Given a system of three first order nonlinear ODEs
$ \begin{align*} x_1' &= f(x_1,x_2,x_3) \\ x_2' &= g(x_1,x_2,x_3) \\ x_3' &= h(x_1,x_2,x_3), \end{align*} $
I'm hoping to be able to show that, given a positive initial condition and, for simplicity, $x_1 = 0$, that $x_2$ and $x_3$ remain in the positive cone. Is there a method commonly used for this? I'm just looking for a starting point.