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Therefore x can be any odd or even \ integer of \[Pi] . 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There are some exceptional trajectories going into and coming out of the \ saddle fixed points (if the system were linear with constant coefficients, \ they would corresponding to the eigenvectors of the Jacobian matrix), which \ are called ", StyleBox["separatrices", FontWeight->"Bold"], ", because they have exceptional behavior and separate the regions of \ generic behavior from each other. The seaparatrices are harder to visualize \ on a Stream Plot (the green curve represents an approximation of one of the \ separatrices)." }], "Text", CellChangeTimes->{{3.700493254649211*^9, 3.700493301884385*^9}, { 3.700493345925662*^9, 3.700493561673851*^9}, {3.7004936269573174`*^9, 3.700493649795751*^9}, {3.700493715437289*^9, 3.700493745926093*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Exercise 2 (predator-prey system): ", "Section", CellChangeTimes->{{3.622790466683278*^9, 3.622790476625743*^9}, { 3.622790511711554*^9, 3.6227905335575237`*^9}, {3.6227906625672903`*^9, 3.622790669390012*^9}, {3.6538711094179993`*^9, 3.65387111076064*^9}, { 3.653874243652176*^9, 3.653874255935793*^9}, {3.654463985629437*^9, 3.654464002198286*^9}, {3.654464069364059*^9, 3.654464102635427*^9}, { 3.655176007208185*^9, 3.655176007686988*^9}, {3.700487686430895*^9, 3.700487687102871*^9}}], Cell[CellGroupData[{ Cell[TextData[{ "Consider the system usually described as a predator - prey model (Voltera, \ 1931):\n ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ FractionBox["dx", "dt"], FontSize->36], StyleBox["=", FontSize->24], StyleBox[ RowBox[{ RowBox[{"0.2", "x"}], "-", RowBox[{"0.7", "xy"}]}], FontSize->24]}], TraditionalForm]], FontSize->36], StyleBox[",", FontSize->36], " ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox["dy", "dt"], StyleBox["=", FontSize->24], StyleBox[ RowBox[{ RowBox[{ RowBox[{"-", "0.4"}], "y"}], "+", RowBox[{"0.5", "xy"}]}], FontSize->24]}], TraditionalForm]], FontSize->36], "." }], "Subsubsection", CellChangeTimes->{{3.6544642408637457`*^9, 3.654464263463875*^9}, { 3.65446429674226*^9, 3.6544644794677*^9}, {3.654464601362609*^9, 3.654464602290267*^9}, {3.654464650514975*^9, 3.654464665223145*^9}}], Cell[TextData[{ "where you can think of the function x(t) as representing a population of \ rabbits that naturally grow at a rate proportional to their population (that \ is, exponential growth in the absence of predators, at the rate ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ FractionBox["dx", "dt"], FontSize->18], StyleBox["=", FontSize->18], StyleBox[ RowBox[{"0.2", "x"}], FontSize->16]}], TraditionalForm]], FontSize->36], ") and the function y(t) as representing a population of foxes that \ naturally decline (that is, exponential decay in the absence of prey, at a \ rate ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ FractionBox["dy", "dt"], FontSize->18], StyleBox["=", FontSize->18], RowBox[{ RowBox[{ StyleBox["-", FontSize->18], StyleBox["0.4", FontSize->16]}], StyleBox["y", FontSize->16]}]}], TraditionalForm]], FontSize->36], "). The term xy in both equations is proportional to the number of likely \ encounters of the two population in a certain environment. Encounters are \ detrimental to the rabbits and beneficial to the foxes." }], "Text", CellChangeTimes->{{3.654464605624176*^9, 3.654464628508052*^9}, { 3.654465226697503*^9, 3.6544652297351027`*^9}, {3.6544652632054167`*^9, 3.654465270382663*^9}, {3.700493797281001*^9, 