The uniqueness of limit cycles in a predator-prey system with Ivlev-type group defense

Jin Liao; André Zegeling; Wentao Huang; Jin Liao; André Zegeling; Wentao Huang

doi:10.3934/math.20241604

AIMS Mathematics

2024, Volume 9, Issue 12: 33610-33631. doi: 10.3934/math.20241604

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Research article

The uniqueness of limit cycles in a predator-prey system with Ivlev-type group defense

College of Mathematics and Statistics, Guangxi Normal University, Guilin, Guangxi 541004, China

Received: 01 September 2024 Revised: 21 October 2024 Accepted: 04 November 2024 Published: 27 November 2024
MSC : 34C07, 34C23, 92D25

This paper discusses the uniqueness of limit cycles in a two-dimensional autonomous Gause predator-prey model with an Ivlev-type group defense introduced by D. M. Xiao, S. G. Ruan, Codimension two bifurcations in a predator-prey system with group defense, Int. J. Bifurcat. Chaos, 11 (2001). We proved their conjecture that the system can exhibit at most one limit cycle. Furthermore, we compared the qualitative differences between this system and two similar systems with group defense: One system with the same Ivlev-type functional response function but with Leslie-Gower predator dynamics and another system with a comparable functional response function. For both systems, we show that two limit cycles can occur.
- limit cycle,
- group defense,
- Liénard system,
- predator-prey,
- Gause system
Citation: Jin Liao, André Zegeling, Wentao Huang. The uniqueness of limit cycles in a predator-prey system with Ivlev-type group defense[J]. AIMS Mathematics, 2024, 9(12): 33610-33631. doi: 10.3934/math.20241604

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Abstract

This paper discusses the uniqueness of limit cycles in a two-dimensional autonomous Gause predator-prey model with an Ivlev-type group defense introduced by D. M. Xiao, S. G. Ruan, Codimension two bifurcations in a predator-prey system with group defense, Int. J. Bifurcat. Chaos, 11 (2001). We proved their conjecture that the system can exhibit at most one limit cycle. Furthermore, we compared the qualitative differences between this system and two similar systems with group defense: One system with the same Ivlev-type functional response function but with Leslie-Gower predator dynamics and another system with a comparable functional response function. For both systems, we show that two limit cycles can occur.

