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Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[ Style["Continuous time models of predator-prey dynamics", 14, Bold]], Manipulate`Dump`ThisIsNotAControl}, { Hold[ Style["\[Copyright] David A. Vasseur\n", 12, Italic]], Manipulate`Dump`ThisIsNotAControl}, { Hold[ Style["Click in the phase plane to set initial conditions.", 10]], Manipulate`Dump`ThisIsNotAControl}, { Hold[ Style[ "Note: Some combinations of growth and functional response models are \ not\n permitted since they produce explosive solutions.", 10, Italic]], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`tmax$$], 25, "Length of Simulation"}, 10, 100}, { Hold[ Style["\nTurn on streams to see sample trajectories", 10]], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`streams$$], 0}, {0, 1}}, { Hold[ Style["Prey Growth Parameters", 12, Italic]], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`gmodel$$], 1, "Model"}, { 1 -> "Exponential", 2 -> "Logistic"}}, {{ Hold[$CellContext`r$$], 1, "Intrinsic Growth Rate r"}, 0, 2}, {{ Hold[$CellContext`KK$$], 2, "Carrying Capacity K"}, 0, 5}, { Hold[ Style["Predator Growth Parameters", 12, Italic]], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`frmodel$$], 1, "Functional Response"}, { 1 -> "Type 1", 2 -> "Type 2", 3 -> "Type 3"}}, {{ Hold[$CellContext`e$$], 0.8, "Conversion Efficiency e"}, 0, 1}, {{ Hold[$CellContext`a$$], 2, "Attack Rate a"}, 0, 5}, {{ Hold[$CellContext`h$$], 0.5, "Handling Time h"}, 0, 2}, {{ Hold[$CellContext`q$$], 1, "Predator Death Rate q"}, 0, 1}, {{ Hold[$CellContext`b$$], 1, "Functional Response Exponent"}, 1, 3}, {{ Hold[$CellContext`startpt$$], {1, 0.75}}}}, Typeset`size$$ = { 906., {200., 206.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = False, $CellContext`tmax$19333$$ = 0, $CellContext`streams$19334$$ = False, $CellContext`gmodel$19335$$ = False, $CellContext`r$19336$$ = 0, $CellContext`KK$19337$$ = 0, $CellContext`frmodel$19338$$ = False, $CellContext`e$19339$$ = 0, $CellContext`a$19340$$ = 0, $CellContext`h$19341$$ = 0, $CellContext`q$19342$$ = 0, $CellContext`b$19343$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`a$$ = 2, $CellContext`b$$ = 1, $CellContext`e$$ = 0.8, $CellContext`frmodel$$ = 1, $CellContext`gmodel$$ = 1, $CellContext`h$$ = 0.5, $CellContext`KK$$ = 2, $CellContext`q$$ = 1, $CellContext`r$$ = 1, $CellContext`startpt$$ = {1, 0.75}, $CellContext`streams$$ = 0, $CellContext`tmax$$ = 25}, "ControllerVariables" :> { Hold[$CellContext`tmax$$, $CellContext`tmax$19333$$, 0], Hold[$CellContext`streams$$, $CellContext`streams$19334$$, False], Hold[$CellContext`gmodel$$, $CellContext`gmodel$19335$$, False], Hold[$CellContext`r$$, $CellContext`r$19336$$, 0], Hold[$CellContext`KK$$, $CellContext`KK$19337$$, 0], Hold[$CellContext`frmodel$$, $CellContext`frmodel$19338$$, False], Hold[$CellContext`e$$, $CellContext`e$19339$$, 