Prey-Predators systems 
Stability of equilibrium points
The importance of the equilibrium points lies in their influence on the neighbouring curves, that is, on the type of stability they exhibit: for initial conditions close to the equilibrium point, do the solution curves remain globally close to the equilibrium point? This information can be obtained from the eigenvalues of the matrix of the partial derivatives of the function \(\left(x,y\right)\rightarrow\left(Ax-Bxy,-Cy+Dxy\right)\) in the point: \[M_{\left(x,y\right)}=\left(\begin{array}{cc} A-By & -Bx\\ Dy & -C+Dx \end{array}\right)\]
The eigenvalues of the diagonalised matrix \(M_{\left(0,0\right)}=\left(\begin{array}{cc} A & 0\\ 0 & -C \end{array}\right)\) are \(A\) e \(C\). Since \(A>0\), we know that \((0,0)\) is unstable, which means that there are initial conditions close to \((0,0)\) whose solution curves move away from \((0,0)\). Since \(-C<0\), there exist initial conditions close to \((0,0)\) whose solution curves approach asymptotically to this equilibrium point - which characterises it as a saddle point.
The eigenvalues \(\lambda\) of \[M_{\left(\frac{C}{D},\frac{A}{B}\right)}=\left(\begin{array}{cc} 0 & -\frac{BC}{D}\\ \frac{AD}{B} & 0 \end{array}\right)\] cancel out the determinant of the matrix \(M_{\left(\frac{C}{D},\frac{A}{B}\right)}-\lambda Identity\), that is, they are zeros of the polynomial equation \[det\left(\begin{array}{cc} -\lambda & -\frac{BC}{D}\\ \frac{AD}{B} & -\lambda \end{array}\right)=0,\] then, \(\lambda^{2}+\frac{ABCD}{BD}=0\) , that is, \(\lambda^{2}=-AC\), which has complex solutions \(\lambda=\pm i\sqrt{AC}.\)
Since both have null real part, the results do not allow us to conclude anything regarding the solutions of initial conditions close to \(\left(\frac{C}{D},\frac{A}{B}\right)\).