References

[1]	J. C. Artés, J. Llibre, D. Schlomiuk, N. Vulpe, Geometric configurations of singularities of planar polynomial differential systems, Birkhäuser, 2021. https://doi.org/10.1007/978-3-030-50570-7
[2]	L. Depalo, C. Gallego, R. Ortells-Fabra, C. Salas, R. Montal, A. Urbaneja, et al., Advancing tomato crop protection: Green leaf volatile-mediated defense mechanisms against Nesidiocoris tenuis plant damage, Biol. Control, 192 (2024), 105517. https://doi.org/10.1016/j.biocontrol.2024.105517 doi: 10.1016/j.biocontrol.2024.105517
[3]	H. I. Freedman, G. S. K. Wolkowicz, Predator-prey systems with group defence: The paradox of enrichment revisited, Bull. Math. Biol., 48 (1986), 493–508. https://doi.org/10.1016/S0092-8240(86)90004-2 doi: 10.1016/S0092-8240(86)90004-2
[4]	A. Gasull, Differential equations that can be transformed into equations of Liénard type, Actas del XVLL colloquio Brasileiro de matematica, 1989.
[5]	G. F. Gause, The struggle for existence, Baltimore: Williams and Wilkins, 1934.
[6]	E. González-Olivares, B. González-Yañez, J. M. Lorca, A. Rojas-Palma, J. D. Flores, Consequences of double Allee effect on the number of limit cycles in a predator-prey model, Comput. Math. Appl., 62 (2011), 3449–3463. https://doi.org/10.1016/j.camwa.2011.08.061 doi: 10.1016/j.camwa.2011.08.061
[7]	C. S. Holling, The components of predation as revealed by a study of small-mammal predation of the European pine sawfly, Can. Entomol., 91 (1959), 293–320. https://doi.org/10.4039/Ent91293-5 doi: 10.4039/Ent91293-5
[8]	J. C. Holmes, W. M. Bethel, Modification of intermediate host behavior by parasites, Zool. J. Linn. Soc., 51 (1972), 123–149.
[9]	X. Y. Hou, Y. Li, X. Y. Zhang, S. J. Ge, Y. Mu, J. Y. Shen, Unraveling the intracellular and extracellular self-defense of Chlorella sorokiniana toward highly toxic pyridine stress, Bioresour. Technol., 385 (2023), 129366. https://doi.org/10.1016/j.biortech.2023.129366 doi: 10.1016/j.biortech.2023.129366
[10]	C. Jost, S. P. Ellner, Testing for predator dependence in predator-prey dynamics: A non-parametric approach, Proc. Roy. Soc. B, 267 (2000), 1611–1620. https://doi.org/10.1098/rspb.2000.1186 doi: 10.1098/rspb.2000.1186
[11]	R. E. Kooij, A. Zegeling, A predator-prey model with Ivlev's functional response, J. Math. Anal. Appl., 198 (1996), 473–489. https://doi.org/10.1006/jmaa.1996.0093 doi: 10.1006/jmaa.1996.0093
[12]	R. E. Kooij, A. Zegeling, Qualitative properties of two-dimensional predator-prey systems, Nonlinear Anal., 29 (1997), 693–715. https://doi.org/10.1016/S0362-546X(96)00068-5 doi: 10.1016/S0362-546X(96)00068-5
[13]	R. E. Kooij, A. Zegeling, Predator-prey models with non-analytical functional response, Chaos Solit. Fractals, 123 (2019), 163–172. https://doi.org/10.1016/j.chaos.2019.03.036 doi: 10.1016/j.chaos.2019.03.036
[14]	P. H. Leslie, J. C. Gower, The properties of a stochastic model for a predatorprey type of interaction between two species, Biometrika, 47 (1960), 219–234. https://doi.org/10.2307/2333294 doi: 10.2307/2333294
[15]	Q. Li, Y. Y. Zhang, Y. N. Xiao, Canard phenomena for a slow-fast predator-prey system with group defense of the prey, J. Math. Anal. Appl., 527 (2023), 127418. https://doi.org/10.1016/j.jmaa.2023.127418 doi: 10.1016/j.jmaa.2023.127418
[16]	Y. L. Li, D. M. Xiao, Bifurcations of a predator-prey system of Holling and Leslie types, Chaos Solit. Fractals, 34 (2007), 606–620. https://doi.org/10.1016/j.chaos.2006.03.068 doi: 10.1016/j.chaos.2006.03.068
[17]	Y. Liu, A. Zegeling, W. T. Huang, The application of Liénard transformations to predator-prey systems, Qual. Theory Dyn. Syst., 23 (2024), 91. https://doi.org/10.1007/s12346-023-00947-0 doi: 10.1007/s12346-023-00947-0