0], Hold[$CellContext`a$$, $CellContext`a$19340$$, 0], Hold[$CellContext`h$$, $CellContext`h$19341$$, 0], Hold[$CellContext`q$$, $CellContext`q$19342$$, 0], Hold[$CellContext`b$$, $CellContext`b$19343$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> Module[{$CellContext`maxN$, $CellContext`maxP$, $CellContext`soln$, \ $CellContext`Niso$, $CellContext`Piso$}, $CellContext`Piso$ := Switch[$CellContext`frmodel$$, 1, $CellContext`q$$/($CellContext`e$$ $CellContext`a$$), 2, $CellContext`q$$/($CellContext`e$$ $CellContext`a$$ - \ $CellContext`a$$ $CellContext`q$$ $CellContext`h$$), 3, ($CellContext`q$$/($CellContext`e$$ $CellContext`a$$ - \ $CellContext`a$$ $CellContext`q$$ $CellContext`h$$))^( 1/$CellContext`b$$)]; $CellContext`Niso$ := Switch[$CellContext`gmodel$$, 1, Switch[$CellContext`frmodel$$, 1, $CellContext`r$$/$CellContext`a$$, 2, $CellContext`r$$/($CellContext`a$$ ( 1 + $CellContext`a$$ $CellContext`h$$ $CellContext`X))], 2, Switch[$CellContext`frmodel$$, 1, ($CellContext`r$$/$CellContext`a$$) ( 1 - $CellContext`X/$CellContext`KK$$), 2, ($CellContext`r$$/$CellContext`a$$) ( 1 - $CellContext`X/$CellContext`KK$$) ( 1 + $CellContext`a$$ $CellContext`h$$ $CellContext`X), 3, $CellContext`X^( 1 - $CellContext`b$$) ($CellContext`r$$/$CellContext`a$$) ( 1 - $CellContext`X/$CellContext`KK$$) ( 1 + $CellContext`a$$ $CellContext`h$$ \ $CellContext`X^$CellContext`b$$)]]; $CellContext`maxN$ := Max[1.25 $CellContext`KK$$, $CellContext`Piso$ 2]; $CellContext`maxP$ := Switch[$CellContext`frmodel$$, 1, 2 ($CellContext`r$$/$CellContext`a$$), 2, 2 $CellContext`r$$ (( 1 + $CellContext`a$$ $CellContext`h$$ $CellContext`KK$$)^2/( 4 $CellContext`a$$^2 $CellContext`h$$ $CellContext`KK$$)), 3, ReplaceAll[ 1.5 $CellContext`Niso$, $CellContext`X -> 0.25 $CellContext`maxN$]]; $CellContext`Plota = ListLinePlot[{{$CellContext`Piso$, 0}, {$CellContext`Piso$, $CellContext`maxP$}}, ImageSize -> {400, 400}, AspectRatio -> 1, Axes -> False, PlotRange -> {{0, $CellContext`maxN$}, {0, $CellContext`maxP$}}, Frame -> True, FrameLabel -> { Style["Prey Density N", 14, FontFamily -> "Helvetica"], Style["Predator Density P", 14, FontFamily -> "Helvetica"]}, FrameTicksStyle -> Directive[12], PlotStyle -> {Thick, Darker[Red]}]; $CellContext`Plotb = Plot[$CellContext`Niso$, {$CellContext`X, -1, 10}, PlotStyle -> {Thick, Darker[Blue]}]; $CellContext`Plotc = If[$CellContext`streams$$ == 1, With[{$CellContext`g$ = If[$CellContext`gmodel$$ == 1, $CellContext`r$$ $CellContext`x, $CellContext`r$$ \ $CellContext`x (1 - $CellContext`x/$CellContext`KK$$)], $CellContext`f$ = Switch[$CellContext`frmodel$$, 1, $CellContext`a$$ $CellContext`x, 2, $CellContext`a$$ ($CellContext`x/( 1 + $CellContext`a$$ $CellContext`h$$ $CellContext`x)), 3, $CellContext`a$$ ($CellContext`x^$CellContext`b$$/( 1 + $CellContext`a$$ $CellContext`h$$ \ $CellContext`x^$CellContext`b$$))]}, StreamPlot[{$CellContext`g$ - $CellContext`f$ $CellContext`y, \ $CellContext`e$$ $CellContext`f$ $CellContext`y - $CellContext`q$$ \ $CellContext`y}, {$CellContext`x, 0, $CellContext`maxN$}, {$CellContext`y, 0, $CellContext`maxP$}, StreamColorFunction -> "Rainbow", StreamPoints -> 10, StreamStyle -> Dashed, PerformanceGoal -> "Speed"]], ListPlot[{-1, -1}]]; $CellContext`soln$ = Quiet[ $CellContext`PP[{$CellContext`r$$, $CellContext`KK$$, \ $CellContext`a$$, $CellContext`h$$, $CellContext`e$$, $CellContext`q$$, \ $CellContext`b$$, $CellContext`startpt$$, $CellContext`gmodel$$, \ $CellContext`frmodel$$}, $CellContext`tmax$$]]; $CellContext`Plotd = ParametricPlot[{ Evaluate[ (Part[#, 1][$CellContext`t]& )[$CellContext`soln$]], Evaluate[ ( Part[#, 2][$CellContext`t]& )[$CellContext`soln$]]}, \ {$CellContext`t, 0, $CellContext`tmax$$}, PlotStyle -> {Thick, Black}]; Row[{ LocatorPane[ Dynamic[$CellContext`startpt$$], Show[$CellContext`Plota, $CellContext`Plotb, $CellContext`Plotc, \ $CellContext`Plotd]], Plot[{ Evaluate[ (Part[#, 1][$CellContext`t]& )[$CellContext`soln$]], Evaluate[ ( Part[#, 2][$CellContext`t]& )[$CellContext`soln$]]}, \ {$CellContext`t, 0, $CellContext`tmax$$}, PlotRange -> {{0, $CellContext`tmax$$}, Automatic}, Frame -> True, PlotStyle -> {{Thick, Darker[Blue]}, {Thick, Darker[Red], Dashed}}, FrameLabel -> { Style["Time", 14, FontFamily -> "Helvetica"], Style[ "Prey (blue) Predator (red)", 14, FontFamily -> "Helvetica"]}, FrameTicksStyle -> Directive[12], ImageSize -> {500, 400}, AspectRatio -> 4/5]}]], "Specifications" :> { Style["Continuous time models of predator-prey dynamics", 14, Bold], Style["\[Copyright] David A. Vasseur\n", 12, Italic], Style["Click in the phase plane to set initial conditions.", 10], Style[ "Note: Some combinations of growth and functional response models are \ not\n permitted since they produce explosive solutions.", 10, Italic], {{$CellContext`tmax$$, 25, "Length of Simulation"}, 10, 100, Appearance -> "Labeled"}, Style[ "\nTurn on streams to see sample trajectories", 10], {{$CellContext`streams$$, 0}, {0, 1}, ControlType -> Checkbox}, Delimiter, Style[ "Prey Growth Parameters", 12, Italic], {{$CellContext`gmodel$$, 1, "Model"}, { 1 -> "Exponential", 2 -> "Logistic"}, Enabled -> Dynamic[$CellContext`frmodel$$ == 1]}, {{$CellContext`r$$, 1, "Intrinsic Growth Rate r"}, 0, 2, Appearance -> "Labeled"}, {{$CellContext`KK$$, 2, "Carrying Capacity K"}, 0, 5, Appearance -> "Labeled", Enabled -> Dynamic[$CellContext`gmodel$$ == 2]}, Delimiter, Style[ "Predator Growth Parameters", 12, Italic], {{$CellContext`frmodel$$, 1, "Functional Response"}, { 1 -> "Type 1", 2 -> "Type 2", 3 -> "Type 3"}, Enabled -> Dynamic[$CellContext`gmodel$$ == 2]}, {{$CellContext`e$$, 0.8, "Conversion Efficiency e"}, 0, 1, Appearance -> "Labeled"}, {{$CellContext`a$$, 2, "Attack Rate a"}, 0, 5, Appearance -> "Labeled"}, {{$CellContext`h$$, 0.5, "Handling Time h"}, 0, 2, Appearance -> "Labeled", Enabled -> Dynamic[$CellContext`frmodel$$ != 1]}, {{$CellContext`q$$, 1, "Predator Death Rate q"}, 0, 1, Appearance -> "Labeled"}, {{$CellContext`b$$, 1, "Functional Response Exponent"}, 1, 3, Appearance -> "Labeled"}, {{$CellContext`startpt$$, {1, 0.75}}, ControlType -> None}}, "Options" :> {SynchronousUpdating -> False}, "DefaultOptions" :> {}], ImageSizeCache->{1340., {260., 265.