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Number and stability of relaxation oscillations for predator-prey systems with small death rates. (English) Zbl 1411.34064

Summary: We consider planar systems of predator-prey models with small predator death rates \(\epsilon>0\). Using geometric singular perturbation theory and Floquet theory, we derive characteristic functions that determine the location and the stability of relaxation oscillations as \(\epsilon\rightarrow 0\). When the prey-isocline has a single interior local extremum, we prove that the system has a unique nontrivial periodic orbit, which forms a relaxation oscillation. For some systems with prey-isocline possessing two interior local extrema, we show that either the positive equilibrium is globally stable, or the system has exactly two periodic orbits. In particular, for a predator-prey model with the Holling type IV functional response we derive a threshold value of the carrying capacity that separates these two outcomes. This result supports the so-called paradox of enrichment.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
34C26 Relaxation oscillations for ordinary differential equations
92D25 Population dynamics (general)
34E15 Singular perturbations for ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations

Software:

XPPAUT; Matlab
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Full Text: DOI arXiv

References:

[1] F. Albrecht, H. Gatzke, N. Wax, and R. M. May, {\it Stable limit cycles in prey-predator populations}, Science, 181 (1973), pp. 1073-1074, .
[2] K.-S. Cheng, {\it Uniqueness of a limit cycle for a predator-prey system}, SIAM J. Math. Anal., 12 (1981), pp. 541-548, . · Zbl 0471.92021
[3] P. De Maesschalck, {\it On maximum bifurcation delay in real planar singularly perturbed vector fields}, Nonlinear Anal., 68 (2008), pp. 547-576, . · Zbl 1135.34035
[4] P. De Maesschalck and S. Schecter, {\it The entry-exit function and geometric singular perturbation theory}, J. Differential Equations, 260 (2016), pp. 6697-6715, . · Zbl 1342.34078
[5] F. Dumortier, R. Roussarie, and C. Rousseau, {\it Hilbert’s 16th problem for quadratic vector fields}, J. Differential Equations, 110 (1994), pp. 86-133, . · Zbl 0802.34028
[6] B. Ermentrout, {\it Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students}, Software Environ. Tools 14, SIAM, Philadelphia, 2002, . · Zbl 1003.68738
[7] N. Fenichel, {\it Geometric singular perturbation theory for ordinary differential equations}, J. Differential Equations, 31 (1979), pp. 53-98, . · Zbl 0476.34034
[8] H. I. Freedman and G. S. K. Wolkowicz, {\it Predator-prey systems with group defence: The paradox of enrichment revisited}, Bull. Math. Biol., 48 (1986), pp. 493-508, . · Zbl 0612.92017
[9] A. Ghazaryan, V. Manukian, and S. Schecter, {\it Travelling waves in the Holling-Tanner model with weak diffusion}, Proc. R. Soc. Lond. Ser. A, 471 (2015), 20150045, . · Zbl 1371.35302
[10] G. W. Harrison, {\it Global stability of predator-prey interactions}, J. Math. Biol., 8 (1979), pp. 159-171, . · Zbl 0425.92009
[11] J. Hofbauer and J. W.-H. So, {\it Multiple limit cycles for predator-prey models}, Math. Biosci., 99 (1990), pp. 71-75, . · Zbl 0701.92015
[12] S.-B. Hsu, {\it On global stability of a predator-prey system}, Math. Biosci., 39 (1978), pp. 1-10, . · Zbl 0383.92014
[13] S.-B. Hsu and T.-W. Huang, {\it Global stability for a class of predator-prey systems}, SIAM J. Appl. Math., 55 (1995), pp. 763-783, . · Zbl 0832.34035
[14] S.-B. Hsu, S. P. Hubbell, and P. Waltman, {\it Competing predators}, SIAM J. Appl. Math., 35 (1978), pp. 617-625, . · Zbl 0394.92025
[15] S.-B. Hsu and J. Shi, {\it Relaxation oscillation profile of limit cycle in predator-prey system}, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), pp. 893-911, . · Zbl 1176.34049
[16] T.-H. Hsu, {\it On bifurcation delay: An alternative approach using geometric singular perturbation theory}, J. Differential Equations, 262 (2017), pp. 1617-1630, . · Zbl 1365.34101
