Global dynamics in a generalized slow-fast predator-prey model

Cheng Wang; Qianqian Zhao; Yanru Xie; Cheng Wang; Qianqian Zhao; Yanru Xie

doi:10.3934/math.2026750

AIMS Mathematics

2026, Volume 11, Issue 6: 18458-18480. doi: 10.3934/math.2026750

Previous Article Next Article

Research article

Global dynamics in a generalized slow-fast predator-prey model

1.
School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing 210023, China
2.
College of Statistics and Mathematics, Hebei University of Economics and Business, Shijiazhuang 050061, China

Received: 08 March 2026 Revised: 17 May 2026 Accepted: 11 June 2026 Published: 23 June 2026
MSC : 34C07, 34C23, 34D15

This paper studies a generalized slow-fast predator-prey model extended from the classic Gause model, where the predator reproduction rate is taken as a small singular perturbation parameter. First, by applying Dulac's criterion, we derive the sufficient conditions under which the local asymptotic stability of the system's unique positive equilibrium implies its global asymptotic stability in the first quadrant. Second, based on the geometric singular perturbation theory, we prove the existence of a unique relaxation oscillation surrounding the positive equilibrium, and show that this relaxation oscillation converges to the transcritical slow-fast cycle in the Hausdorff distance as the perturbation parameter approaches zero. Finally, with relaxation oscillation properties and Zhang Zhifen's limit cycle uniqueness theorem, we further derive sufficient conditions for the existence and uniqueness of stable limit cycles. The results enrich dynamical researches on slow-fast predator-prey systems and are applicable to related ecological dynamic analyses.
- predator-prey model,
- Slow-fast system,
- global asymptotic stability,
- relaxation oscillation,
- limit cycle,
- geometric singular perturbation theory
Citation: Cheng Wang, Qianqian Zhao, Yanru Xie. Global dynamics in a generalized slow-fast predator-prey model[J]. AIMS Mathematics, 2026, 11(6): 18458-18480. doi: 10.3934/math.2026750

Related Papers:

Abstract

This paper studies a generalized slow-fast predator-prey model extended from the classic Gause model, where the predator reproduction rate is taken as a small singular perturbation parameter. First, by applying Dulac's criterion, we derive the sufficient conditions under which the local asymptotic stability of the system's unique positive equilibrium implies its global asymptotic stability in the first quadrant. Second, based on the geometric singular perturbation theory, we prove the existence of a unique relaxation oscillation surrounding the positive equilibrium, and show that this relaxation oscillation converges to the transcritical slow-fast cycle in the Hausdorff distance as the perturbation parameter approaches zero. Finally, with relaxation oscillation properties and Zhang Zhifen's limit cycle uniqueness theorem, we further derive sufficient conditions for the existence and uniqueness of stable limit cycles. The results enrich dynamical researches on slow-fast predator-prey systems and are applicable to related ecological dynamic analyses.

