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$CellContext`tend$192659$$, 0], Hold[$CellContext`p$$, $CellContext`p$192660$$, {0, 0}]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> ($CellContext`pic = VectorPlot[{$CellContext`a$$ $CellContext`x - $CellContext`\[Alpha]$$ \ $CellContext`y $CellContext`x, (-$CellContext`c$$) $CellContext`y + \ $CellContext`\[Gamma]$$ $CellContext`y $CellContext`x}, {$CellContext`x, 0, 7}, {$CellContext`y, 0, 5}, StreamPoints -> {{ Part[$CellContext`p$$, 1], Part[$CellContext`p$$, 2]}}, StreamStyle -> {Red, Thick}, VectorScale -> {Tiny, Automatic, None}, VectorStyle -> "Segment", ImageSize -> {350, 350}, FrameLabel -> { Row[{"rabbits ", Style["x", Italic]}], Row[{"foxes ", Style["y", Italic]}]}]; $CellContext`sol = NDSolve[{{ Derivative[ 1][$CellContext`x1][$CellContext`t] == $CellContext`a$$ \ $CellContext`x1[$CellContext`t] - $CellContext`\[Alpha]$$ \ $CellContext`y1[$CellContext`t] $CellContext`x1[$CellContext`t], Derivative[ 1][$CellContext`y1][$CellContext`t] == (-$CellContext`c$$) \ $CellContext`y1[$CellContext`t] + $CellContext`\[Gamma]$$ \ $CellContext`y1[$CellContext`t] $CellContext`x1[$CellContext`t]}, \ {$CellContext`x1[0] == Part[$CellContext`p$$, 1], $CellContext`y1[0] == Part[$CellContext`p$$, 2]}}, { $CellContext`x1[$CellContext`t], $CellContext`y1[$CellContext`t]}, {$CellContext`t, 0, $CellContext`tend$$}]; $CellContext`x2[ Pattern[$CellContext`t, Blank[]]] = ReplaceAll[ $CellContext`x1[$CellContext`t], Part[$CellContext`sol, 1, 1]]; $CellContext`y2[ Pattern[$CellContext`t, Blank[]]] = ReplaceAll[ $CellContext`y1[$CellContext`t], Part[$CellContext`sol, 1, 2]]; $CellContext`plotx$$ = Plot[ $CellContext`x2[$CellContext`t], {$CellContext`t, 0, $CellContext`tend$$}, PlotLabel -> Row[{"rabbits ", Style["x", Italic]}], PlotStyle -> Red, PlotRange -> {0, 10}, 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SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellID->364836077], Cell[CellGroupData[{ Cell["CAPTION", "Section", CellFrame->{{0, 0}, {1, 0}}, CellFrameColor->RGBColor[0.87, 0.87, 0.87], FontFamily->"Helvetica", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0.597406, 0, 0.0527047]], Cell[TextData[{ "This Demonstration illustrates the predator-prey model with two species, \ foxes and rabbits. Foxes prey on rabbits that live on vegetation. The rabbit \ population is ", Cell[BoxData[ FormBox["x", TraditionalForm]], "InlineMath"], " and the fox population is ", Cell[BoxData[ FormBox["y", TraditionalForm]], "InlineMath"], "; both depend on time ", Cell[BoxData[ FormBox["t", TraditionalForm]], "InlineMath"], "." }], "Text"] }, Close]] }, Open ]], Cell[TextData[{ "1. In the absence of foxes, the rabbit population grows at a rate \ proportional to its current population; thus ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"d", "\[InvisibleSpace]", RowBox[{"x", "/", "d"}], "\[InvisibleSpace]", "t"}], "=", RowBox[{"a", " ", "x"}]}], TraditionalForm]], "InlineMath"], " when ", Cell[BoxData[ FormBox[ RowBox[{"y", "=", "0"}], TraditionalForm]], "InlineMath"], " with ", Cell[BoxData[ FormBox[ RowBox[{"a", ">", "0"}], TraditionalForm]], "InlineMath"], "." }], "Text"], Cell[TextData[{ "2. In the absence of rabbits, the foxes die out; thus ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"d", "\[InvisibleSpace]", RowBox[{"y", "/", "d"}], "\[InvisibleSpace]", "t"}], "=", RowBox[{ RowBox[{"-", "c"}], " ", "y"}]}], TraditionalForm]], "InlineMath"], " when ", Cell[BoxData[ FormBox[ RowBox[{"x", "=", "0"}], TraditionalForm]], "InlineMath"], " with ", Cell[BoxData[ FormBox[ RowBox[{"c", ">", "0"}], TraditionalForm]], "InlineMath"], "." }], "Text"], Cell[TextData[{ "3. The number of encounters between the species is proportional to the \ product of their populations. Each encounter tends to increase ", Cell[BoxData[ FormBox["y", TraditionalForm]], "InlineMath"], " and decrease ", Cell[BoxData[ FormBox["x", TraditionalForm]], "InlineMath"], ". Thus the growth rate of ", Cell[BoxData[ FormBox["x", TraditionalForm]], "InlineMath"], " includes a term of the form ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"-", "\[Alpha]"}], " ", "x", " ", "y"}], TraditionalForm]], "InlineMath"], " and that of ", Cell[BoxData[ FormBox["y", TraditionalForm]], "InlineMath"], " includes a term of the form ", Cell[BoxData[ FormBox[ RowBox[{"\[Gamma]", " ", "x", " ", "y"}], TraditionalForm]], "InlineMath"], ", where ", Cell[BoxData[ FormBox["\[Alpha]", TraditionalForm]], "InlineMath"], " and ", Cell[BoxData[ FormBox["\[Gamma]", TraditionalForm]], "InlineMath"], " positive. The parameters ", Cell[BoxData[ FormBox["a", TraditionalForm]], "InlineMath"], ", ", Cell[BoxData[ FormBox["\[Alpha]", TraditionalForm]], "InlineMath"], ", ", Cell[BoxData[ FormBox["c", TraditionalForm]], "InlineMath"], ", and ", Cell[BoxData[ FormBox["\[Gamma]", TraditionalForm]], "InlineMath"], " are independent of ", Cell[BoxData[ FormBox["t", TraditionalForm]], "InlineMath"], ". 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