The Dynamics of a Predator‐Prey System with State‐Dependent Feedback Control

Hunki Baek

doi:10.1155/2012/101386

Abstract

A Lotka‐Volterra‐type predator‐prey system with state‐dependent feedback control is investigated in both theoretical and numerical ways. Using the Poincaré map and the analogue of the Poincaré criterion, the sufficient conditions for the existence and stability of semitrivial periodic solutions and positive periodic solutions are obtained. In addition, we show that there is no positive periodic solution with period greater than and equal to three under some conditions. The qualitative analysis shows that the positive period‐one solution bifurcates from the semitrivial solution through a fold bifurcation. Numerical simulations to substantiate our theoretical results are provided. Also, the bifurcation diagrams of solutions are illustrated by using the Poincaré map, and it is shown that the chaotic solutions take place via a cascade of period‐doubling bifurcations.

Highlights

In the last decades, some impulsive systems have been studied in population dynamics such as impulsive birth 1, 2, impulsive vaccination 3, 4, and chemotherapeutic treatment of disease 5, 6
Many considerable investigators have studied the dynamic behaviors of system 1.1 without the state feedback control. cf. 23, 24
Under the condition r 0, we show that the semitrivial periodic solution of system 1.1 is stable when −1 < q < q0 and there exists a positive period-one solution of system 1.1

Summary

Introduction

Some impulsive systems have been studied in population dynamics such as impulsive birth 1, 2 , impulsive vaccination 3, 4 , and chemotherapeutic treatment of disease 5, 6. The majority of these studies only consider impulsive control at fixed time intervals to eradiate the prey pest population Such control measure of prey pest management is called fixed-time control strategy, modeled by impulsive differential equations. Abstract and Applied Analysis have started paying attention to another control measure based on the state feedback control strategy, which is taken only when the amount of the monitored prey pest population reaches a threshold value 2, 16–19. In order to investigate the dynamic behaviors of a population model with the state feedback control strategy, an autonomous Lotka-Volterra system, which is one of the most basic and important models, is considered.

Preliminaries

The Existence and Stability of a Periodic Solution When r 0

Numerical Examples

Conclusion