Check if region where $\dot{x} > 0$, $\dot{y} > 0$ is positively/negatively invariant

Question

I have the following linear system $\dot{x} = -x$ and $\dot{y} = -2y + 2x^3$. I need to characterise the regions where $\dot{x} > 0, \dot{y} > 0$, $\dot{x} < 0, \dot{y} > 0$ and so on.

I am trying to do so for the former region given by $L = \{(x,y) \in \mathbb{R}^2 | y < x^3, x < 0 \}$. How do I check it's invariance without computing the flow explicitly.