Consider the following discrete system \begin{align*} x_{t+1} &= (1 + \alpha(1-x_t)) \cdot (1-\alpha y_t) \cdot x_t,\\ y_{t+1} &= (1 + \alpha(1-y_t)) \cdot (1-\alpha x_t) \cdot y_t, \end{align*} with initial values $0 < y_0 < x_0 < 1$ and $\alpha \in (0, 1)$. Based on simulations I believe that there is convergence to the fixed point $\lim\limits_{t \to \infty}~(x_t, y_t) = (1, 0)$ and similarly if $x_0 < y_0$ then $\lim\limits_{t \to \infty}~(x_t, y_t) = (0, 1)$, but I am having trouble proving this and would appreciate any help.
I believe the Jacobian of the map $F : \mathbb{R}^2 \to \mathbb{R}^2$ such that $(x_{t+1}, y_{t+1}) = F(x_t, y_t)$ is given by $$ J(x,y) = \begin{pmatrix} (1 + \alpha(1-2x))(1 - \alpha y) & -\alpha x(1 + \alpha(1-x)) \\ -\alpha y(1 + \alpha(1-y)) &(1 + \alpha(1-2y))(1 - \alpha x) \end{pmatrix} $$ but $J(x,y)$ does not seem to have operator norm less than $1$ on $[0, 1]\times[0,1]$ so I am not sure if I can show contraction. Are there any other techniques for understanding the behavior of systems of this kind?