Yang Yantao, Zhang Xiang, Zu Jian
School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, People's Republic of China.
College of Mathematics and Computer Science, Yan'an University, Yan'an 716000, People's Republic of China.
Chaos. 2024 Oct 1;34(10). doi: 10.1063/5.0204183.
For the Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting, the known results are on the saddle-node bifurcation and the Hopf bifurcation of codimensions 1, the Bogdanov-Takens bifurcations of codimensions 2 and 3, and on the cyclicity of singular slow-fast cycles. Here, we focus on the global dynamics of the model in the slow-fast setting and obtain much richer dynamical phenomena than the existing ones, such as global stability of an equilibrium; an unstable canard cycle exploding to a homoclinic loop; coexistence of a stable canard cycle and an inner unstable homoclinic loop; and, consequently, coexistence of two canard cycles: a canard explosion via canard cycles without a head, canard cycles with a short head and a beard and a relaxation oscillation with a short beard. This last one should be a new dynamical phenomenon. Numerical simulations are provided to illustrate these theoretical results.
对于具有米氏型猎物捕获的莱斯利 - 高尔捕食者 - 猎物模型，已知结果涉及余维1的鞍结分岔和霍普夫分岔、余维2和3的博格达诺夫 - 塔肯斯分岔以及奇异快慢循环的周期性。在此，我们关注该模型在快慢设定下的全局动力学，并获得比现有结果丰富得多的动力学现象，例如平衡点的全局稳定性；不稳定的鸭式循环爆发出同宿环；稳定的鸭式循环与内部不稳定同宿环共存；因此，两个鸭式循环共存：通过无头鸭式循环、短头有须鸭式循环的鸭式爆炸以及短须松弛振荡。最后一种应该是一种新的动力学现象。提供了数值模拟来阐明这些理论结果。
