We explore the local dynamics, N‐S bifurcation, and hybrid control in a discrete‐time Lotka‐Volterra predator‐prey model in ${\mathbb{R}}_{+}^2$. It is shown that $\forall$ parametric values, model has two boundary equilibria: $P_{00}(0,0)$ and $P_{x0}(1,0)$, and a unique positive equilibrium point: $P_{xy}^{+}\left(\frac{d}{c},\frac{r\left(c-d\right)}{bc}\right)$ if $c>d$. We explored the local dynamics along with different topological classifications about equilibria: $P_{00}(0,0)$, $P_{x0}(1,0)$, and $P_{xy}^{+}\left(\frac{d}{c},\frac{r\left(c-d\right)}{bc}\right)$ of the model. It is proved that model cannot undergo any bifurcation about $P_{00}(0,0)$ and $P_{x0}(1,0)$ but it undergoes an N‐S bifurcation when parameters vary in a small neighborhood of $P_{xy}^{+}\left(\frac{d}{c},\frac{r\left(c-d\right)}{bc}\right)$ by using a center manifold theorem and bifurcation theory and meanwhile, invariant close curves appears. The appearance of these curves implies that there exist a periodic or quasiperiodic oscillations between predator and prey populations. Further, theoretical results are verified numerically. Finally, the hybrid control strategy is applied to control N‐S bifurcation in the discrete‐time model.

中文翻译：

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离散Lotka-Volterra模型中的Neimark-Sacker分叉和混合控制

我们在一个离散的Lotka-Volterra捕食者-猎物模型中探索局部动力学，NS分叉和混合控制 ${\mathbb{[R}}_{+}^2$。结果表明$\forall$ 参数值，模型具有两个边界平衡： $P_{00}（0，0）$ 和 $P_{X0}（\mathrm{1个}，0）$，以及唯一的正平衡点： $P_{Xÿ}^{+}\left(\frac{d}{C}，\frac{\mathrm{[R}\left(C-d\right)}{bC}\right)$ 如果 $C>d$。我们探索了局部动力学以及关于平衡的不同拓扑分类：$P_{00}（0，0）$， $P_{X0}（\mathrm{1个}，0）$和 $P_{Xÿ}^{+}\left(\frac{d}{C}，\frac{\mathrm{[R}\left(C-d\right)}{bC}\right)$模型的 证明了模型对$P_{00}（0，0）$ 和 $P_{X0}（\mathrm{1个}，0）$ 但是当参数在 $P_{Xÿ}^{+}\left(\frac{d}{C}，\frac{\mathrm{[R}\left(C-d\right)}{bC}\right)$利用中心流形定理和分岔理论，同时出现不变的闭合曲线。这些曲线的出现意味着在捕食者和被捕食者之间存在周期性或准周期性振荡。此外，对理论结果进行了数值验证。最后，在离散时间模型中，将混合控制策略应用于控制NS分叉。