initial populations {4000, 4000}, {6000, 6000}, and{10000, 8000} \ (starting points 2, 3, and 4), the Blue Whale population decreases in the \ long run, whereas the Fin Whale population increases. \n2) For initial \ populations {7000, 3000} and {12000,6000} (starting points 1 and 5), the Blue \ Whale population increases in the long run, whereas the Fin Whale population \ decreases. \n" }], "Text", TextAlignment->Left], Cell[BoxData[ StyleBox[ ButtonBox[ StyleBox[\(Back\ to\ top\), "IndentedText", FontWeight->"Plain"], ButtonData:>"top", ButtonStyle->"Hyperlink"], "IndentedText", FontWeight->"Plain"]], "Input", Evaluatable->False, FontSize->9], Cell[TextData[{ "\n", StyleBox["SysDyn[{", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\(c\_1\ \)\)], FontWeight->"Bold"], StyleBox[", ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`c\_2\)], FontWeight->"Bold"], StyleBox[", ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\(c\_3\ \)\)], FontWeight->"Bold"], StyleBox[", ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`c\_4\)], FontWeight->"Bold"], StyleBox["}]", FontWeight->"Bold"], " displays the dynamics for the discrete dynamical system of the form\n\n\t\ \t\t\t", StyleBox["x(n+1) = ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\(c\_1\ \)\)]], StyleBox["x(n) + ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`c\_2\)]], StyleBox["x(n) y(n)\n\t\t\t\ty(n+1) = ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\(c\_3\ \)\)]], "y", StyleBox["(n) + ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`c\_4\)]], StyleBox["x(n) y(n).\n\n", FontWeight->"Bold"], "In each of the areas created by the vertical and horizontal lines through \ the equilibrium value(s), the behavior of the system is displayed by a set of \ arrows. These arrows indicate what happens to sequences within in this area. \ By default, only positive values for the variables are displayed (since the \ context for this type of DDS is usually population size, and negative \ population sizes do not make sense). 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The same is true for an \ initial population of {4000, 4000}. However, we do not know which of the two \ trends (up or right) is stronger, so we cannot predict whether the sequence \ of population values will eventually enter the lower right or upper left \ region (see also the example for ", StyleBox["SysGraph", FontWeight->"Bold"], " above, especially the sequences with initial populations labeled 1 and \ 2)." }], "Text"], Cell[BoxData[ StyleBox[ ButtonBox[ StyleBox[\(Back\ to\ top\), "IndentedText", FontWeight->"Plain"], ButtonData:>"top", ButtonStyle->"Hyperlink"], "IndentedText", FontWeight->"Plain"]], "Input", Evaluatable->False, FontSize->9], Cell[TextData[{ StyleBox["range -> All", FontWeight->"Bold"], " is an option for ", StyleBox["SysDyn ", FontWeight->"Bold"], "to display the system dynamics for both positive and negative values of \ the variables. Be careful not to forget to separate this option from the list \ of constants by a comma. 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StyleBox["num", FontWeight->"Bold"], " controls how many different input vectors are used. Each graph lists the \ input vector in the title. In addition, the Eigenvalues and Eigenvectors for \ the 2x2 matrix ", StyleBox["M", FontWeight->"Bold"], " are computed. \n\nAfter the individual graphs have been created, select \ the cell bracket (second from left) that encloses all the graphs. Double \ click on it to collapse the cells so that only the first graph shows. With \ that cell selected use \[ControlKey]", StyleBox["-Y", FontWeight->"Bold"], " (\[CloverLeaf]-", StyleBox["Y", FontWeight->"Bold"], " for Macintosh)", StyleBox[" ", FontWeight->"Bold"], "or select ", StyleBox["Cell -> Animate Selected Graphics", FontWeight->"Bold"], " from the menu to see the interaction between input and output vectors \ dynamically. The speed of the animation can be controlled by using the video \ panel that appears at the lower left corner of the notebook. To stop the \ animation temporarily, use the || button. 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" }], "Text", FontWeight->"Plain", CellTags->"LiveVecs"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"M", "=", " ", RowBox[{"(", GridBox[{ {"2", ".5"}, {"0", \(-.5\)} }], ")"}]}], ";", " ", \(LiveVecs[M, 10]\)}]], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{ TagBox[\(\[Lambda]\_1\), DisplayForm], "\[InvisibleSpace]", \("= "\), "\[InvisibleSpace]", StyleBox["2.`", StyleBoxAutoDelete->True, PrintPrecision->3], "\[InvisibleSpace]", \(" "\), "\[InvisibleSpace]", TagBox[\(v\_1\), DisplayForm], "\[InvisibleSpace]", \(" = "\), "\[InvisibleSpace]", InterpretationBox[ RowBox[{"(", GridBox[{ {"1.`"}, {"0.