I've got this 4x4 system that should model the wine fermentation process. All the $\mu, K_N, k_d$ etc are positive constants. Of course I have no idea of how to solve it. But at least I would like to study its equilibrium points, which should be $(0,b,c,d)$ and $ (a,0,0,0), \forall a,b,c,d. $ I study the Jacobian matrix in such points. In $(0,b,c,d)$, the eigenvalues are $\lambda_1 =0$ with algebraic multiplicity 3, and $\lambda_2 =\frac{\mu b}{K_N + b}-k_d c$, that may be equal to zero. What can I say about the stability in this point? I have no idea of how to find Lyapunov function in this case. For the point $(a,0,0,0)$, the eigenvalues are $\lambda_1 = 0$ a.m.=2, $\lambda_2 = -\frac{\mu a}{YK_N}$ and $\lambda_3 = \frac{\beta a}{k_s Z}$. Once again, I don't know how to proceed. Any suggestion? The system is the following:

$X'=(\frac{\mu N}{K_N + N}-k_d E)X;\\ N'=-\frac{\mu N}{(K_N + N)Y}X; \\ E'=\frac{\beta S}{K_S+S}X;\\S'=\frac{\beta S}{(K_S+S)Z}X\\$