Prey-Predators systems 
Orbits evolve around the equilibrium point
The idea conveyed by the vector field that the orbits go around the equilibrium point is true. To prove it we consider one of these orbits to start, for example, in the region I, that is, \(x(0)>\frac{C}{D}\) and \(y(0)>\frac{A}{B}\). Let us start by verifying that this orbit enters region II.
Let us analyse the first equation of the system, \(x^{'}=Ax-Bxy=(A-By)x\); as \(x\) stays positive it follows that \[\frac{1}{x}x^{'}=A-By\]
Integrating between \(0\) and \(t\), we obtain \(\ln x(t)-\ln x(0)=\intop_{0}^{t}A-By(s)ds.\)
Now, in the region \(I\), \(y\) is increasing, then, \(y(s)>y(0)\) while the orbit holds in the region \(I\). And, therefore, we have \(A-By(s)<A-By(0)\). As \(A-By(0)\) is negative constant, because \(y(0)>\frac{A}{B}\), say \(A-By(0)=-k\) (with \(k>0\)), then \[\begin{array}{ccccccc} \ln x(t)-\ln x(0)<\intop_{0}^{t}-kds & & \ln\frac{x(t)}{x(0)}<-kt \\ \frac{x(t)}{x(0)}<e^{-kt} & & x(t)<x(0)e^{-kt}\end{array}\]
Thus, for every instant \(t\) in which the orbit stays in the region \(I\), we have: \[\frac{C}{D}<x(t)<x(0)e^{-kt}\]
In this way, we can guarantee that the time interval in which the orbit remains in \(I\) is limited, say \([0,T_{MAX}]\) with \(T_{MAX}\) finite (less than \(T^{*}\) as in the figure), because the strictly decreasing function \(x(0)e^{-kt}\) tends to \(0\) when \(t\) take sufficiently large values, which guarantees the existence of \(T^{*}\).
Analogously, starting from the second equation of the system we obtain \[\frac{A}{B}<y(t)<y(0)e^{-k't}\] where \(k^{'}=-C+Dx(0);\) and, since \(t<T_{MAX},y(t)<y(0)e^{-k'T_{MAX}}.\)
Putting together the information about both functions, we can conclude that, while the orbit remains in the region \(I\), it remains in the rectangle \[[\frac{C}{D},x(0)]\times[\frac{A}{B},y(0)e^{-k'T_{MAX}}]\]
In this way, the time interval for which (\(x(t),y(t))\) is defined is \([0,+\infty[\) and therefore the orbit continues beyond the value \(T_{MAX}\), thus not ending in the region \(I\). Then, by continuity and taking into account the monotony of the functions \(x\) and \(y\), extension of the orbit enters in \(II\).
The procedure for the other regions is analogous. Conclusion: the orbits close to \((\frac{C}{D},\frac{A}{B})\) evolve around the equilibrium point.
