Use Results from Manipulate Plot NDSolve to create another plot versus variable used in DE

Question

I created my plot using using this input:

Manipulate[solution[t_] = NDSolve[{
     Derivative[1][w][t] == - k1  S w[t] + k2   x[t] - 
              k7 H  w[t] + k8  y[t], 
     Derivative[1][x][t] == - k2  x[t] + k1 S w[t] - k3 H x[t] + 
              k4  z[t], 
     Derivative[1][y][t] == -k8  y[t] + k7 H w[t] - k5  S y[t] + 
              k6 z[t], 
     Derivative[1][z][t] == - k4  z[t] + k3 H  x[t] - k6  z[t] + 
              k5 S y[t], w[0] == 1, x[0] == 0, y[0] == 0, 
     z[0] == 0}, {w [t],
           x [t], y [t], z [t]}, {t, 0, 10}][[1, All, 2]]; 
  Plot[solution[t], {t, 0, 10}], {{k1, 1}, 0.01, 5, 
    Appearance -> "Labeled"}
   , {{k2, 1}, 0.01, 5, Appearance -> "Labeled"}
   , {{k3, 1}, 0.01, 5, Appearance -> "Labeled"}
   , {{k4, 1}, 0.01, 5, Appearance -> "Labeled"} , {{k5, 1}, 0.01, 5, 
    Appearance -> "Labeled"} , {{k6, 1}, 0.01, 5, 
    Appearance -> "Labeled"} , {{k7, 1}, 0.01, 5, 
    Appearance -> "Labeled"} , {{k8, 1}, 0.01, 5, 
    Appearance -> "Labeled"} , {{H, 1}, 0.1, 5, 
    Appearance -> "Labeled"}, {{S, 1}, 0.01, 5, 
    Appearance -> "Labeled"} , 
  TrackedSymbols :> {k1, k2, k3, k4, k5, k6,  k7, k8, H, S}]

I now want to plot the solutions function versus S. Any suggestions?

Answer 1

S is just a parameter. Normally you'd plot your solutions as a function of t given values of the parameters. If you want to plot it also against one of its parameters (S in this case) you can either choose to plot it for a fixed value of t or choose a 3D plot (Plot3D, ContourPlot, DensityPlot) and plot against both S and t.

The function ParametricNDSolve, introduced in v9, comes in handy here.

Note that I plot only W against S and t, but the other function go likewise.

Manipulate[solution = ParametricNDSolve[{
    Derivative[1][w][t] == - k1  S w[t] + k2   x[t] - 
             k7 H  w[t] + k8  y[t], 
    Derivative[1][x][t] == - k2  x[t] + k1 S w[t] - k3 H x[t] + 
             k4  z[t], 
    Derivative[1][y][t] == -k8  y[t] + k7 H w[t] - k5  S y[t] + 
             k6 z[t], 
    Derivative[1][z][t] == - k4  z[t] + k3 H  x[t] - k6  z[t] + 
             k5 S y[t], w[0] == 1, x[0] == 0, y[0] == 0, 
    z[0] == 0}, {w,
          x , y , z }, {t, 0, 10}, {S}];
 ContourPlot[w[S][t] /. solution // Evaluate, {S, 0.01, 5}, {t, 0, 10}, 
  FrameLabel -> {"S", "t"}], 
 {{k1, 1}, 0.01, 5,   Appearance -> "Labeled"}
   , {{k2, 1}, 0.01, 5, Appearance -> "Labeled"}
   , {{k3, 1}, 0.01, 5, Appearance -> "Labeled"}
   , {{k4, 1}, 0.01, 5, Appearance -> "Labeled"} , {{k5, 1}, 0.01, 5, 
    Appearance -> "Labeled"} , {{k6, 1}, 0.01, 5, 
    Appearance -> "Labeled"} , {{k7, 1}, 0.01, 5, 
    Appearance -> "Labeled"} , {{k8, 1}, 0.01, 5, 
    Appearance -> "Labeled"} , {{H, 1}, 0.1, 5, 
    Appearance -> "Labeled"} , 
  TrackedSymbols :> {k1, k2, k3, k4, k5, k6,  k7, k8, H}]