`"} }], ")"}], MatrixForm[ {1.0, 0.0}]], "\[InvisibleSpace]", RowBox[{\(" "\), " ", TagBox[\(\[Lambda]\_2\), DisplayForm]}], "\[InvisibleSpace]", \("= "\), "\[InvisibleSpace]", RowBox[{"-", StyleBox["0.5`", StyleBoxAutoDelete->True, PrintPrecision->3]}], "\[InvisibleSpace]", \(" "\), "\[InvisibleSpace]", TagBox[\(v\_2\), DisplayForm], "\[InvisibleSpace]", \(" = "\), "\[InvisibleSpace]", InterpretationBox[ RowBox[{"(", GridBox[{ {\(-0.19611613513818412`\)}, {"0.980580675690920244`"} }], ")"}], MatrixForm[ {-.1961161351381841, .98058067569092033}]]}], SequenceForm[ DisplayForm[ SubscriptBox[ "\[Lambda]", "1"]], "= ", 2.0, " ", DisplayForm[ SubscriptBox[ "v", "1"]], " = ", MatrixForm[ {1.0, 0.0}], Times[ " ", DisplayForm[ SubscriptBox[ "\[Lambda]", "2"]]], "= ", -.5, " ", DisplayForm[ SubscriptBox[ "v", "2"]], " = ", MatrixForm[ {-.1961161351381841, .98058067569092033}]], Editable->False]], "Print"] }, Open ]], Cell[BoxData[ StyleBox[ ButtonBox[ StyleBox[\(Back\ to\ top\), "IndentedText", FontWeight->"Plain"], ButtonData:>"top", ButtonStyle->"Hyperlink"], "IndentedText", FontWeight->"Plain"]], "Input", Evaluatable->False, FontSize->9], Cell[TextData[{ StyleBox["GenSol[M] ", FontWeight->"Bold"], "displays the explicit solution for a ", StyleBox["linear", FontVariations->{"Underline"->True}], " DDS that can be written in matrix form as ", Cell[BoxData[ RowBox[{"\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t", RowBox[{ RowBox[{"(", StyleBox[GridBox[{ { StyleBox[\(x \((n + 1)\)\), FontWeight->"Plain"]}, { StyleBox[\(y \((n + 1)\)\), FontWeight->"Plain"]} }], FontWeight->"Bold"], ")"}], "="}], " "}]], Background->None], Cell[BoxData[ RowBox[{ StyleBox["M", FontWeight->"Bold"], "\[CenterDot]"}]], Background->None], Cell[BoxData[ RowBox[{"(", StyleBox[GridBox[{ { StyleBox[\(x \((n)\)\), FontWeight->"Plain"]}, { StyleBox[\(y \((n)\)\), FontWeight->"Plain"]} }], FontWeight->"Bold"], ")"}]], Background->None] }], "Text", CellTags->"GenSol"], Cell[CellGroupData[{ Cell[BoxData[ \(GenSol[M]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{\(\(x \((k)\)\ = \)\ \), \(c\_1\), "*", \(2.`\^k\), "*", FormBox[ RowBox[{"(", GridBox[{ {"1.`"}, {"0.`"} }, ColumnAlignments->{Decimal}], ")"}], "TraditionalForm"], "+", \(c\_2\), "*", \(\((\(-0.5`\))\)\^k\), "*", FormBox[ RowBox[{"(", GridBox[{ {\(-0.196116135138184048`\)}, {"0.980580675690920067`"} }, ColumnAlignments->{Decimal}], ")"}], "TraditionalForm"]}], DisplayForm]], "Print"] }, Open ]], Cell[TextData[{ "Note that the explicit solution involves the Eigenvalues and the \ Eigenvectors of the matrix M (see ", StyleBox["LiveVecs", FontWeight->"Bold"], "). The values for the constants ", Cell[BoxData[ \(TraditionalForm\`c\_1\)]], " and ", Cell[BoxData[ \(TraditionalForm\`c\_2\)]], " need to be determined from the initial values of the variables. Once \ these values are known, we can use the explicit solution to determine what \ happens to the model variables in the long run. " }], "Text"], Cell[TextData[{ StyleBox["sol[k] ", FontWeight->"Bold"], "computes the values of the model variables at time ", StyleBox["k", FontWeight->"Bold"], ". This function is created by ", StyleBox["GenSol", FontWeight->"Bold"], ", i.e., you need to use ", StyleBox["GenSol", FontWeight->"Bold"], " first. Notice that the answer still contains the constants ", Cell[BoxData[ \(TraditionalForm\`c\_1\)]], " and ", Cell[BoxData[ \(TraditionalForm\`c\_2.\)]], " Here are the values of the model variables at time k = 10:" }], "Text", CellTags->"sol"], Cell[CellGroupData[{ Cell[BoxData[ \(sol[10]\)], "Input"], Cell[BoxData[ \({1024.00000000000002`\ "c"\_1 - 0.000191519663220882883`\ "c"\_2, 0.`\ "c"\_1 + 0.000957598316104414237`\ "c"\_2}\)], "Output"] }, Open ]], Cell[BoxData[ StyleBox[ ButtonBox[ StyleBox[\(Back\ to\ top\), "IndentedText", FontWeight->"Plain"], ButtonData:>"top", ButtonStyle->"Hyperlink"], "IndentedText", FontWeight->"Plain"]], "Input", Evaluatable->False, FontSize->9], Cell[TextData[{ StyleBox["Solve[lhs == rhs]", FontWeight->"Bold"], " solves the equation ", StyleBox["lhs", FontWeight->"Bold"], " = ", StyleBox["rhs", FontWeight->"Bold"], ". We can use this function to determine the values of the constants ", Cell[BoxData[ \(TraditionalForm\`c\_1\)]], " and ", Cell[BoxData[ \(TraditionalForm\`c\_2\)]], " in the explicit solution. Since we are given the initial values, we know \ the value of the system at time 0. For example, if the values of the two \ variables are 1 at the beginning, then {", StyleBox["x", FontSlant->"Italic"], "(0), ", StyleBox["y", FontSlant->"Italic"], "(0)} = {1, 1}, which is also the value of sol[0]. Thus, we need to solve \ the following equation: ", StyleBox["sol[0] = {1,1}", FontWeight->"Bold"], ". 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