Global Attractivity in a Genotype Selection Model

Abstract

We obtain a sufficient condition for global attractivity to occur in the delay difference equation x n+1 = x n exp(β n (1 - x n-τ )/(1 + x n-τ )). This leads to the fact that when 0 < β < 3/(τ + 1), the positive equilibrium 1/2 of the genotype selection model y n+1 = y n e β(1-2y n-τ ) / 1 - y n + y n e β(1-2y n-τ ) is a global attractor for all solutions originated from positive initial conditions. Our result matches the computational result 0 < β < 4cos(τπ/(2τ + 1)) suggested in [3].