3 Species Predator-Prey Model

I am trying to solve a 3-species predator-prey system in matlab. Here is the equation:

where Nk is the population of the kth species. If Ek>0 we view it as prey, otherwise if Ek<0 we view it as predator. The eigenvalues of [A] determine if it behaves like a logistic equation (i.e. has finite carrying capacity)

I have found this website that shows how to find the critical points and eigenvalues but my problem is finding a good A matrix. It would be insane to try the infinite numerical values that can go into A, solve it for the critical points, and then solve the eigenvalues SO is there an easier way to find a unique A matrix? Or is there a website with some published A matrices?