We characterize a criterion for the existence of relaxation oscillations in planar systems of the form

where \(k\ge 0\) is an arbitrary constant and \(\varepsilon >0\) is a sufficiently small parameter. Taking into account of possible degeneracy of the “discriminant” function occurred when \(k\ge 0\), this criterion generalizes and strengthens those for the case \(k=0\) obtained by Hsu (SIAM J Appl Dyn Syst 18:33–67, 2019) and Hsu and Wolkowicz (Discrete Contin Dyn Syst Ser B 25:1257–1277, 2020). Differing from the case of \(k=0\), our proof of the criterion is based on the construction of an invariant, thin annular region in an arbitrarily prescribed small neighborhood of a singular closed orbit and the establishment of an asymptotic formula for solutions near the v-axis. As applications of this criterion, we will give concrete conditions ensuring the existence of relaxation oscillations in general predator–prey systems, as well as spatially homogeneous relaxation oscillations and relaxed periodic traveling waves in a class of diffusive predator–prey systems.

我们表征了形式为平面的系统中存在弛豫振荡的准则

其中\（k \ ge 0 \）是任意常数，而\（\ varepsilon> 0 \）是足够小的参数。考虑到\（k \ ge 0 \）时可能发生的“判别式”函数的简并性，该标准对Hsu（SIAM J Appl Dyn Syst 18 ）获得的情况（（k = 0 \））进行了概括和加强： 33-67，2019）和Hsu和Wolkowicz（离散Contin Dyn Syst Ser B 25：1257-1277，2020）。与\（k = 0 \）的情况不同，我们对该准则的证明是基于在任意指定的奇异闭合轨道的小邻域中构造不变的薄环形区域，以及建立解的渐近公式在v附近-轴。作为该标准的应用，我们将给出具体条件，以确保在一般的捕食者－被捕食者系统中存在弛豫振荡，以及一类扩散捕食者－被捕食者系统中的空间均匀弛豫振荡和弛豫的周期性行波。