The following lemma is known
Lemma. Let $p, q, \varepsilon > 0$ satisfy $pq > 1$, and let $0 < T \leq \infty $. Assume that $0 \leq y, z \in C^1(0, T)$, $(y, z) \neq (0, 0)$, and that $(y, z)$ solves
$$ \begin{cases} y'(t) \geq \epsilon z^p \\ z'(t) \geq \epsilon y^q \end{cases} $$ on $(0, T)$. Then $T < \infty$ and there holds $y(t)\leq C_1(T-t)^{-\alpha/2}$ and $z(t)\leq C_1(T-t)^{-\beta/2}$ with $0<t<T$ and $C_1=C_1(p,q,\varepsilon)>0$.
The proof of this lemma can be found in the book Superlinear Parabolic Problems Blow-up, Global Existence and Steady States by Souplet, lemma 32.10 page 365.
I would like to know if the same result holds for a system of the form
$$ \begin{cases} y'(t) \geq z^p - \epsilon z^m-\delta z \\ z'(t) \geq y^q - \epsilon_1 y^n-\delta_1 y \end{cases} $$ where here $\varepsilon_i,\delta_i>0$ and $m,n$ are less than $\min \{p,q \}$. I suspect the same is true since $p$ and $q$ are dominant exponents in the equation, but I don't know how to prove it. In fact, the proof technique of the above lemma does not seem to work for this case. Maybe there is some change of variable that transforms this system into a system equal to the lemma so that the result can be used, but I don't know it either. Anyway, does anyone have any idea what happens with this system, or how to solve it, or a book or article reference that might help?