Existence and asymptotic behavior of positive solutions of a non-autonomous food-limited model with unbounded delay (Q1811437)

A discrete generalized `food-limited' population model \[ \Delta x_n=p_nx_n \left(\frac{1-x_{n-k_n}} {1+\lambda x_{n-k_n}}\right)^r,\, n\geq 0, \] is investigated. Here, \(\Delta x_n=x_{n+1}-x_n\), \(\{p_n\}\) is a sequence of positive real numbers, \(r\) is the ratio of two odd integers, \(\lambda\in [0,1]\), and \(\{k_n\}\) is a sequence of nonnegative integers such that \(\{n-k_n\}\) is nondecreasing. The authors obtain sufficient conditions to ensure that all solutions are positive, and find a uniform bound. Then they prove a result on the convergence of all positive solutions to the positive equilibrium \(\{x_n\}\equiv 1\).