[17] J. Huang, S. Ruan, and J. Song, {\it Bifurcations in a predator-prey system of Leslie type with generalized Holling type III functional response}, J. Differential Equations, 257 (2014), pp. 1721-1752, . · Zbl 1326.34082
[18] C. K. R. T. Jones, {\it Geometric singular perturbation theory}, in Dynamical Systems (Montecatini Terme, 1994), Lecture Notes in Math. 1609, Springer, Berlin, 1995, pp. 44-118, . · Zbl 0840.58040
[19] C. K. R. T. Jones and S.-K. Tin, {\it Generalized exchange lemmas and orbits heteroclinic to invariant manifolds}, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), pp. 967-1023, . · Zbl 1362.34094
[20] R. E. Kooij and A. Zegeling, {\it A predator-prey model with Ivlev’s functional response}, J. Math. Anal. Appl., 198 (1996), pp. 473-489, . · Zbl 0851.34030
[21] Y. Kuang, {\it Nonuniqueness of limit cycles of Gause-type predator-prey systems}, Appl. Anal., 29 (1988), pp. 269-287, . · Zbl 0629.34036
[22] Y. Kuang and H. I. Freedman, {\it Uniqueness of limit cycles in Gause-type models of predator-prey systems}, Math. Biosci., 88 (1988), pp. 67-84, . · Zbl 0642.92016
[23] C. Kuehn, {\it Multiple Time Scale Dynamics}, Appl. Math. Sci. 191, Springer, Cham, 2015. · Zbl 1335.34001
[24] C. Li and H. Zhu, {\it Canard cycles for predator-prey systems with Holling types of functional response}, J. Differential Equations, 254 (2013), pp. 879-910, . · Zbl 1257.34035
[25] M. Y. Li, W. Liu, C. Shan, and Y. Yi, {\it Turning points and relaxation oscillation cycles in simple epidemic models}, SIAM J. Appl. Math., 76 (2016), pp. 663-687, . · Zbl 1356.34053
[26] W. Liu, D. Xiao, and Y. Yi, {\it Relaxation oscillations in a class of predator-prey systems}, J. Differential Equations, 188 (2003), pp. 306-331, . · Zbl 1094.34025
[27] The MathWorks, {\it MATLAB, version 9.3.0 (R2017b)}, The MathWorks, Natick, MA, 2017.
[28] R. M. May, {\it Limit cycles in predator-prey communities}, Science, 177 (1972), pp. 900-902, .
[29] S. H. Piltz, F. Veerman, P. K. Maini, and M. A. Porter, {\it A predator-2 prey fast-slow dynamical system for rapid predator evolution}, SIAM J. Appl. Dyn. Syst., 16 (2017), pp. 54-90, . · Zbl 1382.37101
[30] M. L. Rosenzweig, {\it Paradox of enrichment: Destabilization of exploitation ecosystems in ecological time}, Science, 171 (1971), pp. 385-387, .
[31] R. Roussarie, {\it Bifurcations of Planar Vector Fields and Hilbert’s Sixteenth Problem}, Modern Birkhäuser Classics, Birkhäuser/Springer, Basel, 1998, .
[32] S. Ruan and D. Xiao, {\it Global analysis in a predator-prey system with nonmonotonic functional response}, SIAM J. Appl. Math., 61 (2001), pp. 1445-1472, . · Zbl 0986.34045
[33] S. Schecter, {\it Exchange lemmas 2. General Exchange Lemma}, J. Differential Equations, 245 (2008), pp. 411-441, . · Zbl 1158.34038
[34] G. Seo and G. S. K. Wolkowicz, {\it Sensitivity of the dynamics of the general Rosenzweig-MacArthur model to the mathematical form of the functional response: A bifurcation theory approach}, J. Math. Biol., 76 (2018), pp. 1873-1906, . · Zbl 1390.92123
[35] J. Sugie, {\it Two-parameter bifurcation in a predator-prey system of Ivlev type}, J. Math. Anal. Appl., 217 (1998), pp. 349-371, . · Zbl 0894.34025
[36] G. Teschl, {\it Ordinary Differential Equations and Dynamical Systems}, Grad. Stud. Math. 140, AMS, Providence, RI, 2012, .
[37] G. S. K. Wolkowicz, {\it Bifurcation analysis of a predator-prey system involving group defence}, SIAM J. Appl. Math., 48 (1988), pp. 592-606, . · Zbl 0657.92015
[38] D. M. Wrzosek, {\it Limit cycles in predator-prey models}, Math. Biosci., 98 (1990), pp. 1-12, . · Zbl 0694.92015
[39] D. Xiao and Z. Zhang, {\it On the uniqueness and nonexistence of limit cycles for predator-prey systems}, Nonlinearity, 16 (2003), pp. 1185-1201, . · Zbl 1042.34060
[40] D. Xiao and H. Zhu, {\it Multiple focus and Hopf bifurcations in a predator-prey system with nonmonotonic functional response}, SIAM J. Appl. Math., 66 (2006), pp. 802-819, . · Zbl 1109.34034
[41] H. Zhu, S. A. Campbell, and G. S. K. Wolkowicz, {\it Bifurcation analysis of a predator-prey system with nonmonotonic functional response}, SIAM J. Appl. Math., 63 (2002), pp. 636-682, . · Zbl 1036.34049
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