Bifurcations and hybrid control in a 3 x 3 discrete-time predator-prey model

被引:0
作者
Khan, Abdul Qadeer [1 ]
Kiyani, Azhar Zafar [1 ]
Ahmad, Imtiaz [2 ]
机构
[1] Univ Azad Jammu & Kashmir, Dept Math, Muzaffarabad 13100, Pakistan
[2] Mirpur Univ Sci & Technol MUST, Dept Math, Mirpur 10250, Ajk, Pakistan
关键词
predator-prey model; bifurcation and hybrid control; center manifold theorem; numerical simulation; DIFFERENTIAL-EQUATIONS; BOUNDEDNESS; SYSTEMS;
D O I
10.3934/mbe.2020360
中图分类号
Q [生物科学];
学科分类号
07 ; 0710 ; 09 ;
摘要
In this paper, we explore the bifurcations and hybrid control in a 3 x 3 discrete-time predator-prey model in the interior of R-+(3). It is proved that 3 x 3 model has four boundary fixed points: P-000(0, 0, 0), P0y0(0, r-1/r, 0), P-0yz(0, d/f, rf-f-dr/cf), P-x0z (d/e, 0, a/b) and the unique positive fixed point: P-xyz(+) (br(d-f)+ f(b+ac)/ber br-b-ac/br a/b) under certain restrictions to the involved parameters. By utilizing method of Linearization, local dynamics along with topological classifications about fixed points have been investigated. Existence of prime period and periodic points of the model are also investigated. Further for 3 x 3 model, we have explored the occurrence of possible bifurcations about each fixed point, that gives more insight about the under consideration model. It is proved that the model cannot undergo any bifurcation about P-000(0, 0, 0) and P-x0z (d/e, 0, a/b), but the model undergo P-D and N-S bifurcations respectively about P0y0 (0, r-1/r, 0) and P-0yz(0, d/f, rf-f-dr/cf). For the unique positive fixed point: Pxyz+ (br(d-f)+f(b+ac)/ber, br-b-ac/br, a/b), we have proved the N-S as well as P-D bifurcations by explicit criterion. Further, theoretical results are verified by numerical simulations. We have also presented the bifurcation diagrams and corresponding maximum Lyapunov exponents for the 3 x 3 model. The computation of the maximum Lyapunov exponents ratify the appearance of chaotic behavior in the under consideration model. Finally, the hybrid control strategy is applied to control N-S as well as P-D bifurcations in the discrete-time model.
引用
收藏
页码:6963 / 6992
页数:30
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