I am trying to find out if it is possible to determine the steady state behavior of the predator-prey system defined by the nonlinear equations:
$$\begin{eqnarray} \frac{dx}{dt}&=&ax-bxy \\ \frac{dy}{dt}&=&cry-dy \end{eqnarray}$$
Lotka-Volterra predator-prey model eigenvalue explains that I can understand the stability of the two equilibrium points using the system's eignevalues. However, I wish to know if it is possible to determine whether the state will decay, oscillate, explode, etc. given any arbitrary initial condition, without numerical solving the equations or employing a phase diagram.