Stability analysis for a system of two differential equations I have this system of differential equations:
\begin{equation}
\frac{dx}{dt}=\alpha x-\beta xy\\
\frac{dy}{dt}=\beta xy-\gamma y
\end{equation}
I need to find the critical points and then do a stability analysis. After this I need to find the solutions curves for this system.
I have started like this:
The critical points are $(0,0), (0,\frac{\alpha} {\beta}), (\frac{γ}{β},0)$. Is this correct?
As I understand it I need to create a Jacobian matrix: $g(X)$= $\begin{pmatrix}
\alpha & -\beta x  \\
\beta y & -\gamma 
\end{pmatrix}$, where $X=(x,y)^T$
Then form: (also matrices)
$A_1=g(0,0)$= $\begin{pmatrix}
\alpha & 0  \\
 0 & -\gamma 
\end{pmatrix}$,
$A_2=g\bigl(0,\frac{\alpha} {\beta}\bigr)$= $\begin{pmatrix}
\alpha & 0  \\
 \alpha & -\gamma 
\end{pmatrix}$,
$A_3=g\bigl(\frac{γ}{β},0\bigr)$= $\begin{pmatrix}
\alpha & -\gamma  \\
 0 & -\gamma 
\end{pmatrix}$

My question is how do I know which one is stable? and then how do I form the solution curves. 
 A: The computation of the equilibrium points is not correct. Since you do not give any assumptions on the parameters, let us suppose that $\alpha,\beta,\gamma$ are non zero. you need to solve simultaneously 
\begin{equation}
\alpha x-\beta xy=0; \qquad \beta xy-\gamma y=0.
\end{equation}
The point $(0,0)$ is clearly an equilibrium point. Observe that $x=0$ implies $\gamma y=0$ and that $y=0$ implies $\alpha x=0$. Thus, your last two equilibrium points are not correct. 
Now, suppose $(x,y)$ is not the origin. Then $(x,y)=(\gamma/\beta,\alpha/\beta)$ is the other equilibrium point. 
The Jacobian is the matrix
\begin{equation}
J=\begin{bmatrix}\alpha-\beta y & -\beta x\\
\beta y & \beta x-\gamma
\end{bmatrix},
\end{equation}
then
\begin{equation}
J|_{(0,0)}=\begin{bmatrix}\alpha& 0\\
0 & -\gamma
\end{bmatrix}; \qquad J|_{(\gamma/\beta,\alpha/\beta)}=\begin{bmatrix}0 & -\gamma\\
\alpha & 0 
\end{bmatrix}
\end{equation}
You could construct the following table, depending on the eigenvalues of $J$ at each point. Let $p_1=(0,0)$, $p_2=(\gamma/\beta,\alpha/\beta)$
\begin{array}{|c|c|c|c|}
\hline
\alpha & \beta & \gamma & p_1 & p_2 \\ \hline
 + & + & + & saddle & center \\ \hline
 + & + & - & source & saddle \\ \hline
 + & - & + & saddle & center \\ \hline
 + & - & - & source & saddle \\ \hline
 - & + & + & sink & saddle \\ \hline
 - & + & - & saddle & center \\ \hline
 - & - & + & sink & saddle \\ \hline
 - & - & - & saddle & center \\ \hline
\end{array}
And then, for example, have the following phase portraits (at least for the first 2 cases, you could try to do the rest)


