The following modell describes an eco-system where $X$ is the number of rabbits and $Y$ is the number of vicious wolves. $r>0$ is the growth rate of the rabbits, and similarly $s>0$ is the growth rate of the wolves. $K>0$ and $L>0$ is the maximum population of respectively wolves and rabbits in the absence of preditors / prey.
\begin{align} \dot{X} & = \frac{r}{K}X^2-rX+\alpha X Y = 0 \tag{1} \\ \dot{Y} & = sY-\frac{s}{L}Y^2+\beta X Y = 0 \tag{2} \end{align} a) Use the Jacobian-matrix to determine analytically the dynamic of the system, find the equilibrium, and draw the state space of the system.
I tried solving the problem above, but ran into some difficulties. Stationary points:
\begin{align*} (0,0), \ (0,L), \ (K,0), \ \left(\frac{K s (r - L \alpha)}{Z}, \frac{L r (K \beta + s)}{Z}\right) \end{align*}
Where $Z:= K L \alpha\beta + r s$. Now I need to check if they are max, min or saddlepoints. The Jacobi matrix can be written as
\begin{align*} J(X,Y) = \begin{bmatrix} \partial \dot{X} / \partial X & \partial \dot{X}/ \partial Y \\ \partial \dot{Y} / \partial X & \partial \dot{Y}/ \partial Y \end{bmatrix} = \begin{bmatrix} \cfrac{2 r}{K} \, X - r + \alpha Y & \alpha X \\ \beta Y & s - \cfrac{2 s}{L} \, Y + \beta X \end{bmatrix} \end{align*} Hence the second order derivative test gives
\begin{align*} \bigl|J(0, L)\bigr| & = \begin{vmatrix} \alpha L - r & \phantom{-} 0 \\ \beta L \phantom{- r.} & -s \end{vmatrix} = s(r-\alpha L) \end{align*} and \begin{align} \left|J\left(\frac{K s (r - L \alpha)}{Z}, \frac{L r (K \beta + s)}{Z}\right)\right| & = \frac{1}{Z^2} \begin{vmatrix} r s (r-L\alpha) & \alpha s K (r - \alpha L) \\ \beta r L (K\beta+s) & -rs(\beta K + s) \end{vmatrix} = \frac{r s (\alpha L - r) (\beta K + s)}{Z} \end{align} I did not bother writing down the other ones as they were pretty straight forward to figure out. Now I have to look at the cases $\alpha L > r$, $\alpha L < r$ and $\alpha L = r$.
In the case $r < \alpha L$ then $(0,L)$ and $(K s (r - L \alpha)/Z, L r (K \beta + s)/Z)$ are saddle points. If $r > \alpha L$ then both are maxima, because $\partial\dot{X}/\partial X$ is positive for both points.
My question is: How do I figure out what kind of points I have when $r = \alpha L$?