Faced with this system (epidemiology):
$$
\begin{cases}
\frac{dN}{dt}=(a-b)N-\alpha Y \\
\frac{dY}{dt}=\nu W(N-Y)-(\alpha + b + \gamma)Y\\
\frac{dW}{dt}=\lambda Y-(\mu + \nu N)W
\end{cases}
$$
(from Anderson, R. M., & May, R. M. (1980). Infectious diseases and population cycles of forest insects. Science)
I try to calculate (i) the eigenvalues of the Jacobian and (ii) the period of its cyclic solution.
(i) Using Mathematica I have calculated the Jacobian but the software does not manage to calculate correctly the expression of the eigenvalues. On the other hand by giving numerical values to the parameters these eigenvalues can be calculated: the first is negative real, the second complex with a positive real part and a negative imaginary part, the third is the conjugate of the second. I know that the sign and the domain of apartenance of the eigenvalues allows to describe the dynamics of the stationary solution but in this case what are these dynamics?
(ii) Again with Mathematica I tried to calculate the conserved quantity $E(N, Y, W)$ as described in this other post (Procedure to find a conserved quantity) but without success either, the software not seeming to find one. It is possible that there is none, but how can I be sure? I wonder if a change of variable would unblock the situation.