Section 8.4: Ordinary Differential equations
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1 Section8_4.nb Section 8.4: Ordinary Differential equations Created by Tomas de-camino-beck SIR model Let ihtl be the infected at time t and shtl suceptibles. Note that we are not really interested in recovery: eq = s '@td ã -b s@td i@td; eq2 = i '@td ã b s@td i@td - a i@td; To find a numreical solution define a and b, then we use NDSolve: a =.4; b =.2; solution = NDSolve@8eq, eq2, s@d == 99., i@d ã <, 8i, s<, 8t,, <D 88i Ø InterpolatingFunction@88.,.<<, <>D, s Ø InterpolatingFunction@88.,.<<, <>D<< Plot the numreical solution: ParametricPlot@Evaluate@8t, s@td< ê. solutiond, 8t,, <D;
2 Section8_4.nb 2 ParametricPlot@Evaluate@8t, i@td< ê. solutiond, 8t,, <D; This is a plot of shtl against ihtl: ParametricPlot@Evaluate@8i@tD, s@td< ê. solutiond, 8t,, <D; Logistic Equation Find a solution for the logistic equation sol = DSolve@8x'@tD ã r x@td H - x@tdl, x@d ã x<, x, td Solve::ifun : Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. More 99x Ø FunctionA8t<, r t x ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ - x + r t x E== We can plot a vector field to see the trayectory of solutions: << Graphics`PlotField` r = 2; note that we ommit the fact that x is a function of time i.e. we use the non-dimensionalized logistic equation:
3 Section8_4.nb 3 vectorplot = PlotVectorFieldA8, r x H - xl<, 8t,, 6<, 8x,,.3<, AspectRatio Ø ÅÅÅÅÅÅÅÅÅ, Frame Ø True, FrameTicks Ø AutomaticE Plot some solutions solplot = Plot@Evaluate@8 x@td ê. sol ê. x Ø., x@td ê. sol ê. x Ø <D, 8t,, 6<, PlotRange Ø All, Ticks Ø NoneD Put them toghether:
4 Section8_4.nb 4 Show@vectorplot, solplotd Predator Prey Model Consider the following non-dimensional predator v and prey u model: prey = u'@td == u@td H - u@td ê kl - u@td v@td; pred = v'@td == g Hu@tD - L v@td; Solving for steady-states, we use the Solve function, letting u' HtL, v' HtL = : solution = Solve@8prey ê. 8u'@tD Ø <, pred ê. 8v'@tD Ø <<, 8u@tD, v@td<d 98u@tD Ø, v@td Ø <, 8u@tD Ø k, v@td Ø <, 9v@tD Ø - ÅÅÅÅ, u@td Ø == k The following code, calculates the Jacobian of a system of equations: JacobianMatrix@f_List, var_listd := Outer@D, f, vard; Now lets compute the jacobian of the predator prey-system (note that to get the right hanbd side of the equation we use f P2T: J = JacobianMatrix@8preyP2T, predp2t<, 8u@tD, v@td<d; MatrixForm@JD i j - ÅÅÅÅÅÅÅÅÅÅ 2 u@td - v@td -u@td k y z k g v@td g H- + u@tdl { Evaluated at the steady states: J ê. solution 988, <, 8, -g<<, 88-, -k<, 8, g H- + kl<<, 99- ÅÅÅÅ, -=, 9g J - ÅÅÅÅ N, === k k
5 Section8_4.nb 5 Note that we get our three matrices, that correspond to the steady states. Now, if we choose some parameter values for g and k, we can compute the eigenvalues in order to study the stability of this system: In matrix from: param = 8g Ø, k Ø 2<; evalj = J ê. solution ê. param 988, <, 8, -<<, 88-, -2<, 8, <<, 99- ÅÅÅÅ 2, -=, 9 ÅÅÅÅ 2, === TableForm@MatrixForm êü evaljd J - N - -2 J N i - ÅÅÅ - 2 y j k ÅÅÅ 2 z { We can also study the vector field: << Graphics`PlotField` vp = PlotVectorField@8u H - u ê kl - u v, g Hu - L v< ê. 8g Ø, k Ø 2<, 8u,, 2<, 8v,, <, Frame Ø True, PlotPoints Ø 5D solution = NDSolve@8prey ê. param, pred ê. param, u@d ã 2, v@d ã.<, 8u, v<, 8t,, <D 88u Ø InterpolatingFunction@88.,.<<, <>D, v Ø InterpolatingFunction@88.,.<<, <>D<<
6 Section8_4.nb 6 phase = ParametricPlot@Evaluate@8u@tD, v@td< ê. solutiond, 8t,, <D; Show@vp, phased.5.5 2
Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information.
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