The research presented in this paper compares the occurrence of limit cycles under different bifurcation mechanisms in a simple system of two-dimensional autonomous predator–prey ODEs. Surprisingly two unconventional approaches, for a singular system and for a system with a center, turn out to produce more limit cycles than the traditional Andronov–Hopf bifurcation. The system has a functional response function which is a monotonically increasing cubic function of x for \(0\le x\le 1\) where x represents the prey density, and which is constant for \(x>1\). It acts as a proxy for investigating more general systems. The following results are obtained. For the Andronov–Hopf bifurcation the highest order of the weak focus is 2 and at most 2 small-amplitude limit cycles can be created. In the center bifurcation cases are shown to exist with at least 3 limit cycles. In the singular perturbation cases are shown to exist with at least 4 limit cycles and in some cases an exact upper bound of 2 limit cycles is obtained. Finally we indicate how the conclusions can be extended to more general systems. We show how an arbitrary number of limit cycles can be created by choosing an appropriate functional response function and growth function for the prey. One special situation is the system with group defense: the three bifurcation mechanisms typically produce less limit cycles if a group defense element is included.

本文中提出的研究比较了在二维自主捕食者 - 猎物 ODE 的简单系统中不同分叉机制下极限循环的发生。令人惊讶的是，对于奇异系统和具有中心的系统，两种非常规方法会产生比传统 Andronov-Hopf 分岔更多的极限环。该系统具有函数响应函数，它是x的单调递增三次函数，对于\(0\le x\le 1\)，其中x表示猎物密度，并且对于\(x>1\)是常数。它充当调查更一般系统的代理。得到以下结果。对于Andronov-Hopf 分岔弱焦点的最高阶为 2，最多可以创建 2 个小幅度极限环。在中心分叉的情况下显示存在至少 3 个极限循环。在奇异扰动情况下，显示存在至少 4 个极限循环，在某些情况下，可以获得 2 个极限循环的确切上限。最后，我们指出如何将结论扩展到更一般的系统。我们展示了如何通过为猎物选择适当的功能响应函数和增长函数来创建任意数量的极限循环。一种特殊情况是具有群防的系统：如果包含群防元素，三个分叉机制通常会产生较少的极限环。