# American Institute of Mathematical Sciences

April  2020, 25(4): 1257-1277. doi: 10.3934/dcdsb.2019219

## A criterion for the existence of relaxation oscillations with applications to predator-prey systems and an epidemic model

 1 Department of Mathematics, University of Miami, 1365 Memorial Drive, Coral Gables, FL 33146, USA 2 Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, Ontario L8S 4K1, Canada

Received  February 2019 Revised  June 2019 Published  September 2019

We derive characteristic functions to determine the number and stability of relaxation oscillations for a class of planar systems. Applying our criterion, we give conditions under which the chemostat predator-prey system has a globally orbitally asymptotically stable limit cycle. Also we demonstrate that a prescribed number of relaxation oscillations can be constructed by varying the perturbation for an epidemic model studied by Li et al. [SIAM J. Appl. Math, 2016].

Citation: Ting-Hao Hsu, Gail S. K. Wolkowicz. A criterion for the existence of relaxation oscillations with applications to predator-prey systems and an epidemic model. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1257-1277. doi: 10.3934/dcdsb.2019219
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##### References:
System (1) with $\epsilon = 0$ exhibits a family of heteroclinic orbits on $\Lambda$. The dynamics on the segment $\overline{\Lambda}\cap \{x = 0\}$ is governed by system (5)
Trajectories for the limiting system (9) of (8) with $\epsilon = 0$
A family of heteroclinic orbits, parameterized by $\gamma(s)$, of limiting system (14)
Numerical simulations of $\chi$ and $\lambda$ for Example 3.1. The function $\chi$ has a single root $x_1\approx 6.92$, with $\lambda(x_1)<0$
(A) The trajectory of system (1) for Example 3.1 with $\epsilon = 0.5$ and initial point $(S,x,y)(0) = (6,1,10)$ converges to a periodic orbit $\ell_\epsilon$. (B) The trajectory $\gamma(x_0)$ of (51), where $x_0$ is a root of $\chi$. The simulation shows that $\ell_\epsilon$ is close to $\gamma(x_0)$
(A) For Example 4.1, the function $\chi$ has a root $N_1\approx 377.01$ with $\lambda(N_1)\approx -4.11<0$. (B) The dashed curve is a trajectory for the system with $\epsilon = 10^{-5}$ with the initial condition $(S,I,N)(0) = (60,2,120)$, and the solid curve is the singular orbit $\gamma(N_1)$. The simulation shows that the trajectory approaches a periodic orbit near $\gamma(N_1)$
The perturbation term $\epsilon f(N)$ with $f(N) = N(1-N_{\max})$ in Example 4.1 is replaced by $\epsilon f_1(N)$ with $f_1(N) = f(N)-c_1\exp(-c_2(N-c_3))$ in Example 4.2. Essentially $f_1$ is obtained by dropping the value of $f$ in a small interval right to $N_0$
(A) For Example 4.2, the function $\chi$ has two roots, $N_1\approx 156.89$ and $N_2\approx 342.18$, with $\lambda(N_1)\approx 1.06>0$ and $\lambda(N_2)\approx -2.48<0$. (B) The dashed curves are trajectories for the system with $\epsilon = 10^{-5}$, and the solid curves are the singular orbits $\gamma(N_1)$ and $\gamma(N_2)$. The simulation shows that the trajectory with the initial condition $(S,I,N)(0) = (40,2.5,80)$ approaches a periodic orbit near $\gamma(N_2)$ while trajectory with the initial condition $(S,I,N)(0) = (40,1.3,80)$ approaches the interior equilibrium. Near $\gamma(N_1)$ is an unstable periodic orbit
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