Let $n \geq 2$ be an integer. I am analyzing a set of differential equations as follows: \begin{align*} \dot{x}_1 &:=f_1(x_1,x_2) = x_1^{n} + n x_1^{n-1}(1-x_1-x_2) + n x_1(1-x_1-x_2)^{n-1}-x_1\\ \dot{x}_2 &:=f_2(x_1,x_2) = x_2^{n} - x_2, \end{align*} where $x_1,x_2 \in [0,1].$
It is clear that the restpoints of the above system are $(0,1)$ and $(1,0).$ I verfied that the Jacobian evaluated at $(0,1)$ and $(1,0)$ is given by, \begin{align*} J(0,1) &= \begin{pmatrix} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2}\\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} \end{pmatrix}_{\big|(x_1,x_2)=(0,1)}=\begin{pmatrix} -1 & 0\\ 0 & n-1 \end{pmatrix}\\ J(1,0) &= \begin{pmatrix} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2}\\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} \end{pmatrix}_{\big|(x_1,x_2)=(1,0)}=\begin{pmatrix} -1 & -n\\ 0 & -1 \end{pmatrix}. \end{align*}
With the Jacobian analysis, I can conclude that $(0,1)$ is unstable and $(1,0)$ is locally stable. Numeric analysis which I performed shows that $(1,0)$ is almost globally asymptotically stable (meaning that all the trajectories starting from an interior state converge to $(1,0)$). Could someone point me to a result which I can use to establish global stability of $(1,0)$ analytically? Thanks a lot in advance!