I have a system of differential equations for which I want to find conditions under which a fixed point is stable or not. The system has the following parameters: $b_1, \ldots, b_K, g_1, \ldots, g_K$ and $(m_{lc})_{l, c = 1}^K$. The Jacobian matrix evaluated at the fixed point is given as follows: \begin{equation} J = \begin{pmatrix} l b_1 - (g_1 + \sum_{c=1}^K m_{1c}) & m_{21} & \dots & m_{K1} \\ m_{12} & l b_2 - (g_2 + \sum_{c=1}^K m_{2c}) & \dots & m_{K2} \\ \vdots & \vdots & \ddots & \vdots \\ m_{1K} & m_{2K} & \dots & l b_K - (g_K + \sum_{c=1}^K m_{Kc}) \end{pmatrix} \end{equation} I have applied Gershgorin's circle theorem to $J^T$ and derived the following property of the eigenvalues $\lambda^J$ of $J$: \begin{equation} |\lambda^J - (lb_s - (g_s + \sum_{l = 1}^K m_{sl}))| \leq \sum_{l=1}^K m_{sl} \end{equation} i.e. \begin{equation} \lambda^J \in [lb_s - (g_s + 2 \sum_{l=1}^K m_{sl}), lb_s - g_s] \end{equation} The fixed point will be stable if all eigenvalues are negative. Thus if the condition $l b_s - g_s < 0$ is satisfied, the fixed point will be stable.
Is my logic correct? And is there something else that I can apply?