<s>[INST] Stability of steady states by eigenvalue approach I am taking a systems biology course but as biologist has been complicated to understand the math behind the systems. I am not sure if this questions would be suitable for this forum but any light or direction to interpret the below question would be very appreciated.
The differential equations below describe a positive feed forward loop. From the biological point of view this could represent gene interactions in a development tissue:
\begin{align*}
\dfrac{dx}{dt}
=b \dfrac{(\dfrac{y}{K})^n}{1+(\dfrac{y}{K})^n}-ax
=b\dfrac{(y)^n}{K^n+y^n}-ay
\\~\\
\dfrac{dy}{dt}
=b \dfrac{(\dfrac{x}{K})^n}{1+(\dfrac{x}{K})^n}-ax
=b\dfrac{(x)^n}{K^n+x^n}-ax
\end{align*}
If we take as parameter values: a = 1, K = 1, b = 2, and n = 2
\begin{align*}
f(x,y) = 2\dfrac{y^2}{1+y^2}-x
\\~\\
g(x,y) = 2\dfrac{x^2}{1+x^2}-y
\end{align*}
I know that [0,0]=[x,y] is a steady state in the system. In the next step we then need to check stability. From this point things become a little cloudy to me. I do know that we can use the eigenvalue approach to achieve that. Thus, the first step is to construct the Jacob matrix, which in this example should read (following my course handout):
\begin{align*}
J(x,y) =
\begin{bmatrix}
-1 & \dfrac{4y}{(1+y^2)^2} \\
\dfrac{4x}{(1+x^2)^2} & -1
\end{bmatrix}
\end{align*}
At this point given my incapacity to understand how we ended with this jacobian matrix, I used a function in Matlab to produce a jacobian matrix and surprisingly the result was different:
\begin{align*}
J(x,y) =
\begin{bmatrix}
 -1 & \dfrac{(4y)}{(y^2 + 1)}-\dfrac{(4y^3)}{(y^2 + 1)^2}\\
 \dfrac{(4x)}{(x^2 + 1)}-\dfrac{(4x^3)}{(x^2 + 1)^2} & -1
\end{bmatrix}
\end{align*}
Well, as you can imagine the cloud in my head is even more opaque at this stage. 
Let's believe in the handout, then we have:
\begin{align*}
J(0,0) =
\begin{bmatrix}
 -1 & 0\\
 0 & -1
\end{bmatrix}
\end{align*}
The eigenvalues follow:
\begin{align*}
det
\begin{bmatrix}
 -1-\lambda & 0\\
 0 & -1-\lambda
\end{bmatrix}
=(1+\lambda)^2=0
\end{align*}
The conclusion is that at [x,y]=[0,0] 
\begin{align*}
\lambda1 = \lambda2 =-1
\end{align*}
Then is negative then is stable. I understand negative values here indicate stability, however, the requested mathematical concepts to reach the final result (or even the jacob matrix) really don't ring a bell for a biologist. I would be very grateful if someone could explain (1) what is possibly wrong in the jacob matrix (2) basic math concepts implemented in each stage here to get the final result. 
 [/INST] Note that:
$$
\dfrac{(4y)}{(y^2 + 1)}-\dfrac{(4y^3)}{(y^2 + 1)^2}=\dfrac{4y\left[(y^2+1)-y^2 \right]}{(y^2+1)^2}=\dfrac{4y}{(y^2+1)^2}
$$
and the Jacobian matrix is:
$$
J(x,y) =
\begin{bmatrix}
-1 & \dfrac{4y}{(1+y^2)^2} \\
\dfrac{4x}{(1+x^2)^2} & -1
\end{bmatrix}
$$
( in the second row of OP there is a typo that change $x$ to $y$).
So $J(0,0)$ is the diagonal matrix with eigenvalues $-1$ as in OP and  also the other results are correct.
</s>