Relaxation Oscillations in Predator–Prey Systems

We characterize a criterion for the existence of relaxation oscillations in planar systems of the form dudt=uk+1g(u,v,ε),dvdt=εf(u,v,ε)+uk+1h(u,v,ε),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{du}{dt}= u^{k+1} g(u,v,\varepsilon ), \qquad \frac{dv}{dt}=\varepsilon f(u,v,\varepsilon ) + u^{k+1} h(u,v,\varepsilon ), \end{aligned}$$\end{document}where k≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\ge 0$$\end{document} is an arbitrary constant and ε>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document} is a sufficiently small parameter. Taking into account of possible degeneracy of the “discriminant” function occurred when k≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\ge 0$$\end{document}, this criterion generalizes and strengthens those for the case k=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=0$$\end{document} 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=0$$\end{document}, 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.