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## 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, doi: 10.3934/dcdsb.2019219
##### References:
 [1] T. Bolger, B. Eastman, M. Hill and G. S. K. Wolkowicz, A predator-prey model in the chemostat with Holling type Ⅱ (Monod) response function, preprint. Google Scholar [2] G. J. Butler and P. Waltman, Persistence in dynamical systems, J. Differential Equations, 63 (1986), 255-263.  doi: 10.1016/0022-0396(86)90049-5.  Google Scholar [3] K.-S. Cheng, Uniqueness of a limit cycle for a predator-prey system, SIAM J. Math. Anal., 12 (1981), 541-548.  doi: 10.1137/0512047.  Google Scholar [4] P. De Maesschalck and F. Dumortier, Singular perturbations and vanishing passage through a turning point, J. Differential Equations, 248 (2010), 2294-2328.  doi: 10.1016/j.jde.2009.11.009.  Google Scholar [5] P. De Maesschalck and S. Schecter, The entry-exit function and geometric singular perturbation theory, J. Differential Equations, 260 (2016), 6697-6715.  doi: 10.1016/j.jde.2016.01.008.  Google Scholar [6] N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9.  Google Scholar [7] A. Ghazaryan, V. Manukian and S. Schecter, Travelling waves in the Holling-Tanner model with weak diffusion, Proc. R. Soc. Lond. Ser. A, 471 (2015), 20150045, 16pp. doi: 10.1098/rspa.2015.0045.  Google Scholar [8] J. R. Graef, M. Y. Li and L. Wang, A study on the effects of disease caused death in a simple epidemic model, in Dynamical Systems and Differential Equations (eds. W. Chen and S. Hu), Southwest Missouri State University Press, 1 (1998), 288–300.  Google Scholar [9] S.-B. Hsu and J. Shi, Relaxation oscillation profile of limit cycle in predator-prey system, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 893-911.  doi: 10.3934/dcdsb.2009.11.893.  Google Scholar [10] T.-H. Hsu, On bifurcation delay: An alternative approach using geometric singular perturbation theory, J. Differential Equations, 262 (2017), 1617-1630.  doi: 10.1016/j.jde.2016.10.022.  Google Scholar [11] T.-H. Hsu, Number and stability of relaxation oscillations for predator-prey systems with small death rates, SIAM J. Appl. Dyn. Syst., 18 (2019), 33-67.  doi: 10.1137/18M1166705.  Google Scholar [12] C. K. R. T. Jones, Geometric singular perturbation theory, in Dynamical systems (Montecatini Terme, 1994), vol. 1609 of Lecture Notes in Math., Springer, Berlin, 1995, 44–118. doi: 10.1007/BFb0095239.  Google Scholar [13] Y. Kuang and H. I. Freedman, Uniqueness of limit cycles in Gause-type models of predator-prey systems, Math. Biosci., 88 (1988), 67-84.  doi: 10.1016/0025-5564(88)90049-1.  Google Scholar [14] C. Kuehn, Multiple Time Scale Dynamics, vol. 191 of Applied Mathematical Sciences, Springer, Cham, 2015. doi: 10.1007/978-3-319-12316-5.  Google Scholar [15] C. Li and H. Zhu, Canard cycles for predator-prey systems with Holling types of functional response, J. Differential Equations, 254 (2013), 879-910.  doi: 10.1016/j.jde.2012.10.003.  Google Scholar [16] M. Y. Li, W. Liu, C. Shan and Y. Yi, Turning points and relaxation oscillation cycles in simple epidemic models, SIAM J. Appl. Math., 76 (2016), 663-687.  doi: 10.1137/15M1038785.  Google Scholar [17] L.-P. Liou and K.-S. Cheng, On the uniqueness of a limit cycle for a predator-prey system, SIAM J. Math. Anal., 19 (1988), 867-878.  doi: 10.1137/0519060.  