Question

I have a system of ODE’s. One of the ODE’s has a constant parameter which I want to alter between two different values depending on one of the ODE solutions.

So for example let’s say that I have the following equations:

{
  A'[x] == -q A[x]B[x],
  B'[x] ==  q A[x]B[x] - g B[x],
  C'[x] ==  g B[x]
}

Now I can solve them easily using the NDSolve function when q and g are constant values. What I want to do though is vary the value of q so that it has one value when B[x] is below a certain threshold but then changes in value when B[x] rises above this threshold value.

I’ve tried using If statements and Piecewise functions outside of the NDSolve but I haven’t managed to get it working.

Answer 1

This might do something like what you want. I left out the third equation, which seems superfluous.

Clear[f, g, s, t, x];
s[a_, b_] = Piecewise[{{a*b - b, b < 1}, {2 a*b - b, b >= 1}}];
t[a_, b_] = Piecewise[{{-a*b, b < 1}, {-2 a*b, b >= 1}}]; 
{f[x_], g[x_]} = {f[x], g[x]} /. 
  First[NDSolve[{
    f'[x] == t[f[x], g[x]],
    g'[x] == s[f[x], g[x]],
    f[0] == 10, g[0] == 1},
  {f[x], g[x]}, {x, 0, 2}]]