Lotka Volterra predator prey system I am doing a project work mainly saying the relation between jacobian matrix and lotka volterra predator prey method , and I had a doubt,when I find eigenvalues of the system,I got purely imaginary values.what did it mean and how can I conclude my project
 A: If we use the Predator Prey Model:
$$x' = a x - \alpha x y \\ y' = - b y + \beta x y$$
where $a, b, \alpha, \beta$ are positive constants, we start by finding the critical points. The critical points are:
$$(0, 0), \left( \dfrac{b}{\beta}, \dfrac{a}{\alpha} \right)$$
The Jacobian matrix is given by:
$$J(x, y) = \begin{bmatrix} a - \alpha y & -\alpha x\\ \beta y & -b + \beta x\end{bmatrix}$$
At $(0,0)$, we have:
$$ \begin{bmatrix} a & 0\\ 0 & -b \end{bmatrix}$$
The eigenvalues are $a$ and $-b$, hence an unstable saddle point.
At $\left( \dfrac{b}{\beta}, \dfrac{a}{\alpha} \right)$, we have:
$$\begin{bmatrix} 0 & -\dfrac{\alpha b}{\beta} \\ \dfrac{a \beta}{\alpha} & 0 \end{bmatrix}$$
The eigenvalues are $\pm~ i~ \sqrt{ a b}$, which is a stable center.
A sample system and phase portrait showing this is:

To see both critical points (saddle and center), we have:

We would not care about starting with a population of $(0,0)$, the saddle, or negative populations, so we do not show that.
If we take any initial population of cheetahs and an antelopes, and follow the phase portrait, we see one species starting to dominate, but then the prey decreases, so the predators decrease and we are in a state of equilibrium (where one is not annihilating the other).
