Bifurcation analysis of an ecological model with nonlinear state-dependent feedback control by Poincaré map defined in phase set
type: Item
instance of: scholarly article
title: Bifurcation analysis of an ecological model with nonlinear state-dependent feedback control by Poincaré map defined in phase set
zbMATH Open document ID: 1496.34088
author: Qianqian Zhang
author: Sanyi Tang
published in: Communications in Nonlinear Science and Numerical Simulation
publication date: 2022-02-16T00:00:00Z
review text: In this paper, a predator-prey model with state-dependent pulse interventions, including spraying insecticides and releasing natural enemies, in which the implementation of control measures depends on whether the weighted value of pest density and its change rate reaches the action threshold is proposed to develop qualitative analysis techniques. First, the threshold condition for the existence and stability of the boundary periodic solutions is discussed and then define a one-parameter family of discrete (Poincaré) maps. By analyzing the properties of those discrete maps, it is concluded that if there is no interior equilibrium for the system without control measures and the releasing amount of natural enemies is greater than zero, then there is at least one positive periodic solution. While if the system without control measures has an internal equilibrium and only the chemical tactic is applied, then there could be an unstable positive periodic solution near the boundary periodic solution. As a result, backward bifurcation and bi-stability occur. The analytical techniques developed here could be applied to analyze more generalized models and other fields, including infectious disease control. This paper aims to investigate the periodic solutions and the bifurcations associated with the control strategy for the proposed ecological model (\textit{Lotka-Volterra predator-prey model with state-dependent pulse and the signal that triggers pest control action is a convex combination of size and growth speed of the pest population}) (1.2). In Section 2, the relevant definitions and notations of planar impulsive semi-dynamic system and the dynamical behavior of the ODE model, Model (1.2) without control measures, are summarized. In Section 3, the dynamics of impulsive Model (1.2) are discussed when the ODE model has no positive equilibrium, including the exact ranges of impulsive set and phase set and the existence and stability of the boundary periodic solution. In Sect…
reviewed by: A. Özbekler
Mathematics Subject Classification ID: 34C60
Mathematics Subject Classification ID: 92D40
Mathematics Subject Classification ID: 92D45
Mathematics Subject Classification ID: 34A37
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 93B52
zbMATH DE Number: 7474623
zbMATH Keywords: state-dependent feedback control
zbMATH Keywords: non-linear impulsive set
zbMATH Keywords: action threshold
zbMATH Keywords: Poincaré map
zbMATH Keywords: transcritical bifurcation
MaRDI profile type: Publication
full work available at URL: j.cnsns.2021.106212
OpenAlex ID: W4200178145
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DOI: 10.1016/J.CNSNS.2021.106212
description: scientific article; zbMATH DE number 7474623