Dynamics of a Discrete Allelopathic Phytoplankton Model with Infinite Delays and Feedback Controls

Liang Zhao,Bin Qin,Fengde Chen

doi:10.1155/2020/7023075

Abstract

A discrete allelopathic phytoplankton model with infinite delays and feedback controls is studied in this paper. By applying the discrete comparison theorem, a set of sufficient conditions which guarantees the permanence of the system is obtained. Also, by constructing some suitable discrete Lyapunov functionals, some sufficient conditions for the extinction of the system are obtained. Our results extend and supplement some known results and show that the feedback controls and toxic substances play a crucial role on the permanence and extinction of the system.

Highlights

Given a bounded sequence of real numbers f(k), let fu and fl denote supk∈Zf(k) and infk∈Zf(k), respectively.Many real-world phenomena are studied through discrete mathematical models governed by difference equations which are more appropriate than the continuous ones when the populations have nonoverlapping generations; the study of the dynamic behaviors of discrete time models becomes the subject of intense research in mathematics biology, such topics as permanence and extinction, and existence of positive periodic solution, have been extensively studied by many scholars.Recently, some scholars believed that a more appropriate competition model should be considered with nonlinear interinhibition terms
Some scholars believed that a more appropriate competition model should be considered with nonlinear interinhibition terms
By applying the discrete comparison theorem, a set of sufficient conditions which guarantees the permanence of the system is obtained

Summary

Introduction

Given a bounded sequence of real numbers f(k), let fu and fl denote supk∈Zf(k) and infk∈Zf(k), respectively. Yue [3] investigated system (1) with the second species which could be toxic, while the other one is nontoxic He studied system (2) and gave the sufficient conditions of the extinction of one species and the global attractive of the other one. Zhao et al [8] further considered a discrete Lotka–Volterra competition system with infinite delays and single feedback control variable as follows:. We propose and study the following discrete competition system with infinite delays and feedback control variables:. We mention that this is the first time such kind of model be proposed and studied, and as far as system (5) is concerned, whether the feedback control variables and toxic substances have influence on the permanence and extinction of the system or not is an interesting problem.

Lemmas

Permanence

Extinction

M2 al2 exp

Discussion