Google Scholar [18] W. Liu, D. Xiao and Y. Yi, Relaxation oscillations in a class of predator-prey systems, J. Differential Equations, 188 (2003), 306-331.  doi: 10.1016/S0022-0396(02)00076-1.  Google Scholar [19] N. L. Lundström and G. Söderbacka, Estimates of size of cycle in a predator-prey system, Differential Equations and Dynamical Systems, (2018), 1–29. doi: 10.1007/s12591-018-0422-x.  Google Scholar [20] S. H. Piltz, F. Veerman, P. K. Maini and M. A. Porter, A predator-2 prey fast-slow dynamical system for rapid predator evolution, SIAM J. Appl. Dyn. Syst., 16 (2017), 54-90.  doi: 10.1137/16M1068426.  Google Scholar [21] H. Renato, Predator–prey systems with small predator's death rate, Electronic Journal of Qualitative Theory of Differential Equations, 2018 (2018), 1-16.  doi: 10.14232/ejqtde.2018.1.86.  Google Scholar [22] J. Shen, C.-H. Hsu and T.-H. Yang, Fast–slow dynamics for intraguild predation models with evolutionary effects, Journal of Dynamics and Differential Equations, (2019), 1–26. doi: 10.1007/s10884-019-09744-3.  Google Scholar [23] H. L. Smith and P. Waltman, The Theory of the Chemostat, vol. 13 of Cambridge Studies in Mathematical Biology, Cambridge University Press, Cambridge, 1995, Dynamics of microbial competition. doi: 10.1017/CBO9780511530043.  Google Scholar [24] J. Wang, X. Zhang, J. Shi and Y. Wang, Profile of the unique limit cycle in a class of general predator-prey systems, Appl. Math. Comput., 242 (2014), 397-406.  doi: 10.1016/j.amc.2014.05.020.  Google Scholar [25] G. S. K. Wolkowicz, Bifurcation analysis of a predator-prey system involving group defence, SIAM J. Appl. Math., 48 (1988), 592-606.  doi: 10.1137/0148033.  Google Scholar

show all references

##### References:
 [1] T. Bolger, B. Eastman, M. Hill and G. S. K. Wolkowicz, A predator-prey model in the chemostat with Holling type Ⅱ (Monod) response function, preprint. Google Scholar [2] G. J. Butler and P. Waltman, Persistence in dynamical systems, J. Differential Equations, 63 (1986), 255-263.  doi: 10.1016/0022-0396(86)90049-5.  Google Scholar [3] K.-S. Cheng, Uniqueness of a limit cycle for a predator-prey system, SIAM J. Math. Anal., 12 (1981), 541-548.  doi: 10.1137/0512047.  Google Scholar [4] P. De Maesschalck and F. Dumortier, Singular perturbations and vanishing passage through a turning point, J. Differential Equations, 248 (2010), 2294-2328.  doi: 10.1016/j.jde.2009.11.009.  Google Scholar [5] P. De Maesschalck and S. Schecter, The entry-exit function and geometric singular perturbation theory, J. Differential Equations, 260 (2016), 6697-6715.  doi: 10.1016/j.jde.2016.01.008.  Google Scholar [6] N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9.  Google Scholar [7] A. Ghazaryan, V. Manukian and S. Schecter, Travelling waves in the Holling-Tanner model with weak diffusion, Proc. R. Soc. Lond. Ser. A, 471 (2015), 20150045, 16pp. doi: 10.1098/rspa.2015.0045.  Google Scholar [8] J. R. Graef, M. Y. Li and L. Wang, A study on the effects of disease caused death in a simple epidemic model, in Dynamical Systems and Differential Equations (eds. W. Chen and S. Hu), Southwest Missouri State University Press, 1 (1998), 288–300.  Google Scholar [9] S.-B. Hsu and J. Shi, Relaxation oscillation profile of limit cycle in predator-prey system, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 893-911.  doi: 10.3934/dcdsb.2009.11.893.  