Consider the following system of first order nonlinear difference equation: $$ \begin{cases} x_{k+1} = x_k + \alpha(1-x_ky_k^2) \\ y_{k+1} = y_k + \alpha(1-x_k^2y_k) \\ \end{cases} $$ with a given initial condition $x_0>y_0$. How to show that if we fix a small enough constant $\alpha>0$, then $(x_k,y_k)_{k\in\mathbb{N}}$ is a bounded sequence?
Notice that numerical experiments show that the bound of this sequence may get larger and larger if we take $\alpha$ smaller and smaller, but as long as $\alpha$ is a small enough constant, neither $|x_k|$ nor $|y_k|$ will go to infinity as $k\to\infty$.
I don't know what is the standard approach to analyze such kind of nonlinear difference equation without Lipschitz continuous gradient. I would appreciate it if one can provide some possible approaches that I can have a try.