In this paper, two prey–predator models with distributed delays are presented based on the growth and loss rates of the predator, which are much smaller than that of the prey, leading to a singular perturbation problem. It is obtained that Hopf bifurcation can occur, where the coexistence equilibrium becomes unstable leading to a stable limit cycle. Subsequently, considering the perturbation parameter $0<𝜀\ll 1$, the fact that the solution crossing the transcritical point converges to a stable equilibrium is discussed for the model with Holling type I using the linear chain criterion, center-manifold reduction, the geometric singular perturbation theory and entry–exit function. The existence and uniqueness of relaxation oscillation cycle for the model with Holling type II are obtained. In addition, numerical simulations are provided to verify the analytical results.

在本文中，基于捕食者的生长率和损失率，提出了两种具有分布式延迟的捕食者-捕食者模型，捕食者的生长率和损失率远小于猎物的生长率和损失率，从而导致奇异扰动问题。结果表明，可能发生 Hopf 分岔，共存平衡变得不稳定，导致稳定的极限环。随后，考虑扰动参数$0<𝜀\ll 1$，利用线性链准则、中心流形约简、几何奇异摄动理论和入口-出口函数，讨论了Holling I型模型的跨跨临界点解收敛到稳定平衡的事实。得出Holling II型模型弛豫振荡周期的存在性和唯一性。此外，还提供了数值模拟来验证分析结果。