Equilibrium point

det(J)

tr(J)

Results

$O\left(0,0\right)$

det(J) > 0

tr(J) < 0

Stable

$A\left(0,1\right)$

det(J) > 0

tr(J) > 0

Unstable

$B\left(1,1\right)$

det(J) > 0

tr(J) < 0

Stable

$C\left(1,0\right)$

det(J) > 0

tr(J) > 0

Unstable

$D\left(y^{*},x^{*}\right)$

det(J) > 0

tr(J) = 0

Saddle point