This paper studies the dynamics of the generalist predator–prey systems modeled in [E. Alexandra, F. Lutscher and G. Seo, Bistability and limit cycles in generalist predator–prey dynamics, Ecol. Complex. 14 (2013) 48–55]. When prey reproduces much faster than predator, by combining the normal form theory of slow-fast systems, the geometric singular perturbation theory and the results near non-hyperbolic points developed by Krupa and Szmolyan [Relaxation oscillation and canard explosion, J. Differential Equations 174(2) (2001) 312–368; Extending geometric singular perturbation theory to nonhyperbolic points—fold and canard points in two dimensions, SIAM J. Math. Anal. 33(2) (2001) 286–314], we provide a detailed mathematical analysis to show the existence of homoclinic orbits, heteroclinic orbits and canard limit cycles and relaxation oscillations bifurcating from the singular homoclinic cycles. Moreover, on global stability of the unique positive equilibrium, we provide some new results. Numerical simulations are also carried out to support the theoretical results.

中文翻译：

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奇异扰动的通才捕食者-猎物系统中的鸭式爆炸、同宿和异宿轨道

本文研究了 [E. Alexandra，F. Lutscher 和 G. Seo，泛泛捕食者-猎物动力学中的双稳态和极限循环，Ecol。复杂的。14 (2013) 48-55]。当猎物的繁殖速度比捕食者快得多时，通过结合慢速系统的范式理论、几何奇异摄动理论以及 Krupa 和 Szmolyan 开发的非双曲线点附近的结果 [松弛振荡和鸭式爆炸，J. 微分方程 174 (2) (2001) 312–368；将几何奇异微扰理论扩展到非双曲点——二维中的折叠点和鸭式点，SIAM J. Math。肛门。33（2）（2001）286-314]，我们提供了详细的数学分析来显示同宿轨道的存在，异宿轨道和鸭式极限环以及从奇异同宿周期分叉的弛豫振荡。此外，关于唯一正均衡的全局稳定性，我们提供了一些新的结果。还进行了数值模拟以支持理论结果。