{"title":"Slow–Fast Dynamics of a Piecewise-Smooth Leslie–Gower Model with Holling Type-I Functional Response and Weak Allee Effect","authors":"Xiao Wu, Feng Xie","doi":"10.1142/s021812742450086x","DOIUrl":null,"url":null,"abstract":"<p>The slow–fast Leslie–Gower model with piecewise-smooth Holling type-I functional response and weak Allee effect is studied in this paper. It is shown that the model undergoes singular Hopf bifurcation and nonsmooth Hopf bifurcation as the parameters vary. The theoretical analysis implies that the predator’s food quality and Allee effect play an important role and lead to richer dynamical phenomena such as the globally stable equilibria, canard explosion phenomenon, a hyperbolically stable relaxation oscillation cycle enclosing almost two canard cycles with different stabilities and so on. Moreover, the predator and prey will coexist as multiple steady states or periodic oscillations for different positive initial populations and positive parameter values. Finally, we present some numerical simulations to illustrate the theoretical analysis such as the existence of one, two or three limit cycles.</p>","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":"21 1","pages":""},"PeriodicalIF":1.9000,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Journal of Bifurcation and Chaos","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1142/s021812742450086x","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0
Abstract
The slow–fast Leslie–Gower model with piecewise-smooth Holling type-I functional response and weak Allee effect is studied in this paper. It is shown that the model undergoes singular Hopf bifurcation and nonsmooth Hopf bifurcation as the parameters vary. The theoretical analysis implies that the predator’s food quality and Allee effect play an important role and lead to richer dynamical phenomena such as the globally stable equilibria, canard explosion phenomenon, a hyperbolically stable relaxation oscillation cycle enclosing almost two canard cycles with different stabilities and so on. Moreover, the predator and prey will coexist as multiple steady states or periodic oscillations for different positive initial populations and positive parameter values. Finally, we present some numerical simulations to illustrate the theoretical analysis such as the existence of one, two or three limit cycles.
本文研究了具有片滑霍林 I 型功能响应和弱阿利效应的慢-快莱斯利-高尔模型。结果表明,随着参数的变化,模型会发生奇异霍普夫分岔和非光滑霍普夫分岔。理论分析表明,捕食者的食物质量和阿利效应起着重要作用,并导致了更丰富的动力学现象,如全局稳定平衡、卡纳德爆炸现象、一个双曲线稳定弛豫振荡周期包围了几乎两个不同稳定性的卡纳德周期等。此外,对于不同的正初始种群和正参数值,捕食者和猎物将以多种稳态或周期振荡的形式共存。最后,我们通过一些数值模拟来说明理论分析,如存在一个、两个或三个极限周期。
期刊介绍:
The International Journal of Bifurcation and Chaos is widely regarded as a leading journal in the exciting fields of chaos theory and nonlinear science. Represented by an international editorial board comprising top researchers from a wide variety of disciplines, it is setting high standards in scientific and production quality. The journal has been reputedly acclaimed by the scientific community around the world, and has featured many important papers by leading researchers from various areas of applied sciences and engineering.
The discipline of chaos theory has created a universal paradigm, a scientific parlance, and a mathematical tool for grappling with complex dynamical phenomena. In every field of applied sciences (astronomy, atmospheric sciences, biology, chemistry, economics, geophysics, life and medical sciences, physics, social sciences, ecology, etc.) and engineering (aerospace, chemical, electronic, civil, computer, information, mechanical, software, telecommunication, etc.), the local and global manifestations of chaos and bifurcation have burst forth in an unprecedented universality, linking scientists heretofore unfamiliar with one another''s fields, and offering an opportunity to reshape our grasp of reality.