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, Initialization:>({$CellContext`soln = NDSolve[{ Derivative[1][{}][$CellContext`t] == 0.5 (1 - {}[$CellContext`t]) {}[$CellContext`t] - \ ($CellContext`y[$CellContext`t] {}[$CellContext`t])/(1 + 2 {}[$CellContext`t]), Derivative[1][$CellContext`y][$CellContext`t] == -(Graphics[ GraphicsComplex[CompressedData[" 1:eJxTTMoPSmViYGAwAmIQPQpGwSjADQD2ewHg "], {}], { Axes -> True, AxesOrigin -> {0, 0}, PlotRange -> {{0., 1.}, {0., 1.}}, PlotRangeClipping -> True, PlotRangePadding -> { Scaled[0.02], Scaled[0.02]}}] $CellContext`y[$CellContext`t]) + ( 0.5 $CellContext`y[$CellContext`t] {}[$CellContext`t])/(1 + 2 {}[$CellContext`t]), {{}[0] == 0.435, $CellContext`y[0] == 0.56}}, {{}, $CellContext`y}, {$CellContext`t, 0, 500}], Attributes[Derivative] = {NHoldAll, ReadProtected}, $CellContext`Niso := Switch[FE`gmodel$$1382, 1, Switch[ FE`frmodel$$1382, 1, FE`r$$1382/FE`a$$1382, 2, FE`r$$1382/( FE`a$$1382 (1 + (FE`a$$1382 FE`h$$1382) $CellContext`X))], 2, Switch[ FE`frmodel$$1382, 1, (FE`r$$1382/FE`a$$1382) (1 - $CellContext`X/FE`KK$$1382), 2, ((FE`r$$1382/FE`a$$1382) (1 - $CellContext`X/FE`KK$$1382)) ( 1 + (FE`a$$1382 FE`h$$1382) $CellContext`X), 3, ((FE`r$$1382/(FE`a$$1382 $CellContext`X)) (1 - $CellContext`X/ FE`KK$$1382)) (1 + (FE`a$$1382 FE`h$$1382) $CellContext`X^2)]], FE`gmodel$$1382 = 1, FE`frmodel$$1382 = 3, FE`r$$1382 = 1, FE`a$$1382 = 2, FE`h$$1382 = 0.5, FE`KK$$1382 = 2, $CellContext`Plota = Graphics[{{}, {{{}, {}, { Hue[0.67, 0.6, 0.6], Directive[ PointSize[0.019444444444444445`], AbsoluteThickness[1.6], Thickness[Large], RGBColor[2/3, 0, 0]], Line[{{0.625, 0.}, {0.625, 1.}}]}}}, {}, {}, {{}, {}}}, { DisplayFunction -> Identity, PlotRangePadding -> {{0, 0}, {0, 0}}, AxesOrigin -> {0, 0}, PlotRange -> {{0, 2.5}, {0, 1}}, PlotRangeClipping -> True, ImagePadding -> All, DisplayFunction -> Identity, AspectRatio -> 1, Axes -> {False, False}, AxesLabel -> {None, None}, AxesOrigin -> {0, 0}, DisplayFunction :> Identity, Frame -> {{True, True}, {True, True}}, FrameLabel -> {{ Style["Predator Density P", 14, FontFamily -> "Helvetica"], None}, { Style["Prey Density N", 14, FontFamily -> "Helvetica"], None}}, FrameTicks -> {{Automatic, Automatic}, {Automatic, Automatic}}, FrameTicksStyle -> Directive[12], GridLines -> {None, None}, GridLinesStyle -> Directive[ GrayLevel[0.5, 0.4]], ImageSize -> {400, 400}, Method -> {"CoordinatesToolOptions" -> {"DisplayFunction" -> ({ ( Part[{{Identity, Identity}, {Identity, Identity}}, 