Global Stability for a Discrete Space-Time Lotka–Volterra System with Feedback Control

Li Xu,Ruiwen Han

doi:10.1155/2020/2960503

Abstract

In this paper, a discrete space-time Lotka–Volterra model with the periodic boundary conditions and feedback control is proposed. By means of a discrete version of comparison theorem, the boundedness of the nonnegative solution of the system is proved. By the combination of the Volterra-type and quadratic Lyapunov functions, the global asymptomatic stability of the unique positive equilibrium is investigated. Finally, numerical simulations are presented to verify the effectiveness of the main results.

Highlights

For population dynamical systems with feedback controls, an important and interesting subject is to study the effects of feedback controls to the persistence, permanence, and extinction of species, the stability, and dynamical complexity of systems [7]. ere are lots of important and interesting results on stability research for continuous time population dynamical models [8,9,10,11,12,13,14,15,16]
Complexity models for numerical simulations. erefore, it is reasonable to study discrete-time models governed by difference equations, and there has been some work done on the study of the persistence, permanence, and global stability for various discrete-time nonlinear population systems with feedback when the effect of spatial factors is not considered [5, 7, 17,18,19]
The main purpose of this paper is to study the global asymptomatic stability of an one-dimensional spatially discrete reaction diffusion Lotka–Volterra model with the periodic boundary conditions and feedback control

Summary

Introduction

It is well known that the ecosystem in the real world is often distributed by unpredictable forces or interference factors, such as natural disturbances (floods, fires, disease outbreaks, and droughts), human-caused interference factors (oil spills), and slowly changing long-term stresses (nutrient enrichment), which may result into changes in the biological parameters such as survival rates [1,2,3]. e presence of the unpredictable forces or interference factors in an ecological system raises the following essential and basic question from the practical interest in ecology: “Can the ecosystem withstand those unpredictable forces which persist for a finite period of time?” e question has motivated the development of some control mechanisms for managing populations to ensure that the interacting species can coexist, such as impulsive control, optimal vibration control, intermittent control, and feedback control. [4,5,6]. Erefore, it is reasonable to study discrete-time models governed by difference equations, and there has been some work done on the study of the persistence, permanence, and global stability for various discrete-time nonlinear population systems with feedback when the effect of spatial factors is not considered [5, 7, 17,18,19]. In reference [19], some sufficient conditions on the permanence and the global stability of the system of a n-species Lotka–Volterra discrete system with delays and feedback control by constructing the suitable discrete type Lyapunov functionals are obtained. The main purpose of this paper is to study the global asymptomatic stability of an one-dimensional spatially discrete reaction diffusion Lotka–Volterra model with the periodic boundary conditions and feedback control.

Model and Preliminaries

Global Stability

Example and Numerical Simulations