Global Attractivity in a Genotype Selection Model

where k∈N and {βn} is a sequence of positive real numbers. When k= 0 and βn ≡ β for alln∈N, (1.1) was introduced by May [2, pages 513–560] as an example of a map generated by a simple model for frequency-dependent natural selection. The local stability of the equilibrium ȳ = 1/2 of (1.1) was investigated by May [2]. In [1] (see also [3]), Grove further investigated the stability of the equilibrium ȳ = 1/2 of (1.1) and proved that when βn ≡ β, the equilibrium ȳ = 1/2 of (1.1) is locally asymptotically stable if 0 < β < 4cos(kπ/(2k+1)) and is unstable if 0 < β < 4cos(kπ/(2k+1)). Furthermore, if