References

[1]	Y. Liu, A. Zegeling, W. Huang, The application of Liénard transformations to predator-prey systems, Qual. Theory Dyn. Syst., 23 (2024), Paper No. 91, 33. https://doi.org/10.1007/s12346-023-00947-0
[2]	D. Zhang, P. Yan, Necessary and sufficient conditions for the nonexistence of limit cycles of Leslie-Gower predator-prey models, Appl. Math. Lett., 71 (2017), 1–5. https://doi.org/10.1016/j.aml.2017.03.008 doi: 10.1016/j.aml.2017.03.008
[3]	Y. Dai, Y. Zhao, B. Sang, Four limit cycles in a predator-prey system of Leslie type with generalized Holling type Ⅲ functional response, Nonlinear Anal. Real World Appl., 50 (2019), 218–239. https://doi.org/10.1016/j.nonrwa.2019.04.003 doi: 10.1016/j.nonrwa.2019.04.003
[4]	D. Xiao, Z. Zhang, On the uniqueness and nonexistence of limit cycles for predator-prey systems, Nonlinearity, 16 (2003), 1185–1201. https://doi.org/10.1088/0951-7715/16/3/321 doi: 10.1088/0951-7715/16/3/321
[5]	J. Huang, S. Ruan, J. Song, Bifurcations in a predator-prey system of Leslie type with generalized Holling type Ⅲ functional response, J. Differential Equations, 257 (2014), 1721–1752. https://doi.org/10.1016/j.jde.2014.04.024 doi: 10.1016/j.jde.2014.04.024
[6]	R. E. Kooij, A. Zegeling, Co-existence of a period annulus and a limit cycle in a class of predator-prey models with group defense, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 31 (2021), Paper No. 2150154, 11. https://doi.org/10.1142/S0218127421501546
[7]	E. González-Olivares, H. Meneses-Alcay, B. González-Yañez, J. Mena-Lorca, A. Rojas-Palma, R. Ramos-Jiliberto, Multiple stability and uniqueness of the limit cycle in a Gause-type predator-prey model considering the Allee effect on prey, Nonlinear Anal. Real World Appl., 12 (2011), 2931–2942. https://doi.org/10.1016/j.nonrwa.2011.04.003 doi: 10.1016/j.nonrwa.2011.04.003
[8]	K. Hasík, On a predator-prey system of Gause type, J. Math. Biol., 60 (2010), 59–74. https://doi.org/10.1007/s00285-009-0257-8 doi: 10.1007/s00285-009-0257-8
[9]	T. W. Hwang, Uniqueness of the limit cycle for Gause-type predator-prey systems, J. Math. Anal. Appl., 238 (1999), 179–195. https://doi.org/10.1006/jmaa.1999.6520 doi: 10.1006/jmaa.1999.6520
[10]	S. Ruan, D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61 (2000/01), 1445–1472. https://doi.org/10.1137/S0036139999361896
[11]	L. Zhao, J. Shen, Canards and homoclinic orbits in a slow-fast modified May-Holling-Tanner predator-prey model with weak multiple Allee effect, Discrete Contin. Dyn. Syst. Ser. B, 27 (2022), 6745–6769. https://doi.org/10.3934/dcdsb.2022018 doi: 10.3934/dcdsb.2022018
[12]	Z. Zhu, X. Liu, Canard cycles and relaxation oscillations in a singularly perturbed Leslie-Gower predator-prey model with Allee effect, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 32 (2022), Paper No. 2250071, 23. https://doi.org/10.1142/S0218127422500717
[13]	W. Su, X. Zhang, Global stability and canard explosions of the predator-prey model with the sigmoid functional response, SIAM J. Appl. Math., 82 (2022), 976–1000. https://doi.org/10.1137/21M1437755 doi: 10.1137/21M1437755
[14]	X. Chen, X. Zhang, Dynamics of the predator-prey model with the Sigmoid functional response, Stud. Appl. Math., 147 (2021), 300–318. https://doi.org/10.1111/sapm.12382 doi: 10.1111/sapm.12382
[15]	P. R. Chowdhury, S. Petrovskii, M. Banerjee, Oscillations and pattern formation in a slow-fast prey-predator system, Bull. Math. Biol., 83 (2021), Paper No. 110, 41. https://doi.org/10.1007/s11538-021-00941-0
[16]	P. R. Chowdhury, M. Banerjee, S. Petrovskii, Canards, relaxation oscillations, and pattern formation in a slow-fast ratio-dependent predator-prey system, Appl. Math. Model., 109 (2022), 519–535. https://doi.org/10.1016/j.apm.2022.04.022 doi: 10.1016/j.apm.2022.04.022
[17]	J. Shen, C. H. Hsu, T. H. Yang, Fast-slow dynamics for intraguild predation models with evolutionary effects, J. Dynam. Differential Equations, 32 (2020), 895–920. https://doi.org/10.1007/s10884-019-09744-3 doi: 10.1007/s10884-019-09744-3
[18]	T. H. Hsu, S. Ruan, Relaxation oscillations and the entry-exit function in multidimensional slow-fast systems, SIAM J. Math. Anal., 53 (2021), 3717–3758. https://doi.org/10.1137/19M1295507 doi: 10.1137/19M1295507
[19]	S. Li, X. Wang, X. Li, K. Wu, Relaxation oscillations for Leslie-type predator-prey model with Holling type Ⅰ response functional function, Appl. Math. Lett., 120 (2021), Paper No. 107328, 6. https://doi.org/10.1016/j.aml.2021.107328
[20]	K. Li, S. Li, K. Wu, Relaxation Oscillations and Canard Explosion of a Leslie-Gower Predator-Prey System with Addictive Allee Effect, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 32 (2022), Paper No. 2250168, 9. https://doi.org/10.1142/S0218127422501681
[21]	S. Chen, J. Duan, J. Li, Effective reduction of a three-dimensional circadian oscillator model, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 5407–5419. https://doi.org/10.3934/dcdsb.2020349 doi: 10.3934/dcdsb.2020349
