The stability of two area-preserving biological models

This paper will focus on two area-preserving biological models: a host-parasite model$ \begin{align*} x_{n+1} & = ax_nf(y_n), \\ y_{n+1} & = bx_ny_nf(y_n), \end{align*} $where $ f $ is a smooth, monotonically increasing positive function, and a maternal-quality single species population model$ \begin{align*} x_{n+1} & = x_ng(y_{n+1}), \\ y_{n+1} & = y_nf(x_n) \end{align*} $with $ f $ being a smooth, monotonically increasing positive function and $ g $ a smooth, monotonically decreasing positive function.By computing the first twist coefficient, we will find sufficient conditions for non-resonant elliptic fixed points to be non-degeneracy, and thus utilize KAM theory to establish the stability of these fixed points.