I am working on a predator-prey model and the linearization about and equilibrium point $(0,e_2)$ has Jacobian matrix as follows $$\mathcal{J} = \begin{pmatrix} 0 & 0\\ b& -b \end{pmatrix},$$ where the parameters $e_2$ and $b$ are positive. I never have dealt with this kind of system and I tried to find something online, still cant find anything. Can someone show me how the system behaves? or what method of analysis is implemented?
Thanks in advance
Such a Jacobian matrix does not suffice to determine the stability of the fixed point. Compare the behaviour of the differential systems $$x'=ax^3\qquad y'=x-y+c$$ around their fixed point $(0,c)$ for some positive $a$ (unstable) and for some negative $a$ (stable).