Analysis of a nonautonomous Nicholson Blowfly model

doi:10.1016/S0378-4371(99)00355-6

Physica A: Statistical Mechanics and its Applications

Volume 273, Issues 1–2, 1 November 1999, Pages 198-211

https://doi.org/10.1016/S0378-4371(99)00355-6 Get rights and content

Abstract

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, N_n+1=F(P,δ,N_n,…,N_n−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 N_n+1=F(P_n,δ_n,N_n,…,N_n−k) where P_n and δ_n 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.

Section snippets

Introduction and preliminaries

Many dynamical systems that model biological phenomena contain several parameters. Biologists are tasked to determine the exact parameter values in order to use the model for prediction purposes. Unfortunately, in the real world, parameters are not fixed constants. Typically, the parameters are estimated using statistical methods and at each stage in time the estimate will be improved. Therefore, the parameters in a model are actually sequences that converge to a constant parameter value as

Numerical results

We ran many numerical experiments for Eq. (2) where {P_n} and {δ_n} were chosen to be monotone, or oscillating sequences. Our numerical data visually represent the global asymptotic results in the next section. The only conditions we applied to the sequential parameters are that $lim_{n→∞} δ_{n} =δ>0$ and $lim_{n→∞} P_{n} =P>0$ . The sequence terms were restricted to δ_n,P_n∈[0,∞). Fig. 1shows that if the limiting values of the sequential parameters satisfy, δ<P and the inequality $((1−δ)^{−(k+1)} −1) ln P δ ⩽1$ then the

Analytical results

We will now prove several properties of the nonautonomous Nicholson Blowfly model where the parameters satisfy the conditions, $δ>0,P>0 and a>0,$ and the varying constants satisfy δ_n⩾0 and P_n⩾0 for all n⩾0. These conditions imply that although δ and P must be strictly positive, a finite number of zero terms in the sequences are allowed. Under these parameter conditions, Eq. (2) yields bounded and positive solutions as stated in the following proposition.

Proposition 1

Consider system (2) with $lim_{n→∞} P_{n} =P$ and $lim_{n→∞} δ_{n}$

Acknowledgements

The authors would like to thank Gerasimos Ladas for helpful discussions on this problem. We would also like to thank Deborah Bennett for proofreading our work.

References (4)

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On the stability and uniform persistence of a discrete model of Nicholson blowflies
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Nature
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There are more references available in the full text version of this article.

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