Variable structure control for a singular biological economic model with time delay and stage structure

Yi Zhang,Yudan Wei

doi:10.1186/s13662-017-1436-3

Abstract

A singular biological economic model which considers a prey-predator system with time delay and stage structure is proposed in this paper. The local stability at the equilibrium point and the dynamic behavior of the model are studied. Local stability analysis of the model without time delay reveals that there is a phenomenon of singularity-induced bifurcation due to the economic equilibrium. Furthermore, the phenomenon of Hopf bifurcation of the model at the boundary equilibrium point occurs as the time delay satisfies certain conditions. In order to apply variable structure control to eliminate the complex behaviors caused by singularity-induced bifurcation, the singular model is transformed into a single-input and single-output model with parameter varying within definite intervals. Then variable structure control with sliding mode based on a power reaching law is designed to stabilize the model. Numerical simulations are given to verify the effectiveness of the conclusions.

Highlights

1 Introduction The stage structure of the biological population is the whole life course of the biological population consisting of some non-overlapping stages [1]
We know there is a bifurcation at the positive equilibrium point for model (3), which is shown in the following theorem
Remark 5.1 When the variable structure control is used, the singular biological economic model is transformed into a linear system with parameters varying within finite time intervals

Summary

Introduction

The stage structure of the biological population is the whole life course of the biological population consisting of some non-overlapping stages [1]. The bifurcation analysis and control of a class of singularity biological economic models with stage structure were researched in [6]. A singular predator-prey bioeconomic model with time delay and stage structure is established and studied.

Results

Conclusion