Existence and exponential stability of positive almost periodic solution for Nicholson's blowflies models on time scales

TL;DR

This paper introduces new definitions of almost periodicity on time scales and proves the existence and exponential stability of positive almost periodic solutions for Nicholson's blowflies models, unifying continuous and discrete cases.

Contribution

It develops novel definitions of almost periodic functions on time scales and applies fixed point theory to establish solution stability for Nicholson's models.

Findings

Existence of positive almost periodic solutions under simple conditions

Exponential stability of these solutions

Equivalence of continuous and discrete models' dynamics

Abstract

In this paper, we first give a new definition of almost periodic time scales, two new definitions of almost periodic functions on time scales and investigate some basic properties of them. Then, as an application, by using the fixed point theorem in Banach space and the time scale calculus theory, we obtain some sufficient conditions for the existence and exponential stability of positive almost periodic solutions for a class of Nicholson's blowflies models on time scales. Finally, we present an illustrative example to show the effectiveness of obtained results. Our results show that under a simple condition the continuous-time Nicholson's blowflies models and their discrete-time analogue have the same dynamical behaviors.