Bifurcations and hybrid control in a \(3 \times 3\) discrete-time predator-prey model
type: Item
instance of: scholarly article
title: Bifurcations and hybrid control in a \(3 \times 3\) discrete-time predator-prey model
zbMATH Open document ID: 1485.92093
author: A. Q. Khan
author: Azhar Zafar Kiyani
author: Imtiaz Ahmad
published in: Mathematical Biosciences and Engineering
publication date: 2021-09-03T00:00:00Z
review text: The authors are concerned with the analysis of a discrete model describing the dynamics of one predator and two prey populations suggested by \textit{M. R. Sagaya Raj} et al. [``Dynamical behavior in a three species discrete model of prey-predator interactions'', Int. J. Comput. Sci. Math. 5, 11--20 (2013)], \[ x_{n+1}=\left( 1+a\right) x_{n}-bx_{n}z_{n},\quad y_{n+1}=ry_{n}\left( 1-y_{n}\right) -cy_{n}z_{n},\quad z_{n+1}=\left( 1-d\right) z_{n} +ex_{n}z_{n}+fy_{n}z_{n}. \] In the cited paper, five equilibria of this system, \begin{align*} E_{0} & =\left( 0,0,0\right) ,\qquad E_{1}=\left( 0,\frac{r-1} {r},0\right) ,\qquad E_{2}=\left( 0,\frac{d}{f},\frac{rf-f-dr}{cf}\right) ,\\ E_{3} & =\left( \frac{d}{e},\frac{r-1}{r},0\right) ,\qquad E_{4}=\left( \frac{br\left( d-f\right) +f\left( b+ac\right) }{ber},\frac{br-b-ac} {br},\frac{a}{b}\right) , \end{align*} were found and the first three were classified with respect to their stability. Relevant details were provided and supported with numerical simulations. The authors of the paper under review argued that Sagaya Raj et al. ``cannot give complete local dynamical classifications to interested readers, which is still further consideration and improvements towards local dynamical properties along with topological classifications and bifurcation analysis for the model under consideration''. \ Complementing the results in the cited paper, they revisited stability properties of all five equilibria. Surprisingly, not all characterizations of the equilibria in Lemmas 4.1 and 4.2 agree with those reported by Sagaya Raj et al. Lemma 4.1 characterizes the equilibrium \(E_{0}\) similarly to Proposition 2 in the cited paper except for the claim that \(E_{0}\) cannot be a sink, contrary to the statement that \(E_{0}\) is a sink if \(r<1,\) \(-2<a<0,\) and \(0<d<2\) in the latter result. Likewise, Lemma 4.2 claims that \(E_{1}\) ``is never sink'', contrary to Proposition 3 proved by Sagaya Raj et al.\ Unfortunately, neither explanati…
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 39A13
Mathematics Subject Classification ID: 39A23
Mathematics Subject Classification ID: 39A28
Mathematics Subject Classification ID: 39A30
Mathematics Subject Classification ID: 39A60
zbMATH DE Number: 7390442
zbMATH Keywords: discrete model
zbMATH Keywords: predator-prey systems
zbMATH Keywords: equilibria
zbMATH Keywords: stability
zbMATH Keywords: bifurcations
zbMATH Keywords: hybrid control
Wikidata QID: Q104619351
reviewed by: Svitlana P. Rogovchenko
MaRDI profile type: Publication
full work available at URL: mbe.2020360
OpenAlex ID: W3092765753
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DOI: 10.3934/MBE.2020360
description: scientific article; zbMATH DE number 7390442