1, 2][#]& )[ Part[#, 1]], ( Part[{{Identity, Identity}, {Identity, Identity}}, 2, 2][#]& )[ Part[#, 2]]}& ), "CopiedValueFunction" -> ({ ( Part[{{Identity, Identity}, {Identity, Identity}}, 1, 2][#]& )[ Part[#, 1]], ( Part[{{Identity, Identity}, {Identity, Identity}}, 2, 2][#]& )[ Part[#, 2]]}& )}}, PlotRange -> {{0, 2.5}, {0, 1}}, PlotRangeClipping -> True, PlotRangePadding -> {{0, 0}, {0, 0}}, Ticks -> {Automatic, Automatic}}], $CellContext`Plotb = Graphics[{{{{}, {}, { Directive[ Opacity[1.], AbsoluteThickness[1.6], Thickness[Large], RGBColor[0, 0, 2/3]], Line[CompressedData[" 1:eJxTTMoPSmViYGAwAWIQXRdd1f7///v9DGDwwN4z45XBnO43cL5IadQtC/GX cP5kIcf2VTMfwvl+jmcfxJpdhfNTmg/9nfJvL5yver96e1fTXnsY32n9BqWj Tlfh/JL1uwqt1R/B+dOefuD+2vkSzp8k+uXLWbMPcH5K2Rsz1s+f4fw3cUel Nud/h/Nnm3VEX7z1C84PPeOeHzzhH5z/6EuxkBozowOMvyDpUFjIYyY4/+Px 1pO7drHA+bKFlr0xRWxwvn9ltpWBPwecH3O018KehwvOzyvQU2h4xw3niwi7 mP08zgvnT/3pOXdtIz+c3yf//JvnBgE43y5l4fFb/YJw/g6VaevZs4Xg/K0i Xj1nI4Th/PlqZYIbzUTgfJ6MxG5ddVE431m1XcZNWAzOT2dRDat5j+DzCTe1 eN8Th/MXO4dHW+yVgPMZP6R5HpwuCec/Om2q87hFCs7feLJmu1KSNJzP9L3J /HegDJzv6MaY8NFQFs73Etw1v5lXDs6/xm10Zd1vBN9fQ2nj95vycL7qroyD OakKcD7HvLtbV99G8B9N6qv76akI59fKzz84+SCCz7/39Q0bQyU4v23G1d+b 5yH4+7nlZ5aLKsP57/qXcHm1IPiXpfoYVQ8h+DuWlv/4/x/BBwCrj/aN "]]}}}, {}, {}}, { DisplayFunction -> Identity, Ticks -> {Automatic, Automatic}, AxesOrigin -> {0, 0}, FrameTicks -> {{Automatic, Automatic}, {Automatic, Automatic}}, GridLines -> {None, None}, DisplayFunction -> Identity, PlotRangePadding -> {{ Scaled[0.02], Scaled[0.02]}, { Scaled[0.05], Scaled[0.05]}}, PlotRangeClipping -> True, ImagePadding -> All, DisplayFunction -> Identity, AspectRatio -> GoldenRatio^(-1), Axes -> {True, True}, AxesLabel -> {None, None}, AxesOrigin -> {0, 0}, DisplayFunction :> Identity, Frame -> {{False, False}, {False, False}}, FrameLabel -> {{None, None}, {None, None}}, FrameTicks -> {{Automatic, Automatic}, {Automatic, Automatic}}, GridLines -> {None, None}, GridLinesStyle -> Directive[ GrayLevel[0.5, 0.4]], Method -> { "DefaultBoundaryStyle" -> Automatic, "DefaultMeshStyle" -> AbsolutePointSize[6], "ScalingFunctions" -> None, "CoordinatesToolOptions" -> {"DisplayFunction" -> ({ ( Part[{{Identity, Identity}, {Identity, Identity}}, 1, 2][#]& )[ Part[#, 1]], ( Part[{{Identity, Identity}, {Identity, Identity}}, 2, 2][#]& )[ Part[#, 2]]}& ), "CopiedValueFunction" -> ({ ( Part[{{Identity, Identity}, {Identity, Identity}}, 1, 2][#]& )[ Part[#, 1]], ( Part[{{Identity, Identity}, {Identity, Identity}}, 2, 2][#]& )[ Part[#, 2]]}& )}}, PlotRange -> {{-1, 10}, {0., 1.}}, PlotRangeClipping -> True, PlotRangePadding -> {{ Scaled[0.02], Scaled[0.02]}, { Scaled[0.02], Scaled[0.02]}}, Ticks -> {Automatic, Automatic}}], $CellContext`Plotc = Graphics[{{}, {{{}, { Hue[0.67, 0.6, 0.6], Directive[ PointSize[0.012833333333333334`], RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6]], Point[{{1., -1.