Figure 1.
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)
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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 |
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].
Keywords:
Relaxation oscillations,
limit cycles,
chemostat predator-prey models,
epidemic models,
periodicity in disease incidence.
Mathematics Subject Classification:
Primary: 34C26; Secondary: 92D25.
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 and Continuous Dynamical Systems - B,
2020, 25
(4)
: 1257-1277.
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [14] |
C. Kuehn, Multiple Time Scale Dynamics, vol. 191 of Applied Mathematical Sciences, Springer, Cham, 2015.
doi: 10.1007/978-3-319-12316-5. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [14] |
C. Kuehn, Multiple Time Scale Dynamics, vol. 191 of Applied Mathematical Sciences, Springer, Cham, 2015.
doi: 10.1007/978-3-319-12316-5. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
Figure 2.
Trajectories for the limiting system (9) of (8) with $ \epsilon = 0 $
Figure 3.
A family of heteroclinic orbits, parameterized by $ \gamma(s) $ , of limiting system (14)
Figure 4.
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 $
Figure 5.
(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) $
Figure 6.
(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) $
Figure 7.
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 $
Figure 8.
(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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