[18]	M. Lu, D. Z. Gao, J. C. Huang, H. Wang, Relative prevalence-based dispersal in an epidemic patch model, J. Math. Biol., 86 (2023), 52. https://doi.org/10.1007/s00285-023-01887-8 doi: 10.1007/s00285-023-01887-8
[19]	M. Lu, J. C. Huang, S. G. Ruan, P. Yu, Bifurcation analysis of an SIRS epidemic model with a generalized nonmonotone and saturated incidence rate, J. Differ. Equ., 267 (2019), 1859–1898. https://doi.org/10.1016/j.jde.2019.03.005 doi: 10.1016/j.jde.2019.03.005
[20]	M. Lu, J. C. Huang, S. G. Ruan, P. Yu, Global dynamics of a susceptible-infectious-recovered epidemic model with a generalized nonmonotone incidence rate, J. Dyn. Differ. Equ., 33 (2021), 1625–1661. https://doi.org/10.1007/s10884-020-09862-3 doi: 10.1007/s10884-020-09862-3
[21]	J. Morcuende, J. Martín-García, P. Velasco, T. Sánchez-Gómez, Ó. Santamaría, V. M. Rodríguez, J. Poveda, Effective biological control of chickpea rabies (Ascochyta rabiei) through systemic phytochemical defenses activation by Trichoderma roots colonization: From strain characterization to seed coating, Biol. Control, 193 (2024), 105530. https://doi.org/10.1016/j.biocontrol.2024.105530 doi: 10.1016/j.biocontrol.2024.105530
[22]	F. Rothe, D. S. Shafer, Multiple bifurcation in a predator-prey system with nonmonotonic predator response, Proc. R. Soc. Edinb. A: Math., 120 (1992), 313–347. https://doi.org/10.1017/S0308210500032169 doi: 10.1017/S0308210500032169
[23]	Y. L. Tang, F. Li, Multiple stable states for a class of predator-prey systems with two harvesting rates, J. Appl. Anal. Comput., 14 (2024), 506–514. https://doi.org/10.11948/20230295 doi: 10.11948/20230295
[24]	J. S. Tener, Muskoxen in Canada, a biological and taxonomic review, Canadian Government, Ottawa, 1965.
[25]	G. S. K. Wolkowicz, Bifurcation analysis of a predator-prey system involving group defence, SIAM J. Appl. Math., 48 (1988), 592–606. https://doi.org/10.1137/0148033 doi: 10.1137/0148033
[26]	D. M. Xiao, S. G. Ruan, Codimension two bifurcations in a predator-prey system with group defense, Int. J. Bifurc. Chaos, 11 (2001), 2123–2131. https://doi.org/10.1142/S021812740100336X doi: 10.1142/S021812740100336X
[27]	D. M. Xiao, H. P. Zhu, Multiple focus and Hopf bifurcations in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 66 (2006), 802–819. https://doi.org/10.1137/050623449 doi: 10.1137/050623449
[28]	R. C. Yang, L. Q. Yang, Y. L. Tang, Some research on limit cycles of Liénard system, Math. Theory Appl., 41 (2021), 59–95.
[29]	Y. Q. Ye, Theory of limit cycles, American Mathematical Society, 1986.
[30]	A. Zegeling, R. E. Kooij, Uniqueness of limit cycles in polynomial systems with algebraic invariants, Bull. Aust. Math. Soc., 49 (1994), 7–20. https://doi.org/10.1017/S0004972700016026 doi: 10.1017/S0004972700016026
[31]	A. Zegeling, R. E. Kooij, Several bifurcation mechanisms for limit cycles in a predator-prey system, Qual. Theory Dyn. Sys., 20 (2021), 65. https://doi.org/10.1007/s12346-021-00501-w doi: 10.1007/s12346-021-00501-w
[32]	A. Zegeling, H. L. Wang, G. Z Zhu, Uniqueness of limit cycles in a predator-prey model with sigmoid functional response, J. Nonlinear Model. Anal., 5 (2023), 790–802. https://doi.org/10.12150/jnma.2023.790 doi: 10.12150/jnma.2023.790
[33]	Z. F. Zhang, T. R. Ding, W. Z. Huang, Z. X. Dong, Qualitative theory of differential equations, American Mathematical Society, 1992.
[34]	Y. R. Zhou, C. W. Wang, D. Blackmore, The uniqueness of limit cycles for Liénard system, J. Math. Anal. Appl., 304 (2005), 473–489. https://doi.org/10.1016/j.jmaa.2004.09.037 doi: 10.1016/j.jmaa.2004.09.037
[35]	B. Zimmerman, H. Sand, P. Wabakken, O. Liberg, H. P. Andreassen, Predator-dependent functional response in wolves: From food limitation to surplus killing, J. Anim. Ecol., 84 (2015), 102–112. https://doi.org/10.1111/1365-2656.12280 doi: 10.1111/1365-2656.12280

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