Is this node proper or improper We have the Jacobian matrix (note: $c > b$ and $a,b,c>0$): $$J\left(\frac{a}{b}, 0\right) = \begin{pmatrix} -a & -\frac { ca }{ b }  \\ 0 & a-\frac { ca }{ b }  \end{pmatrix}$$ which is lower triangular so has eigenvalues $\lambda_{1} = -a < 0, \lambda_{2} = a-\frac{ca}{b}$. Since $c > b$ we have that $\lambda_{2} < 0$. So this is a stable node, since $\lambda_{1}, \lambda_{2} < 0$ it is an improper node.
Is this correct?

We have the eigenvalues as:
$$\lambda_{1,2} = \left\{-a,\frac{a (b-c)}{b}\right\}$$
We have:


*

*For the first eigenvalue, $a \gt 0 \implies \lambda_1 \lt 0$.

*For the second eigenvalue, we have cases:

*Case 1: $c \le 0, c \gt b, ~\lambda_2 \gt 0$. This is a saddle.

*Case 2: $c \gt 0 \gt b, \lambda_2 \lt 0$. Eigenvalues are real, unequal, both negative, we have an improper node (asymptotically stable).


Note: We can have an improper node when we have real, unequal, both positive (unstable)/negative (asymptotically stable). We can have a proper or improper node when we have real and equal, both positive (unstable) / negative (asymptotically stable).
For example, if we have a real, double eigenvalue (positive or negative) with only one linearly independent eigenvector, the critical point is called an improper or degenerate node. 
Here is a nice summary.
