A global attractivity in a genotype selection model (Q1430444)

The authors offer a sufficient condition for global attractivity of the delay difference equation \[ x_{n+1}=x_n\exp(\beta_n (1-x_{n-\tau})/(1+x_{n-\tau})). \] This leads to the fact that when \(0<\beta\leq 3/(\tau+1)\), the positive equilibrium 1/2 of the genotype selection model \[ y_{n+1}=\frac{y_n\exp(\beta(1-2y_{n-\tau})}{1-y_n+y_n\exp(\beta(1-2y_{n-\tau}))} \] is a global attractor for all solutions originated from positive initial conditions, which matches the computational result \(0<\beta\leq 4\cos (\tau\pi/(2\tau+1))\) suggested by \textit{V. L. Kocic} and \textit{G. Ladas} [Global behavior of nonlinear difference equations of higher order with applications (1993; Zbl 0787.39001)].