Global stability in a nonautonomous genotype selection model

Consider the nonautonomous difference equation \[ y_{n + 1} = \frac { y_n \exp \left ( \beta _n \left ( 1 - \sum \nolimits _{i = 0}^k \alpha _i y_{n - i} \right ) \right )} {1 - y_n + y_n \exp \left ( \beta _n \left ( 1 - \sum \nolimits _{i = 0}^k \alpha _i y_{n - i} \right ) \right )}, n = 0, 1,..., \qquad \left ( 0.1 \right )\] where $k$ is a nonnegative integer, $\alpha _0, \alpha _1, \dots , \alpha _{k - 1}$ are nonnegative constants, $\alpha _k$ is a positive constant and $\left \{ \beta _n \right \}$ is a nonnegative sequence, which is used as a genotype selection model. In this paper, we first establish some criteria for the positive equilibrium of Eq. (0.1) to be globally asymptotically stable. Then some special cases of Eq. (0.1) are investigated further and more global stability results are obtained. Our results also extend and improve some known results in the literature.