Existence and Asymptotic Behavior of Positive Solutions of a Non-Autonomous Food-Limited Model with Unbounded Delay

Consider the non-autonomous logistic model ∆xn = pnxn 1− xn−kn 1 + λxn−kn r (n ≥ 0) where ∆xn = xn+1 − xn, {pn} is a sequence of positive real numbers, {kn} is a sequence of non-negative integers such that {n − kn} is non-decreasing, λ ∈ [0, 1], and r is the ratio of two odd integers. We obtain new sufficient conditions for the attractivity of the equilibrium x = 1 of the model and conditions that guarantee the solution to be positive, which improve and generalize some recent results established by Phios and by Zhou and Zhang.