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

Abstract

Consider the non-autonomous logistic model

where $\Delta x_n=x-n+1–x_n$, {$p_n$} is a sequence of positive real numbers, {$k_n$} is a sequence of non-negative integers such that {$n–k_n$} is non-decreasing, $\lambda \in [0,1]$, 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.