In this paper the authors have the dynamical system
$\begin{align} T_f \dot{y}_f & = -y_f + (1-\alpha(v))\varphi(z,d) \\ T_r \dot{y}_r & = -y_r + \alpha(v) \varphi(z,d) \\ \dot{z} & = -\varphi(z,d) + y_r + u \end{align}$
and they state in eqns (8-10) that the eigenvalues of the linearization at the equilibrium points $(\overline{y}_f, \overline{y}_r, \overline{z})$ are
$\begin{align} \lambda_1 & = -T_f^{-1} \\ \lambda_2 + \lambda_3 & = -\varphi_z(\overline{z},d) - T_r^{-1} \\ \lambda_2 \lambda_3 & = T_r^{-1} \phi_z(\overline{z},d)(1-\alpha(\overline{v})) \end{align}$
Can someone explain to me how these are derived?