}, {2., -1.}}]}, {}}}, {}, {}, {{}, {}}}, { DisplayFunction -> Identity, PlotRangePadding -> {{ Scaled[0.02], Scaled[0.02]}, { Scaled[0.05], Scaled[0.02]}}, AxesOrigin -> {0., 0}, PlotRange -> {{0., 2.}, {-2., 0}}, PlotRangeClipping -> True, ImagePadding -> All, DisplayFunction -> Identity, AspectRatio -> GoldenRatio^(-1), Axes -> {True, True}, AxesLabel -> {None, None}, AxesOrigin -> {0., 0}, DisplayFunction :> Identity, Frame -> {{False, False}, {False, False}}, FrameLabel -> {{None, None}, {None, None}}, FrameTicks -> {{Automatic, Automatic}, {Automatic, Automatic}}, GridLines -> {None, None}, GridLinesStyle -> Directive[ GrayLevel[0.5, 0.4]], Method -> {"CoordinatesToolOptions" -> {"DisplayFunction" -> ({ ( Part[{{Identity, Identity}, {Identity, Identity}}, 1, 2][#]& )[ Part[#, 1]], ( Part[{{Identity, Identity}, {Identity, Identity}}, 2, 2][#]& )[ Part[#, 2]]}& ), "CopiedValueFunction" -> ({ ( Part[{{Identity, Identity}, {Identity, Identity}}, 1, 2][#]& )[ Part[#, 1]], ( Part[{{Identity, Identity}, {Identity, Identity}}, 2, 2][#]& )[ Part[#, 2]]}& )}}, PlotRange -> {{0., 2.}, {-2., 0}}, PlotRangeClipping -> True, PlotRangePadding -> {{ Scaled[0.02], Scaled[0.02]}, { Scaled[0.05], Scaled[0.02]}}, Ticks -> {Automatic, Automatic}}], $CellContext`g = { 0, 8677/50000, 30183/500000, 0, 0, 0, 3343/250000, 0, 1499/1000000, 0, 0, 0, 100201/1000000, 0, 6793/200000, 0, 1107/1000000, 0, 169647/ 1000000, 5439/200000}, $CellContext`f = {-2.0598309857514763`, 3.3599596986759543`, 1.8599349142092054`, -2.0598309857514754`, 1.859934914209545, -2.059830985751476, 1.8599349142092665`, -0.020408889913875114`, -0.02040888991388056, 4.223354321428102, -2.059830985751475, 1.859934914210106, 0.5924267875444242, 1.8599349142093065`, -0.02040888991387896, 1.8599349142092239`, 1.859934914209177, 4.223354321428099, -1.227893603493188, 1.8599349142094892`}, $CellContext`PP[{ Pattern[$CellContext`r, Blank[]], Pattern[$CellContext`KK, Blank[]], Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`h, Blank[]], Pattern[$CellContext`e, Blank[]], Pattern[$CellContext`q, Blank[]], Pattern[$CellContext`startpt, Blank[]], Pattern[$CellContext`gmodel, Blank[]], Pattern[$CellContext`frmodel, Blank[]]}, Pattern[$CellContext`tmax, Blank[]]] := Module[{$CellContext`NN, $CellContext`P, $CellContext`t, \ $CellContext`g, $CellContext`f}, $CellContext`g := If[$CellContext`gmodel == 1, $CellContext`r $CellContext`NN[$CellContext`t], \ ($CellContext`r $CellContext`NN[$CellContext`t]) ( 1 - $CellContext`NN[$CellContext`t]/$CellContext`KK)]; \ $CellContext`f := Switch[$CellContext`frmodel, 1, $CellContext`a $CellContext`NN[$CellContext`t], 2, $CellContext`a ($CellContext`NN[$CellContext`t]/( 1 + ($CellContext`a $CellContext`h) \ $CellContext`NN[$CellContext`t])), 3, $CellContext`a ($CellContext`NN[$CellContext`t]^2/( 1 + ($CellContext`a $CellContext`h) \ $CellContext`NN[$CellContext`t]^2))]; Flatten[ ReplaceAll[{$CellContext`NN, $CellContext`P}, Quiet[ NDSolve[{ Derivative[ 1][$CellContext`NN][$CellContext`t] == $CellContext`g - \ $CellContext`f $CellContext`P[$CellContext`t], Derivative[ 1][$CellContext`P][$CellContext`t] == ($CellContext`e \ $CellContext`f) $CellContext`P[$CellContext`t] - $CellContext`q \ $CellContext`P[$CellContext`t], Thread[{ $CellContext`NN[0], $CellContext`P[ 0]} == $CellContext`startpt]}, {$CellContext`NN, \ $CellContext`P}, {$CellContext`t, 0, $CellContext`tmax}]]]]], $CellContext`PP[{ Pattern[$CellContext`r, Blank[]], Pattern[$CellContext`KK, Blank[]], Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`h, Blank[]], Pattern[$CellContext`e, Blank[]], Pattern[$CellContext`q, Blank[]], Pattern[$CellContext`b, Blank[]], Pattern[$CellContext`startpt, Blank[]], Pattern[$CellContext`gmodel, Blank[]], Pattern[$CellContext`frmodel, Blank[]]}, Pattern[$CellContext`tmax, Blank[]]] := Module[{$CellContext`NN, $CellContext`P, $CellContext`t, \ $CellContext`g, $CellContext`f}, $CellContext`g := If[$CellContext`gmodel == 1, $CellContext`r $CellContext`NN[$CellContext`t], $CellContext`r \ $CellContext`NN[$CellContext`t] ( 1 - $CellContext`NN[$CellContext`t]/$CellContext`KK)]; \ $CellContext`f := Switch[$CellContext`frmodel, 1, $CellContext`a $CellContext`NN[$CellContext`t], 2, $CellContext`a ($CellContext`NN[$CellContext`t]/( 1 + $CellContext`a $CellContext`h \ $CellContext`NN[$CellContext`t])), 3, $CellContext`a \ ($CellContext`NN[$CellContext`t]^$CellContext`b/( 1 + $CellContext`a $CellContext`h \ $CellContext`NN[$CellContext`t]^$CellContext`b))]; Flatten[ ReplaceAll[{$CellContext`NN, $CellContext`P}, Quiet[ NDSolve[{ Derivative[ 1][$CellContext`NN][$CellContext`t] == $CellContext`g - \ $CellContext`f $CellContext`P[$CellContext`t], Derivative[ 1][$CellContext`P][$CellContext`t] == $CellContext`e \ $CellContext`f $CellContext`P[$CellContext`t] - $CellContext`q \ $CellContext`P[$CellContext`t], Thread[{ $CellContext`NN[0], $CellContext`P[ 0]} == $CellContext`startpt]}, {$CellContext`NN, \ $CellContext`P}, {$CellContext`t, 0, $CellContext`tmax}]]]]], $CellContext`Plotd = Graphics[{{{}, {}, { Directive[ Opacity[1.], AbsoluteThickness[1.6], FaceForm[ Opacity[0.3]], Thickness[Large], GrayLevel[0]], Line[CompressedData[" 1:eJwcl3c81e/7xw+FiiTRIGVkZiS78JI0JJWR7GRlNI200JBRiUgfiSSrktIx jr333nvvvdc5Gb/z/Z1/PF7u933d933d131dz4vH/K62FS2BQMigJxD+9zfk 0/9+aSD8/28UnxbLlwqepSGeczT5desIcsvXe57ZpMHzhde+y39HcDqwSOOn Qho0NQM/e1iMQNvSVfpFfyokQyls4eXDqCKI2m+rTkWkYuLhxahhtFkH3Nqd noq05ivNz92HEfWqIjoqMBViX4tiz8oMo37Deir8Yiqq2Qc07kcOYXZlQYFJ MRWqRr9EvNyHwK6ltXX6aCr2XWo1bjQawpYMOQGmnan4LzhzbQ87VUsSPr5u JEH2a38z7+tBHD78X9rjEhII6gLrUzcHcb2IVrUpnQTzzNtj02cGsV/wv5zC byRs7Pih+J52EOq/EnDLhQThpv9GtNwG8JNUG/zuNgnxxjY6RSYDINKmKLJa krBSIlZzS2kA2U5+ZxW0SZhSymO7tt6PP5kWYe3qJPiJRN3+1NWPBULU0bZT 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