We then determine the stability of these points by finding a linear approximation using partial derivatives. The theory is worthwhile, so check out this link to find out about linearization.
The Jacobian of our equations is $J(x,y)=\begin{pmatrix} {\lambda - \alpha y } & { -\alpha x} \\ {\beta y} & { \beta x -\gamma} \end{pmatrix}$, which we evaluate at the equilibrium points:
$ J(0,0)=\begin{pmatrix} {\lambda} & 0 \\ 0 & {-\gamma} \end{pmatrix}$ which always has eigenvalues of different signs $\Rightarrow$ saddle point.
$J\big(x=\frac{\gamma}{\beta}, y=\frac{\lambda}{\alpha}\big)=\begin{pmatrix} 0 & {\frac{-\alpha \gamma }{ \beta}} \\ {\frac{\lambda \beta }{\alpha}} & { 0} \end{pmatrix}$ which has imaginary eigenvalues of different signs $\Rightarrow$ oscillatory motion.