I have the following system of equations:
\begin{eqnarray}\frac{dx}{dt} = x(1 - x^2 - y^2) \\ \frac{dy}{dt} = y(4 - x^2 - y^2) \end{eqnarray}
I want to prove that if a solutions starts (at time $t = 0$) in the region
$$ G = \{(x, y) \in \mathbb{R}^2 : x > 0, y > 0 \text{ and } 1 < x^2 + y^2 < 4\} $$
that it remains there for all $t \geq 0$. I have already proven that, from $G$, a solution cannot cross the $x$-axis where $0 < x < 2$ or the $y$-axis where $0 < y < 2$ (since those line segments are orbits), but the pieces of the circles with radius 1 and 2 (where $x > 0, y > 0$) are the lines I haven't done yet. In this question we can focus on just the circle with radius 1 since the proof for the other circle is completely analogous to it.
In my book, there is a similar problem where they show that if a solution crosses a certain line, that solution attains its maximum or minimum value for $x$ and $y$, and then they derive a contradiction. However, in this case, for $n \geq 1$ the $n$-th derivative of $x$ equals 0 on the circle with radius 1. (If the second derivative were negative, $x$ would attain its maximum on the circle, a contradiction since $\frac{dx}{dt} < 0$ in $G$, so $x$ is decreasing.)
I don't see why a solution can't cross the circle at all. On the circle, $\frac{dx}{dt} = 0$ and $\frac{dy}{dt} > 0$, so after crossing the circle, the solution would go back up to $G$.
Is the circle itself an orbit?
EDIT I think I have found the answer. Using polar coordinates, I found $$ \frac{dr}{dt} = r - r^3 + 3r \sin^2(\theta) $$ Now, $$ \frac{dr}{dt} > 0 \Leftrightarrow 0 < r - r^3 + 3r \sin^2(\theta) \leq r - r^3 + 3r = 4r - r^3 $$ since $0 \leq \sin^2(\theta) \leq 1$. Note that $0 < r \leq 1$ satisfies $4r - r^3 > 0$, and thus $\frac{dr}{dt} > 0$. Therefore, if a solution goes from $G$ to the edge of the unit circle, $r$ must, on the edge, either be decreasing or constant (i.e., its derivative is less than or equal to 0), contradicting the fact that the derivative is positive.
Can someone tell me if this is correct?
Note: This is a plot of the velocity vectors, mostly for me to get an idea what is going on.
(Large Version) (Source, I used scale $0.1$)