A System of differential Equations How can I analyze the phase diagram for this system of differential eqs?

This field is not my area of my expertise, so please be generous with the answers. I appreciate quick references as well. Thank you.
 A: Well, you start by finding the zeroes of the vector field
$X(x, y) = \begin{pmatrix} X_x \\ X_y \end{pmatrix} = \begin{pmatrix} x(7 - x - 2y) \\ y(5 - y - x) \end{pmatrix}, \tag{1}$
which corresponds to the given differential equation since we may write
$\mathbf r = \begin{pmatrix} x \\ y \end{pmatrix} \tag{2}$
so that the ODE becomes
$\dot{\mathbf r} = X(\mathbf r). \tag{3}$
It's pretty easy to see that the zeroes of $X$ occur at $(0, 0)$, $(0, 5)$, $(7, 0)$, and $(3, 2)$.  Next you can work out the Jacobean matrix $J_X$ of $X$ in general terms, that is
$J_X = \begin{bmatrix} \dfrac{\partial X_x}{\partial x} &  \dfrac{\partial X_x}{\partial y} \\  \dfrac{\partial X_y}{\partial x} &  \dfrac{\partial X_y}{\partial y} \end{bmatrix}. \tag{4}$
which, using the specific form of $X(x, y)$ from (1) yields
$J_X = \begin{bmatrix} 7 - 2x - 2y & -2x \\ -y & 5 - 2y - x \end{bmatrix}. \tag{5}$
Then we plug in the numbers at each zero of $X$, and find the eigenvalues of $J_X$ at each zero.  The eigenvalues are easily found using the characteristic polynomial of $J_X$ at each point, a simple quadratic equation.  I tabulate the results below:
$J_X(0, 0) = \begin{bmatrix} 7 & 0 \\ 0 & 5 \end{bmatrix}; \; p_J(\lambda) = \lambda^2 - 12\lambda + 35; \; \lambda = 5, 7. \tag{7}$
$J_X(0, 5) = \begin{bmatrix} -3 & 0 \\ -5 & -5 \end{bmatrix}; \; p_J(\lambda) = \lambda^2 + 8\lambda + 15; \; \lambda = -3, -5. \tag{8}$
$J_X(7, 0) = \begin{bmatrix} -7 & -14 \\ 0 & -2 \end{bmatrix}; \; p_J(\lambda) = \lambda^2 + 9\lambda + 14; \; \lambda = -7, -2. \tag{9}$
$J_X(3, 2) = \begin{bmatrix} -3 & -6 \\ -2 & -2 \end{bmatrix}; \; p_J(\lambda) = \lambda^2 + 5\lambda -6; \; \lambda = -6, 1. \tag{10}$
Finally, using (7)-(10) we can describe the types of the zeroes of $X$ and the flow surrounding them.  $(0, 0)$ is a repelling point; all trajectories starting near $(0, 0)$ move away from it as $t \to \infty$; $(0, 5)$ and $(7, 0)$ are attractors, points sufficiently near each of them converge toward the one they start near under the action of the flow, as $t \to \infty$; the remaining zero of $X$, at $(3, 2)$, is a saddle and there are, locally (that is, sufficiently near), exactly two trajectories which approach $(3, 2)$ as $t \to \infty$ and two which approach $(3, 2)$ as $t \to -\infty$.  Near $(3, 2)$ the rest of the integral curves of $X$ have a vaguely hyperbolic appearance, heading towards $(3, 2)$ for awhile and then abruptly turning away.  Taking the trouble to calculate (and sketch) all the eigenvectors for the matrices in (7)-(10) strongly enhances the purely verbal imagery I exploit as I give this somewhat crude description of the phase portrait of $X$, but I type these words as I concurrently work on several other projects, and I am not really in a mood for more arithmetic at the moment.  Perhaps I'll leave my old buddy Amzoti an note in the comments and he'll come on down here and get his programs to make us a nice phase portrait; I don't have the graphics SW to do this at the moment.  But I think anyone who works out the eigenvectors of these four $J_X$ will quickly get my meaning.  It should be observed that there is one special trajectory $\gamma(t)$ such that $\gamma(t) \to (0, 0) \; \text{as} \; t \to -\infty$ and $\gamma(t) \to (3, 2) \; \text{as} \; t \to \infty$; this is the seperatrix which effectively serves as a "switch point" between integral curves emanating from $(0, 0)$ ultimately headed for $(0, 5)$ and those which are destined for $(7, 0)$.  Well, I hope this verbage was at least a tad on the clear side; I hope Amzoti will show up with a phase portrait.
A good book for this kind of stuff is Differential Equations, Dynamical Systems and an Introduction to Chaos by Hirsch, Smale, and Devaney, Academic Press, 2004.  A more advanced treatment of many of these topics may be found in Ordinary Differential Equations by Jack Hale, available from Dover for a song.  Finally,the theorem which puts teeth into the derivation of these phase portraits is called The Stable Manifold Theorem; it can be found in wikipedia here.  There is also a somewhat cursory discussion of it in my answers to this question.
Well, even if my answer is not the $\Omega$ of this topic, it is at least an $A$!
Hope this helps.  Cheerio,
and as always,
Fiat Lux!!!
Thanks to Amzoti for the fine graphics below!
Update
Here is the systems phase portrait using two different tools.


