Positive periodic solution of a discrete Lotka-Volterra commensal symbiosis model

Sufficient conditions are obtained for the existence of positive periodic solution of the following discrete Lotka-Volterra commensal symbiosis model arrayrcl x₁(k+1)&=& x₁(k)exp\ a₁(k)-b₁(k)x₁(k)+c₁(k)x₂(k)\, x₂(k+1)&=& x₂(k)exp\ a₂(k)-b₂(k)x₂(k)\, array where ᵢ(k)\, i=1, 2, ₁(k)\ are all positive ω-periodic sequences, ω is a fixed positive integer, ᵢ(k)\ are ω-periodic sequences, which satisfies āᵢ=1/ω∑ₖ₌₀ω-1aᵢ(k)>0, i=1,2.