We have the following nonlinear ODE:
$$ f' = af-bg -(f+g)^k \bigl(f'(0) +g'(0)\bigr) + f'(0), $$
$$ \bigl(G-T(x)\bigr) g' = -af+bg - g'(0), $$
where $a,b,k,G$ are constants, $f'(0)$ and $g'(0)$ are the initial conditions of the first order derivatives, and we know that $f(0) = g(0) = 0$. Further, $T(x)$ is a nonlinear function that we know everything about, especially that $T(x) \to G$ when $x \to \infty$.
Are we able to find a meaningful limit for $f'(x)/g'(x)$ when $x \to \infty$ ?