[22]	S. Chen, J. Duan, J. Li, Dynamics of the Tyson-Hong-Thron-Novak circadian oscillator model, Phys. D, 420 (2021), Paper No. 132869, 16. https://doi.org/10.1016/j.physd.2021.132869
[23]	P. De Maesschalck, G. Kiss, A. Kovacs, Relaxation oscillations and canards of a regulated two-gene model, J. Math. Anal. Appl., 502 (2021), Paper No. 125144, 18. https://doi.org/10.1016/j.jmaa.2021.125144
[24]	S. Ai, S. Sadhu, The entry-exit theorem and relaxation oscillations in slow-fast planar systems, J. Differential Equations, 268 (2020), 7220–7249. https://doi.org/10.1016/j.jde.2019.11.067 doi: 10.1016/j.jde.2019.11.067
[25]	T. H. Hsu, G. S. K. Wolkowicz, A criterion for the existence of relaxation oscillations with applications to predator-prey systems and an epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 1257–1277. https://doi.org/10.3934/dcdsb.2019219 doi: 10.3934/dcdsb.2019219
[26]	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. https://doi.org/10.1137/18M1166705 doi: 10.1137/18M1166705
[27]	W. Liu, D. Xiao, Y. Yi, Relaxation oscillations in a class of predator–prey systems, J. Differential Equations, 188 (2003), 306–331. https://doi.org/10.1016/S0022-0396(02)00076-1 doi: 10.1016/S0022-0396(02)00076-1
[28]	C. Li, H. Zhu, Canard cycles for predator–prey systems with Holling types of functional response, J. Differential Equations, 254 (2013), 879–910. http://dx.doi.org/10.1016/j.jde.2012.10.003 doi: 10.1016/j.jde.2012.10.003
[29]	Y. M. Tai, Z. Zhang, Relaxation oscillations in a spruce-budworm interaction model with Holling's type Ⅱ functional response, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 2173–2199. https://doi.org/10.3934/dcdsb.2021027 doi: 10.3934/dcdsb.2021027
[30]	S. Li, C. Wang, K. Wu, Relaxation oscillations of a slow-fast predator-prey model with a piecewise smooth functional response, Appl. Math. Lett., 113 (2021), Paper No. 106852, 6. https://doi.org/10.1016/j.aml.2020.106852
[31]	A. Ghazaryan, V. Manukian, S. Schecter, Travelling waves in the Holling-Tanner model with weak diffusion, Proc. A., 471 (2015), 20150045, 16. https://doi.org/10.1098/rspa.2015.0045 doi: 10.1098/rspa.2015.0045
[32]	C. Wang, X. Zhang, Relaxation oscillations in a slow-fast modified Leslie-Gower model, Appl. Math. Lett., 87 (2019), 147–153. https://doi.org/10.1016/j.aml.2018.07.029 doi: 10.1016/j.aml.2018.07.029
[33]	L. Zhong, J. Shen, Degenerate transcritical bifurcation point can be an attractor: a case study in a slow-fast modified Leslie-Gower model, Qual. Theory Dyn. Syst., 21 (2022), Paper No. 76, 11. https://doi.org/10.1007/s12346-022-00608-8
[34]	Y. Fu, X. Liu, Dynamics of a spatio-temporal slow-fast predator-prey system with reproductive Allee effect and a generalist predator, Qual. Theory Dyn. Syst., 24 (2025), Paper No. 223, 49. https://doi.org/10.1007/s12346-025-01386-9
[35]	H. Dong, X. Liu, Traveling front in a diffusive predator-prey model with Beddington-DeAngelis functional response, Math. Methods Appl. Sci., 48 (2025), 3682–3711. https://doi.org/10.1002/mma.10508 doi: 10.1002/mma.10508
[36]	Z. Zhu, X. Liu, Canard cycle and nonsmooth bifurcation in a piecewise-smooth continuous predator-prey model, Math. Comput. Simulation, 227 (2025), 477–499. https://doi.org/10.1016/j.matcom.2024.08.017 doi: 10.1016/j.matcom.2024.08.017
[37]	Z. Zhu, X. Liu, Existence of traveling wave solutions in a singularly perturbed predator-prey equation with spatial diffusion, Discrete Contin. Dyn. Syst., 44 (2024), 78–115. https://doi.org/10.3934/dcds.2023097 doi: 10.3934/dcds.2023097
[38]	X. Wu, P. Zhang, F. Xie, Canard explosion of a singularly perturbed SIS epidemic model with a non-monotone incidence rate and a logistic-type population birth rate, Commun. Nonlinear Sci. Numer. Simul., 154 (2026), Paper No. 109547, 21. https://doi.org/10.1016/j.cnsns.2025.109547
[39]	X. Wu, F. Xie, Slow-fast dynamics of a piecewise-smooth Leslie-Gower model with Holling type-Ⅰ functional response and weak Allee effect, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 34 (2024), Paper No. 2450086, 28. https://doi.org/10.1142/S021812742450086X
[40]	X. Wu, S. Lu, F. Xie, Relaxation oscillations of a piecewise-smooth slow-fast Bazykin's model with Holling type Ⅰ functional response, Math. Biosci. Eng., 20 (2023), 17608–17624. https://doi.org/10.3934/mbe.2023782 doi: 10.3934/mbe.2023782
[41]	J. Huang, R. Huzak, J. Yao, Relaxation oscillations in predator-prey systems with piecewise smooth functional responses, J. Differential Equations, 453 (2026), Paper No. 113907, 33. https://doi.org/10.1016/j.jde.2025.113907
[42]	M. Krupa, P. Szmolyan, Extending geometric singular perturbation theory to nonhyperbolic points–fold and canard points in two dimensions, SIAM J. Math. Anal., 33 (2001), 286–314. http://dx.doi.org/10.1137/S0036141099360919 doi: 10.1137/S0036141099360919
[43]	Z. F. Zhang, T. R. Ding, W. Z. Huang, Z. X. Dong, Qualitative theory of differential equations, vol. 101 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1992. Translated from the Chinese by Anthony Wing Kwok Leung. https://doi.org/10.1090/mmono/101

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)