3.700493884501216*^9}}, TextJustification->1., FontSize->16], Cell[CellGroupData[{ Cell[TextData[{ "Plot the vector field using ", StyleBox["StreamPlot[...]", FontWeight->"Bold"], " in the region that makes sense for the given problem. On the same plot, \ plot the trajectory starting at the point (0, 1) in Red, the trajectory \ starting at the point (1, 0) in Green, the trajectory starting at (1,1) in \ Orange, and the trajectory starting at (5, 2) in Purple. What do you notice? \ How do the solutions look like?" }], "ItemNumbered", CellChangeTimes->{{3.622814562760298*^9, 3.622814564601383*^9}, 3.6228147114882507`*^9, {3.654465494368629*^9, 3.654465595364773*^9}, { 3.70028038372861*^9, 3.7002804037473516`*^9}}, TextJustification->1., FontSize->16], Cell["\<\ Find the critical points of the system and describe their nature.\ \>", "ItemNumbered", CellChangeTimes->{{3.622814562760298*^9, 3.622814588339142*^9}, { 3.6228146853418417`*^9, 3.6228146932370367`*^9}, {3.622816137363825*^9, 3.622816145483821*^9}, {3.654465777331684*^9, 3.654465791122643*^9}, { 3.7002804090977716`*^9, 3.7002804296650677`*^9}}, TextJustification->1., FontSize->16], Cell[TextData[{ "Assume that initially there are 5 rabbits and 2 foxes. Use ", StyleBox["NDSolve[...]", FontWeight->"Bold"], " or ", StyleBox["DSolve[...]", FontWeight->"Bold"], " to find the solution corresponding to this initial condition. Try plotting \ the solution that NDSolve returns. 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One of the (possibly multiple) solutions of \ the equation \n", StyleBox[" ", FontSize->10], Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ RowBox[{ RowBox[{"0.2", "x"}], "-", RowBox[{"0.7", "xy"}]}], "=", "0"}], FontSize->18], TraditionalForm]], FontSize->10], ",", StyleBox[" ", FontSize->10], Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", "0.4"}], "y"}], "+", RowBox[{"0.5", "xy"}]}], "=", "0"}], FontSize->18], TraditionalForm]], FontSize->10], ". is x=y=0. \n The linear part of the nonlinear system at (0,0) is simply \ given by ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ FractionBox["dx", "dt"], FontSize->36], StyleBox["=", FontSize->24], StyleBox[ RowBox[{"0.2", "x"}], FontSize->24]}], TraditionalForm]], FontSize->36], StyleBox[",", FontSize->36], " ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox["dy", "dt"], StyleBox["=", FontSize->24], StyleBox[ RowBox[{ RowBox[{"-", "0.4"}], "y"}], FontSize->24]}], TraditionalForm]], FontSize->36], "." }], "Text", CellChangeTimes->{{3.700494492997148*^9, 3.7004947349517517`*^9}, { 3.700495121842352*^9, 3.7004951375866632`*^9}}], Cell[TextData[{ "The Jacobian matrix at (0, 0) is therefore ", Cell[BoxData[ RowBox[{"(", GridBox[{ {"0.2", "0"}, {"0", RowBox[{"-", "0.4"}]} }], ")"}]]], " and has eigenvalues 0.2 and -0.4. Therefore the origin is a ", StyleBox["saddle fixed point", FontWeight->"Bold"], ". In this case, it is very easy to identify the exceptional trajectories \ going into and coming out of the saddle fixed points: they are located on the \ y-axis, and respectively on the x-axis. The positive y-axis describes the \ evolution of a habitat with y foxes and 0 rabbits. In the absence of rabbits, \ the population of foxes converges to 0. The positive x-axis describes the \ evolution of a habitat with 0 foxes and x rabbits. In the absence of foxes, \ the population of rabbits grows to infinity. 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We have already studied the eigenvalues of the Jacobian matrix of the \ system at (0, 0), and decided that the origin is a saddle fixed point. 