Behavior of solutions of a discrete population model with mutualistic interaction

We focus on the stability analysis of two types of discrete dynamic models: a discrete dynamic equation and a discrete dynamics system consisting of two equations with mutualistic interaction given by x n + 1 = a + b x n λ − ( x n − 1 + x n − k ) c + x n − 1 + x n − k {x}_{n+1}=a+\frac{b{x}_{n}{\lambda }^{-\left({x}_{n-1}+{x}_{n-k})}}{c+{x}_{n-1}+{x}_{n-k}} and x n + 1 = a 1 + b 1 y n λ − ( y n − 1 + x n − k ) c 1 + y n − 1<