Google Scholar [10] T.-H. Hsu, On bifurcation delay: An alternative approach using geometric singular perturbation theory, J. Differential Equations, 262 (2017), 1617-1630.  doi: 10.1016/j.jde.2016.10.022.  Google Scholar [11] T.-H. Hsu, Number and stability of relaxation oscillations for predator-prey systems with small death rates, SIAM J. Appl. Dyn. Syst., 18 (2019), 33-67.  doi: 10.1137/18M1166705.  Google Scholar [12] C. K. R. T. Jones, Geometric singular perturbation theory, in Dynamical systems (Montecatini Terme, 1994), vol. 1609 of Lecture Notes in Math., Springer, Berlin, 1995, 44–118. doi: 10.1007/BFb0095239.  Google Scholar [13] Y. Kuang and H. I. Freedman, Uniqueness of limit cycles in Gause-type models of predator-prey systems, Math. Biosci., 88 (1988), 67-84.  doi: 10.1016/0025-5564(88)90049-1.  Google Scholar [14] C. Kuehn, Multiple Time Scale Dynamics, vol. 191 of Applied Mathematical Sciences, Springer, Cham, 2015. doi: 10.1007/978-3-319-12316-5.  Google Scholar [15] C. Li and H. Zhu, Canard cycles for predator-prey systems with Holling types of functional response, J. Differential Equations, 254 (2013), 879-910.  doi: 10.1016/j.jde.2012.10.003.  Google Scholar [16] M. Y. Li, W. Liu, C. Shan and Y. Yi, Turning points and relaxation oscillation cycles in simple epidemic models, SIAM J. Appl. Math., 76 (2016), 663-687.  doi: 10.1137/15M1038785.  Google Scholar [17] L.-P. Liou and K.-S. Cheng, On the uniqueness of a limit cycle for a predator-prey system, SIAM J. Math. Anal., 19 (1988), 867-878.  doi: 10.1137/0519060.  Google Scholar [18] W. Liu, D. Xiao and Y. Yi, Relaxation oscillations in a class of predator-prey systems, J. Differential Equations, 188 (2003), 306-331.  doi: 10.1016/S0022-0396(02)00076-1.  Google Scholar [19] N. L. Lundström and G. Söderbacka, Estimates of size of cycle in a predator-prey system, Differential Equations and Dynamical Systems, (2018), 1–29. doi: 10.1007/s12591-018-0422-x.  Google Scholar [20] S. H. Piltz, F. Veerman, P. K. Maini and M. A. Porter, A predator-2 prey fast-slow dynamical system for rapid predator evolution, SIAM J. Appl. Dyn. Syst., 16 (2017), 54-90.  doi: 10.1137/16M1068426.  Google Scholar [21] H. Renato, Predator–prey systems with small predator's death rate, Electronic Journal of Qualitative Theory of Differential Equations, 2018 (2018), 1-16.  doi: 10.14232/ejqtde.2018.1.86.  Google Scholar [22] J. Shen, C.-H. Hsu and T.-H. Yang, Fast–slow dynamics for intraguild predation models with evolutionary effects, Journal of Dynamics and Differential Equations, (2019), 1–26. doi: 10.1007/s10884-019-09744-3.  Google Scholar [23] H. L. Smith and P. Waltman, The Theory of the Chemostat, vol. 13 of Cambridge Studies in Mathematical Biology, Cambridge University Press, Cambridge, 1995, Dynamics of microbial competition. doi: 10.1017/CBO9780511530043.  Google Scholar [24] J. Wang, X. Zhang, J. Shi and Y. Wang, Profile of the unique limit cycle in a class of general predator-prey systems, Appl. Math. Comput., 242 (2014), 397-406.  doi: 10.1016/j.amc.2014.05.020.  Google Scholar [25] G. S. K. Wolkowicz, Bifurcation analysis of a predator-prey system involving group defence, SIAM J. Appl. Math., 48 (1988), 592-606.  doi: 10.1137/0148033.  Google Scholar
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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