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It is however true, as we saw from the ", StyleBox["StreamPlot[..]. ", FontWeight->"Bold"], "We need to prove this mathematically. We will turn the system of two \ differential equations into one differential equation, which models y as a \ function of x. " }], "Text", CellChangeTimes->{{3.7004967943611403`*^9, 3.7004968583182*^9}, { 3.70049691452099*^9, 3.700496924695356*^9}, {3.700497712410466*^9, 3.700497712824829*^9}, {3.700510985947879*^9, 3.700511004171193*^9}, { 3.700511046252925*^9, 3.700511046708914*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox[" ", FontSize->72], Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ FractionBox["dx", "dt"], "=", RowBox[{ RowBox[{"0.2", "x"}], "-", RowBox[{"0.7", "xy"}]}]}], FontSize->48], TraditionalForm]], FontSize->36], StyleBox[", ", FontSize->72], Cell[BoxData[ FormBox[ RowBox[{ FractionBox["dy", "dt"], "=", RowBox[{ RowBox[{ RowBox[{"-", "0.4"}], "y"}], "+", RowBox[{"0.5", "xy"}]}]}], TraditionalForm]], FontSize->48], StyleBox[".", FontSize->72] }], "Subsubsection", CellChangeTimes->{{3.6544642408637457`*^9, 3.654464263463875*^9}, { 3.65446429674226*^9, 3.6544644794677*^9}, {3.654464601362609*^9, 3.654464602290267*^9}, {3.654464650514975*^9, 3.654464665223145*^9}, { 3.700498085577426*^9, 3.700498087369012*^9}}], Cell["We will compute", "Text", CellChangeTimes->{{3.700498089905698*^9, 3.700498094714859*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ " ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ FractionBox["dy", "dx"], "=", RowBox[{ FractionBox[ FractionBox["dy", "dt"], FractionBox["dx", "dt"]], "=", FractionBox[ StyleBox[ RowBox[{ RowBox[{ RowBox[{"-", "0.4"}], "y"}], "+", RowBox[{"0.5", "xy"}]}], FontSize->48], RowBox[{ RowBox[{"0.2", "x"}], "-", RowBox[{"0.7", "xy"}]}]]}]}], FontSize->72], TraditionalForm]], FontSize->36] }], "Subsubsection", CellChangeTimes->{{3.6544642408637457`*^9, 3.654464263463875*^9}, { 3.65446429674226*^9, 3.6544644794677*^9}, {3.654464601362609*^9, 3.654464602290267*^9}, {3.654464650514975*^9, 3.654464665223145*^9}, { 3.7004981066062326`*^9, 3.700498159522065*^9}, {3.700498227591193*^9, 3.700498230774542*^9}, {3.700509172186968*^9, 3.7005091790257797`*^9}}], Cell["\<\ The new differential equation contains less information than the initial \ system, as it only models the population of foxes (y) as a function of the \ population of rabbits (x), but does not describe how both populations evolve \ as functions of time (t). Nonetheless, this is a separable differential \ equation that we can solve explicitly. We are only interested in positive \ values of x and y so we may assume that x>0 and y>0. We first separate the \ variables and write the equation as:\ \>", "Text", CellChangeTimes->{{3.700498178364253*^9, 3.700498238970131*^9}, { 3.700498326217168*^9, 3.7004984816180363`*^9}, {3.700509086673628*^9, 3.700509103015628*^9}, {3.7005091960980177`*^9, 3.700509260987915*^9}, { 3.700509295629703*^9, 3.7005093098768883`*^9}}, TextJustification->1.] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox[" ", FontSize->24], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ StyleBox[ FractionBox[ RowBox[{"0.2", "-", RowBox[{"0.7", "y"}]}], "y"], FontSize->36], StyleBox["dy", FontSize->24]}], StyleBox["=", FontSize->24], RowBox[{ FractionBox[ RowBox[{ RowBox[{"-", "0.4"}], "+", RowBox[{"0.5", "x"}]}], "x"], StyleBox["dx", FontSize->24]}]}], TraditionalForm]], FontSize->36] }], "Subsubsection", CellChangeTimes->{{3.6544642408637457`*^9, 3.654464263463875*^9}, { 3.65446429674226*^9, 3.6544644794677*^9}, {3.654464601362609*^9, 