Bifurcations and hybrid control in a \(3 \times 3\) discrete-time predator-prey model
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 explanations nor the proofs are provided for both lemmas. Lemmas 4.3 - 4.6 complement the classification in the cited paper, still without proofs. Proposition 4 characterizes \(E_{2}\) as a sink, source or saddle for different combinations of parameters whereas Lemma 4.3 refers to stable or unstable nodes, saddle nodes, or non-hyperbolic points. The equilibrium \(E_{3}\) is classified in Lemma 4.4 as a stable or unstable focus-node or a saddle focus node and \(E_{4}\) is claimed to be a sink in Lemma 4.5; both results lack proofs. The authors go on discussing further properties of equilibria including periodicity (Section 5, proofs of Theorems 5.1 and 5.2 are sketched), bifurcations (Sections 6 and 7, some explanations are provided in the latter one). Numerical simulations are considered in Section 8 and hybrid control of bifurcations is the subject of Section 9. Conclusions and further work are discussed in the final Section 10. Lack of details makes the comprehension of the results challenging.
scientific article; zbMATH DE number 7390442