Dynamic behaviors for a delay Lasota\u2013Wazewska model with feedback control on time scales

Xiaoying Chen,Danhong Wang,Chunling Shi

doi:10.1186/s13662-019-2483-8

Xiaoying Chen, Danhong Wang + Show 1 more

Open Access

https://doi.org/10.1186/s13662-019-2483-8

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Abstract

In this paper, a delay Lasota–Wazewska system with feedback control on time scales is proposed. Firstly, by using some differential inequalities on time scales, sufficient conditions which ensure the permanence of the system are obtained. Secondly, by means of the fixed point theory and the exponential dichotomy of linear dynamic equations on time scales, some sufficient conditions for the existence of unique almost periodic solution are obtained. Moreover, exponential stability of the almost periodic positive solution is investigated by applying the Gronwall inequality. Finally, numeric simulations are carried out to show the feasibility of the main results.

Highlights

Today the dynamical systems are classified into some types, among which discrete systems and continuous systems are two important types
We aim to investigate the permanence of the Lasota–Wazewska timescale model with multiple time-varying delays and feedback control
Using the exponential dichotomy of linear dynamic equations on time scales and a fixed point theorem on time scales, we study the existence and uniqueness of almost periodic solutions for the model

Summary

Introduction

Today the dynamical systems are classified into some types, among which discrete systems and continuous systems are two important types. Motivated by the aforementioned discussion, in the present paper, we first propose an almost periodic Lasota–Wazewska model with feedback control and multiple time-varying delays on time scales:. Using the exponential dichotomy of linear dynamic equations on time scales and a fixed point theorem on time scales, we study the existence and uniqueness of almost periodic solutions for the model. Lemma 2.3 ([5]) If the linear system (2.1) admits an exponential dichotomy, the almost periodic system (2.2) has an almost periodic solution x(t) as follows: t. Set. Theorem 3.1 Let (x(t), u(t))T be any positive solution of system (1.2) with initial condition (3.1). Theorem 4.2 Assume that (H1) and (H2) holds, the unique positive almost periodic solution in the region X∗ is exponentially stable. Proof By Theorem 4.1, system (1.2) has a positive almost periodic solution w∗(t) = (x∗(t), u∗(t))T in the region X∗.

Numerical simulation Example 1 Consider the following system on time scales:

Discussion