Codimension-two bifurcations and bifurcation controls in a discrete biological system with weak Allee effect.
Codimension-two bifurcations and bifurcation controls in a discrete biological system with weak Allee effect.
复制标题
具有弱 Allee 效应的离散生物系统中的余维二分岔和分岔控制。
DOI：
10.1142/s0218127422500365
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发表时间：
2022
影响因子：
2.2
通讯作者：
Guangyuan Liao
中科院分区：
文献类型：
--
作者：
Limin Zhang;Yike Xu;Guangyuan Liao
研究了一类具有弱Allee效应的离散捕食系统的余维2分支和分支控制问题。通过对所有不动点的分析，我们证明了在唯一的内不动点处存在1：2和1：3强共振。qualitative.properties结合伴随算子方法和近似恒等变换，我们得到了这些强共振的规范形。同时计算了共振点附近的分岔曲线的近似表达式，包括Neimark-Sacker分岔曲线和倍周期分岔曲线。并给出了这些分岔为超临界或亚临界的条件。为了研究系统在共振点附近的全局分岔，我们用一个常微分方程来近似原离散系统，并应用时间一近似流理论。在生物学中，分岔和不稳定的涨落可能对生物种群的繁殖产生不利影响。然后将混合控制策略推广到控制1：2和1：3强共振。数值模拟结果与理论计算结果吻合较好。利用MatContM软件包，我们进行了数值分岔分析，进一步支持了我们的理论结果，其中的分岔点和规范形系数与我们的理论完全一致。根据计算结果，分析了弱Allee效应对系统的影响.并对这些结论给出了合理的生物学解释。
In this paper, the codimension-two bifurcations and bifurcation controls of a discrete predatory system with weak Allee effect on prey are investigated. After analyzing the qualitative.properties of all the fixed points, we explore the existences of 1:2 and 1:3 strong resonances at.the unique interior fixed point. Combining the adjoint operator method with an approximate.identity transformation, we obtain the normal forms of these strong resonances. Meanwhile, we.calculate the approximate expressions of bifurcation curves around the resonance point, which.include Neimark–Sacker and period doubling bifurcation curves. And we give the conditions.under which these bifurcations are supercritical or subcritical. To study global bifurcations near.the resonance point, we use an ordinary differential equation to approximate the original discrete.system by the time-one approximation flow theory. In biology, bifurcations and unstable fluctuations may be harmful for the breeding of biological population. Then we extend the hybrid.control strategy to control 1:2 and 1:3 strong resonances. Numerical simulations are in good.agreement with our theoretical results. By using the MatContM package, we perform numerical.bifurcation analyses to further support our theoretical results, in which the bifurcation points.and the normal form coefficients are completely consistent with our theories. According to the.results, we finally analyze the impact of weak Allee effect on the system. And a reasonable.biological explanation is given for these conclusions.