Almost periodic solutions for a delayed Nicholson’s blowflies model with a nonlinear density-dependent mortality term

Bingwen Liu

doi:10.1186/1687-1847-2014-72

Abstract

A delayed Nicholson’s blowflies model with a nonlinear density-dependent mortality term is studied in this paper. Some sufficient conditions are obtained to guarantee the existence and global exponential stability of positive almost periodic solutions of this model. An example with numerical simulations is given to illustrate our main results.

Highlights

1 Introduction In population dynamics, the classic Nicholson’s blowflies equation developed by Gurney et al [ ] takes the following form: x (t) = Px(t – τ )e–αx(t–τ) – γ x(t), where x(t) denotes the population of sexually mature adults at time t, P is the maximum possible per capita egg production rate, /α is the population size at which the whole population reproduces at its maximum rate, τ is the generation time, and the mortality rate γ is assumed to be a constant
A straightforward extension assumes that the mortality rate is density-dependent, for instance, Berezansky et al [ ] proposed the following Nicholson’s blowflies equation: x (t) = –D x(t) + Px(t – τ )e–x(t–τ), ( . )
Motivated by the above discussions, in this paper, we employ a novel method to establish some criteria on the existence and global exponential stability of almost periodic solutions for the nonlinear density-dependent mortality Nicholson’s blowflies model given by m x (t) = –a(t) + b(t)e–x(t) + βj(t)x t – τj(t) e–γj(t)x(t–τj(t)), j=

Summary

Introduction

The classic Nicholson’s blowflies equation developed by Gurney et al [ ] takes the following form:. The effects of almost periodic environment on evolutionary theory have been the object of intensive analysis by numerous authors, and some of these results on Nicholson’s blowflies model without nonlinear density-dependent mortality term can be found in [ – ] These results were obtained by using exponential dichotomy theory on almost periodic differential equations or functional differential equations with linear part. ). Motivated by the above discussions, in this paper, we employ a novel method to establish some criteria on the existence and global exponential stability of almost periodic solutions for the nonlinear density-dependent mortality Nicholson’s blowflies model given by m x (t) = –a(t) + b(t)e–x(t) + βj(t)x t – τj(t) e–γj(t)x(t–τj(t)), j=.

It will be assumed that r max

Since the function

We claim that