Stability of equilibrium of a nonlinear system of ODE's Suppose we have the nonlinear system of ODE's 
$$\begin{cases}
\dot{x_1} = -\beta x_1 x_2 \\
\dot{x_2} = \beta x_1 x_2 - \gamma x_2
\end{cases}
$$
Where we take $\beta, \gamma > 0$ arbitrary for now. In particular I am interested in the equilibrium point $(x_1, x_2) = (1, 0)$. I first linearized the system around the point $(1, 0)$ by using the Jacobian 
$$J(x_1, x_2) = \begin{pmatrix} -\beta x_2 & -\beta x_1 \\ \beta x_2 & \beta x_1 - \gamma \end{pmatrix}.$$
So the linearized system around $(1, 0)$ is given by 
$$\begin{pmatrix} \dot{x_1} \\ \dot{x_2} \end{pmatrix} = \begin{pmatrix} 0 & -\beta \\ 0 & \beta - \gamma \end{pmatrix}\begin{pmatrix} x_1 \\ x_2 \end{pmatrix}.$$
Hence, it follows we have eigenvalues $\lambda_1 = 0$ and $\lambda_2 = \beta - \gamma$. Now if $\beta > \gamma$ we know that the nonlinear system is unstable. However, if we let $\beta \leq \gamma$ we can not determine the stability of the nonlinear system by linearization. 
The system seems relatively simple and I would expect the equilibrium to be stable or even asymptotically stable in the case $\beta \leq \gamma$, but how would one prove this when linearization fails to provide a conclusive answer? Or did I make some error in my reasoning?
 A: I'd attack this system in s few steps, ending with analysis of a 2nd order ODE:
Let $u_1 = \beta x_i$ to turn the equations into 
$$
\left\{ \begin{array}{c} 
\dot{u_1} = -u_1u_2 \\
\dot{u_2} = u_1u_2-\gamma u_2\\
u_1(0) = \beta + \epsilon_1 \\
u_2(0) = \epsilon_2\end{array}\right.
$$
Then get a system about $(0,0)$ by letting $y_1 =  u_1-\beta, y_2 = u_2$:
$$
\left\{ \begin{array}{c} 
\dot{y_1} = -y_2(y_1+\beta) \\
\dot{y_2} =y_2(y_1+\beta) -\gamma y_2\\
y_1(0) =  \epsilon_1 \\
y_2(0) = \epsilon_2\end{array}\right.
$$
Then let $z_1 = y_1+y_2, z_2=y_2$:
$$
\left\{ \begin{array}{c} 
\dot{z_1} =-\gamma z_2\\
\dot{z_2} = z_2(z_1-z_2+\beta) \\
z_1(0) =  \epsilon_1 +\epsilon_2 \equiv \epsilon_3\\
z_2(0) = \epsilon_2\end{array}\right.
$$
Next, use the first equation to write $z_2 = -\dot{z_1}/\gamma$ and multiply the second line through by $-\gamma$ to get a 2nd-order ODE:
$$
\left\{ \begin{array}{c} 
\ddot{z_1} = \dot{z_1}(z_1+\dot{z_1}/\gamma + \beta) \\
z_1(0) =  \epsilon_3\\
\dot{z_1}(0) = \epsilon_2\end{array}\right.
$$
The given point in the original equation is stable if and only if $z_1(0) = \dot{z_1}(0) = 0$ is a stable fixed point in this ODE.  But for $\beta > 0$ and ignorably small $z_1$ and $\dot{z_1}$,  $\dot{z_1}$ grows exponentially.  
So it seems the system is unstable unless $\beta \leq 0$.
