Analysis of a nonautonomous Nicholson Blowfly model

Most dynamic models describing population evolution contain one or more parameters. The parameters are treated as fixed constants and qualitative results, such as stability of equilibria, are calculated using this assumption. In reality, however, the parameters are mathematically evaluated by statistical methods in which the error is decreased over a number of calculations. Therefore, the parameter is a sequence converging to the actual parameter value as time goes to infinity. In this article we consider the kth-order discrete Nicholson Blowfly model, Nn+1=F(P,δ,Nn,…,Nn−k) where δ and P are parameters. For a particular range of parameter values, global stability results are well known. The general form of the discrete dynamical system is now rewritten as Nn+1=F(Pn,δn,Nn,…,Nn−k) where Pn and δn converge to the parametric values P and δ. We show that when the parameters are replaced by sequences, the stability results of the original system still hold. This technique may be of general interest to those studying evolutionary systems in which the parameters are not fundamental constants but sequences.