Conduct stability analysis by finding the Jacobian of this system and then evaluate each steady state for stability. \begin{aligned} \frac{\text d A}{\text d t} &= g\left(1-\frac{A}{k}\right)A\\ \frac{\text d N}{\text d t} &= s A - \delta N \end{aligned}
So far I have been able to solve for $(0,0)$ and $(k, s k/\delta)$ as the steady states, but I'm stuck on the Jacobian and solving for $\lambda$ then seeing if they are stable or unstable. I know sometimes you can use trace and determinate to make it easier but I'm still having trouble solving it.
The Jacobian matrix is given by $$ J(A,N) = \begin{pmatrix} g \left( 1 -2 \frac{A}{k}\right) & 0 \\ s & -\delta \end{pmatrix} . $$ Therefore, $$ J(0,0) = \begin{pmatrix} g & 0 \\ s & -\delta \end{pmatrix} , $$ with characteristic polynomial $\det(J(0,0)-\lambda I) = -(g-\lambda)(\delta+\lambda)$. Thus, $J(0,0)$ has eigenvalues $g$ and $-\delta$. Similarly, $$ J(k,sk/\delta) = \begin{pmatrix} -g & 0 \\ s & -\delta \end{pmatrix} , $$ has eigenvalues $-g$ and $-\delta$. Assuming that all parameters are positive, one can deduce that $(0,0)$ is unstable and $(k,sk/\delta)$ is asymptotically stable.