Suppose, I have three ODEs $$ \dot{x} = f_1(x,y),\\ \dot{y} = f_2(x,y,z),\\ \dot{z} = f_3(y,z) $$ where $x, y,$ and $z$ are function of $t$. Could you please provide the guideline, how to prove that the solution is always positive or condition for a positive solution for all values of $t$.
I need to know the guideline for the prove as well as if you can direct me to the resources/textbook to look for would be appreciated.
Thanks.
Assuming the vector field $F = (f_1, f_2, f_3)$ is sufficiently well-behaved (continuously differentiable is enough), then a sufficient condition is for the vector field $F$ to be pointing "away from" the negative parts of the space on the boundary between negative and positive. For example, if $f_3(x,y,z)>0$ whenever $z=0$, a solution to the ODE which starts with $z>0$ can never cross into the $z\leq0$ region.
You could look for worked examples of the Lotka-Volterra equations, in which positivity is shown using this method.
If your vector field doesn't turn out to have this property, then you need more sophisticated arguments which will depend on your specific equations - this is a very broad question.