Lotka-Volterra predator-prey model eigenvalue I have a question about the eigenvalues of the prey-predator model called Lotka-Volterra. The model itself consists of 2 nonlinear differential equations of first order:
$$\begin{eqnarray}
 \frac{dx}{dt}&=&ax-bxy \\
 \frac{dy}{dt}&=&cry-dy
\end{eqnarray}$$
I found the stationary points to be both $(0,0)$ and $(d/c, a/b).$ Then I found the eigenvalues (e) to be:
\begin{eqnarray}
e_1&=&a,\\
e_2&=&-d,\\
e_3&=&\pm i*\sqrt{ad}.
\end{eqnarray}
I  know the eigenvalues tells how the system behaves around the stationary points but my question is why are these eigenvalues important for the model? Can they tell something specific about the interaction between the species x and y in the model? I found the eigenvalues but I don't really know to do with them. 
Regards Rasmus O.J
(English is not my native language, so I hope its understandable)
 A: The eigenvalues in this case, will tell you how the system behave as $t \to \infty$, for instance, in the case  of this model $(0,0)$ is a saddle point. On the other hand, the positive equilibrium $e_3 = \pm i*\sqrt{ad}$ will predict that the system will oscillate. From an ecological point of view, it is important. Hope this picture helps.
A: As you said, the eigenvalues tell you about the behaviour near the stationary points.  In particular, they say something about stability
of those points.
The stationary point $(0,0)$ has one positive and one negative eigenvalue, so it is a saddle point and unstable.  There is a curve (which happens to be the $y$ axis in this case) along which solutions approach $(0,0)$ as $t \to \infty$, but no other solutions will have this behaviour.  
The stationary point $(d/c, a/b)$ has purely imaginary eigenvalues.  This is actually a borderline case: in the linearized system the stationary point is a "centre", stable but not asymptotically stable, and the solutions near the stationary point form closed curves so as $t \to \infty$ they neither approach the stationary point nor move away from it.
In general, in a nonlinear system a stationary point with imaginary eigenvalues could be asymptotically stable or unstable.  In this case, it turns out that the nonlinear system is also a centre, but you can't tell that just from the eigenvalues. 
As it happens, I have some on-line notes about this: Phase Portraits of Linear Systems, Phase Portraits of Nonlinear Systems, and 
Predator and Prey.
