I have following differential equations:
$\dot{x} = - x + y\,x \qquad \dot{y} = p - b\,y -2\,x\,y$
I made an analysis of fixed points and their stability. We assume $b > 1$, so we have the points:
$\vec{f}_1=\begin{pmatrix} \frac{p-b}{2} \\ 1 \end{pmatrix}$
Stable node for $p>b$ and saddle point for $p<b$. (No complex eigenvalues of the jacobian possible due to $b>1$.)
$\vec{f}_1=\begin{pmatrix} 0 \\ \frac{p}{b} \end{pmatrix}$
Stable node for $p<b$ and saddle point for $p>b$.
Now I want to analyze the bifurcation happening at $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$ if we assume $b$ constant. My problem now is that I read in books about these problem often just for one ODE and not for a system (of two). I guess I have a transcritical Bifurcation, because before and after passing $b=p$ we have two fixed point and at $b=p$ we have just one strange fixed point.
How can I show the classification mathematically?