We consider the discrete-time dynamics of a one-locus, two-allele (two-gene) model of a genetic population experiencing natural selection in a random environment. The dynamics is described by a stochastic recursion relation. We construct hierarchies of upper and lower bounds for the nth generation average gene frequency 〈Xn〉. The bounds provide very sharp estimates of 〈Xn〉 (n = 1,2,…) and may be interpreted in the spirit of mean-field and spin-cluster approximations. We also find a phase boundary in the space of viability parameters and environment probabilities. On one side of the boundary one of the two genes eventually becomes extinct (1 - 〈Xn〉å 0 for n å ∞); whereas, on the other side of the boundary both genes survive. For the special case where the environment is restricted to be a two-valued random variable, the average gene frequency 〈Xn〉 is, in principle, derivable from a partition function for an n-spin, Ising model.
