Permanence and Almost Periodic Solution for an Enterprise Cluster Model Based on Ecology Theory with Feedback Controls on Time Scales

Yuanhong Zhi,Zunling Ding,Yongkun Li

doi:10.1155/2013/639138

Abstract

We present a model with feedback controls based on ecology theory, which effectively describes the competition and cooperation of enterprise cluster in real economic environments. Applying the comparison theorem of dynamic equations on time scales and constructing a suitable Lyapunov functional, sufficient conditions which guarantee the permanence and the existence of uniformly asymptotically stable almost periodic solution of the system are obtained.

Highlights

A few researchers have presented some models about enterprise clusters based on ecology theory, which arouse growing interest in applying the methods of ecology and dynamic system theory to study enterprise clusters, for example [1,2,3,4,5,6,7,8,9] and references cited therein
In [3], the developing strategy of enterprise clusters was analyzed based on the logistic model, and the suggestions of constructing cooperative relation and choosing generalization or specialization tactics for commodity were put forward
Based on the theoretical model of ecological population science, Wang and Pan [6] made a detailed analysis to the equilibrium mechanism of enterprise clusters, including net model and center halfback model and drew a conclusion that the relationship of pierce competition and beneficial cooperation among enterprise clusters was the crucial factor for them to keep stability

Summary

Introduction

A few researchers have presented some models about enterprise clusters based on ecology theory, which arouse growing interest in applying the methods of ecology and dynamic system theory to study enterprise clusters, for example [1,2,3,4,5,6,7,8,9] and references cited therein. It is necessary to study models with control variables which are so-called disturbance functions [14,15,16,17] As well known, both continuous and discrete systems are very important in implementation and applications, but it is troublesome to study the permanence and the existence of almost periodic solutions for continuous system and discrete system, respectively. There have been extensive results on existence of almost periodic solutions of differential equations in the literature. Our main purpose of this paper is by using the comparison theorem of dynamic equations on time scales and constructing a suitable Lyapunov functional to study the permanence and the existence of almost periodic solutions of (3). Where R+ is the set of positively regressive functions from T to R

Preliminaries

Persistence

Existence of Uniformly Asymptotically Stable Almost Periodic Solution

An Example