3.654464602290267*^9}, {3.654464650514975*^9, 3.654464665223145*^9}, { 3.7004981066062326`*^9, 3.700498159522065*^9}, {3.700498227591193*^9, 3.700498230774542*^9}, {3.7004985328690367`*^9, 3.7004985806995068`*^9}}], Cell["\<\ Now we can integrate the left hand side with respect to y and the right hand \ side with respect to x. We obtain\ \>", "Text", CellChangeTimes->{{3.700498602427967*^9, 3.700498639666071*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ " ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[" ", FontSize->48], RowBox[{ RowBox[{ RowBox[{ StyleBox["\[DoubleRightArrow]", FontSize->24], StyleBox[" ", FontSize->24], StyleBox[ RowBox[{ RowBox[{"0.2", " ", "ln", RowBox[{"(", "y", ")"}]}], "-", RowBox[{"0.7", "y"}]}], FontSize->36]}], StyleBox["=", FontSize->36], StyleBox[ RowBox[{ RowBox[{ RowBox[{"-", "0.4"}], RowBox[{"ln", "(", "x", ")"}]}], "+", RowBox[{"0.5", "x"}], "+", "c"}], FontSize->36]}], StyleBox["\[IndentingNewLine]", FontSize->24], StyleBox[" ", FontSize->24], RowBox[{ StyleBox["\[DoubleRightArrow]", FontSize->24], StyleBox[" ", FontSize->36], StyleBox[ RowBox[{ RowBox[{"ln", " ", RowBox[{"(", RowBox[{ SuperscriptBox["y", "0.2"], SuperscriptBox["x", "0.4"]}], ")"}]}], "=", RowBox[{ RowBox[{"0.5", "x"}], "+", RowBox[{"0.7", "y"}], "+", "c"}]}], FontSize->36]}], StyleBox["\[IndentingNewLine]", FontSize->24], StyleBox[" ", FontSize->24], RowBox[{ RowBox[{ StyleBox["\[DoubleRightArrow]", FontSize->24], StyleBox[" ", FontSize->24], RowBox[{ StyleBox[ SuperscriptBox["y", "0.2"], FontSize->48], SuperscriptBox["x", "0.4"], SuperscriptBox["e", StyleBox[ RowBox[{"-", RowBox[{"(", RowBox[{ RowBox[{"0.5", "x"}], "+", RowBox[{"0.7", "y"}]}], ")"}]}], FontSize->36]]}]}], StyleBox["=", FontSize->72], StyleBox["c", FontSize->72]}]}]}], TraditionalForm]], FontSize->36] }], "Subsubsection", CellChangeTimes->{{3.6544642408637457`*^9, 3.654464263463875*^9}, { 3.65446429674226*^9, 3.6544644794677*^9}, {3.654464601362609*^9, 3.654464602290267*^9}, {3.654464650514975*^9, 3.654464665223145*^9}, { 3.7004981066062326`*^9, 3.700498159522065*^9}, {3.700498227591193*^9, 3.700498230774542*^9}, {3.7004985328690367`*^9, 3.7004985806995068`*^9}, { 3.700498714878598*^9, 3.700498926028034*^9}, {3.700508881660157*^9, 3.700508954933877*^9}, {3.700509126855336*^9, 3.700509128759943*^9}, { 3.7005103428049927`*^9, 3.700510347725747*^9}}], Cell[TextData[{ "where ", Cell[BoxData[ FormBox[ RowBox[{"C", "=", SuperscriptBox["e", "c"]}], TraditionalForm]]], " is a random positive constant. Therefore the trajectories of the Voltera \ system in the first quadrant are the level curves of the function ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"F", "(", RowBox[{"x", ",", "y"}], ")"}], "=", RowBox[{ SuperscriptBox["y", "0.2"], SuperscriptBox["x", "0.4"], SuperscriptBox[ StyleBox["e", FontSize->36], StyleBox[ RowBox[{"-", RowBox[{"(", RowBox[{ RowBox[{"0.5", "x"}], "+", RowBox[{"0.7", "y"}]}], ")"}]}], FontSize->36]]}]}], TraditionalForm]], FontSize->48], ". \nIn conclusion, to understand the trajectories (but not how a point \ moves along a trajectory as a function of time t), all we need to do is \ understand the function F." }], "Text", CellChangeTimes->{{3.700508992729352*^9, 3.7005090295261917`*^9}, 3.7005091472830353`*^9, {3.7005093736393347`*^9, 3.7005095080393*^9}, { 3.7005095518585567`*^9, 3.700509553058717*^9}, {3.700509584562724*^9, 3.700509643212484*^9}, {3.700510351184143*^9, 3.700510355512344*^9}, 3.7009123470831213`*^9}], Cell["\<\ The trajectory corresponding 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