(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 10.4' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 21974, 570] NotebookOptionsPosition[ 20953, 529] NotebookOutlinePosition[ 21296, 544] CellTagsIndexPosition[ 21253, 541] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Cobwebbing", "Title", CellChangeTimes->{{3.776600831050974*^9, 3.7766008318453026`*^9}, { 3.7771303814556313`*^9, 3.7771303897679653`*^9}, {3.777130431756935*^9, 3.7771304426145535`*^9}, {3.7771538104310255`*^9, 3.7771538107522936`*^9}}], Cell["Adam Rumpf, 11/4/2014", "Text", CellChangeTimes->{{3.7766008347881403`*^9, 3.776600838290375*^9}, { 3.7771303785432096`*^9, 3.7771303802837615`*^9}}], Cell[CellGroupData[{ Cell["Introduction", "Section", CellChangeTimes->{{3.7766008459498987`*^9, 3.776600848547045*^9}}], Cell[TextData[{ "The initialization code below defines a function called ", StyleBox["cobweb[]", "Code"], ", which accepts a pure function, a parameter ", Cell[BoxData[ FormBox["r", TraditionalForm]], FormatType->"TraditionalForm"], ", an initial value, and a number of iterations. It then produces a cobweb \ plot for the given function. This is used for the main Manipulate environment \ at the end of the Notebook, which has a number of simple built-in discrete \ dynamical systems, each of which depends on a single parameter ", Cell[BoxData[ FormBox["r", TraditionalForm]], FormatType->"TraditionalForm"], ". Note that the meaning of the parameter is different in each system." }], "Text", CellChangeTimes->{{3.776600856235587*^9, 3.776600860481224*^9}, { 3.7771601060172977`*^9, 3.777160276365533*^9}, {3.7771607213398085`*^9, 3.7771607394812503`*^9}}], Cell["\<\ Some of thesefunctions are arbitrary and only meant for showing off how \ different shapes of function can behave, but a few have some significance:\ \>", "Text", CellChangeTimes->{{3.776600856235587*^9, 3.776600860481224*^9}, { 3.7771601060172977`*^9, 3.777160276365533*^9}, {3.7771607213398085`*^9, 3.77716072716536*^9}}], Cell[CellGroupData[{ Cell[TextData[{ "The discrete logistic map, of the general form ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["x", RowBox[{"n", "+", "1"}]], "=", RowBox[{"r", " ", RowBox[{ SubscriptBox["x", "n"], "(", RowBox[{"1", "-", SubscriptBox["x", "n"]}], ")"}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], ", is usually thought of as describing a population whose size is limited by \ finite resources. ", Cell[BoxData[ FormBox[ SubscriptBox["x", "n"], TraditionalForm]], FormatType->"TraditionalForm"], " is the population at time step ", Cell[BoxData[ FormBox["n", TraditionalForm]], FormatType->"TraditionalForm"], ", scaled so that ", Cell[BoxData[ FormBox["1", TraditionalForm]], FormatType->"TraditionalForm"], " is the holding capacity. ", Cell[BoxData[ FormBox[ RowBox[{"r", ">", "0"}], TraditionalForm]], FormatType->"TraditionalForm"], " is the intrinsic growth rate. This system is famous for exhibiting chaotic \ behavior when ", Cell[BoxData[ FormBox["r", TraditionalForm]], FormatType->"TraditionalForm"], " is sufficiently large. Try increasing the value of ", Cell[BoxData[ FormBox["r", TraditionalForm]], FormatType->"TraditionalForm"], " slowly to watch as the nonzero fixed point goes from being stable, to \ being surrounded by periodic orbits, to complete chaos." }], "Item", CellChangeTimes->{{3.777160287433982*^9, 3.7771604336737385`*^9}, { 3.777160482102011*^9, 3.7771605580480304`*^9}}], Cell[TextData[{ "The discrete logistic map with proportional harvesting has the general form \ ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["x", RowBox[{"n", "+", "1"}]], "=", RowBox[{ RowBox[{"r", " ", RowBox[{ SubscriptBox["x", "n"], "(", RowBox[{"1", "-", SubscriptBox["x", "n"]}], ")"}]}], "-", RowBox[{"p", " ", SubscriptBox["x", "n"]}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], ". 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