当前位置: X-MOL 学术Math. Biosci. Eng. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Bifurcations and hybrid control in a 3×3 discrete-time predator-prey model
Mathematical Biosciences and Engineering ( IF 2.6 ) Pub Date : 2020-10-16 , DOI: 10.3934/mbe.2020360
Abdul Qadeer Khan 1 , Azhar Zafar Kiyani 1 , Imtiaz Ahmad 2
Affiliation  

In this paper, we explore the bifurcations and hybrid control in a $3\times3$ discrete-time predator-prey model in the interior of $\mathbb{R}_+^3$. It is proved that $3\times3$ model has four boundary fixed points: $P_{000}(0,0,0)$, $P_{0y0}\left(0,\frac{r-1}{r},0\right)$, $P_{0yz}\left(0,\frac{d}{f},\frac{rf-f-dr}{cf}\right)$, $P_{x0z}\left(\frac{d}{e},0,\frac{a}{b}\right)$, and the unique positive fixed point: $P^+_{xyz}\left(\frac{br(d-f)+f(b+ac)}{ber},\frac{br-b-ac}{br},\frac{a}{b}\right)$ 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 $3\times3$ 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 $P_{x0z}\left(\frac{d}{e},0,\frac{a}{b}\right)$, but the model undergo P-D and N-S bifurcations respectively about $P_{0y0}\left(0,\frac{r-1}{r},0\right)$ and $P_{0yz}\left(0,\frac{d}{f},\frac{rf-f-dr}{cf}\right)$. For the unique positive fixed point: $P^+_{xyz}\left(\frac{br(d-f)+f(b+ac)}{ber},\frac{br-b-ac}{br},\frac{a}{b}\right)$, 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 $3\times3$ 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.

中文翻译:

3×3离散捕食-被捕食模型的分叉与混合控制

在本文中,我们探索了$ \ mathbb {R} _ + ^ 3 $内部的$ 3 \ times3 $离散捕食者-被捕食模型的分支和混合控制。证明$ 3 \ times3 $模型具有四个边界固定点:$ P_ {000}(0,0,0)$,$ P_ {0y0} \ left(0,\ frac {r-1} {r}, 0 \ right)$,$ P_ {0yz} \ left(0,\ frac {d} {f},\ frac {rf-f-dr} {cf} \ right)$,$ P_ {x0z} \ left( \ frac {d} {e},0,\ frac {a} {b} \ right)$,以及唯一的正定点:$ P ^ + _ {xyz} \ left(\ frac {br(df)+ f(b + ac)} {ber},\ frac {br-b-ac} {br},\ frac {a} {b} \ right)$在涉及的参数受到一定限制的情况下。利用线性化方法,研究了局部动力学以及关于固定点的拓扑分类。还研究了模型的黄金时段和周期点的存在。对于$ 3 \ times3 $模型,我们已经探究了每个固定点可能出现的分叉现象,从而为考虑中的模型提供了更多信息。证明该模型不会对$ P_ {000}(0,0,0)$和$ P_ {x0z} \ left(\ frac {d} {e},0,\ frac {a} { b} \ right)$,但模型分别经历了PD和NS分叉,分别约为$ P_ {0y0} \ left(0,\ frac {r-1} {r},0 \ right)$和$ P_ {0yz} \左(0,\ frac {d} {f},\ frac {rf-f-dr} {cf} \ right)$。对于唯一的正定点:$ P ^ + _ {xyz} \ left(\ frac {br(df)+ f(b + ac)} {ber},\ frac {br-b-ac} {br}, \ frac {a} {b} \ right)$,我们已经通过明确的准则证明了NS和PD的分歧。此外,理论结果通过数值模拟得到了验证。我们还给出了$ 3 \ times3 $模型的分叉图和相应的最大Lyapunov指数。最大李雅普诺夫指数的计算证明了考虑模型中混沌行为的出现。最后,在离散时间模型中,将混合控制策略应用于控制NS和PD分叉。
更新日期:2020-10-17
down
wechat
bug