Hopf bifurcation problems near double positive equilibrium points for a class of quartic Kolmogorov model

Chaoxiong Du; Wentao Huang; Chaoxiong Du; Wentao Huang

doi:10.3934/math.20231367

AIMS Mathematics

2023, Volume 8, Issue 11: 26715-26730. doi: 10.3934/math.20231367

Previous Article Next Article

Research article Special Issues

Hopf bifurcation problems near double positive equilibrium points for a class of quartic Kolmogorov model

Chaoxiong Du ^{1
,
,},
Wentao Huang ²

1.
School of Mathematics, Changsha Normal University, Changsha 410100, China
2.
College of Mathematics and Statistics, Guangxi Normal University, Guilin 541006, China

Received: 23 July 2023 Revised: 03 September 2023 Accepted: 05 September 2023 Published: 19 September 2023
MSC : 34C07, 34C23

The Kolmogorov model is a class of significant ecological models and is initially introduced to describe the interaction between two species occupying the same ecological habitat. Limit cycle bifurcation problem is close to Hilbertis 16th problem. In this paper, we focus on investigating bifurcation of limit cycle for a class of quartic Kolmogorov model with two positive equilibrium points. Using the singular values method, we obtain the Lyapunov constants for each positive equilibrium point and investigate their limit cycle bifurcations behavior. Furthermore, based on the analysis of their Lyapunov constants' structure and Hopf bifurcation, we give the condition that each one positive equilibrium point of studied model can bifurcate 5 limit cycles, which include 3 stable limit cycles.
- limit cycles,
- Kolmogorov model,
- Poincar$ \acute{e} $ succession function,
- stable cycles
Citation: Chaoxiong Du, Wentao Huang. Hopf bifurcation problems near double positive equilibrium points for a class of quartic Kolmogorov model[J]. AIMS Mathematics, 2023, 8(11): 26715-26730. doi: 10.3934/math.20231367

Related Papers:

Abstract

The Kolmogorov model is a class of significant ecological models and is initially introduced to describe the interaction between two species occupying the same ecological habitat. Limit cycle bifurcation problem is close to Hilbertis 16th problem. In this paper, we focus on investigating bifurcation of limit cycle for a class of quartic Kolmogorov model with two positive equilibrium points. Using the singular values method, we obtain the Lyapunov constants for each positive equilibrium point and investigate their limit cycle bifurcations behavior. Furthermore, based on the analysis of their Lyapunov constants' structure and Hopf bifurcation, we give the condition that each one positive equilibrium point of studied model can bifurcate 5 limit cycles, which include 3 stable limit cycles.

References

[1]	A. Algaba, C. García, J. Giné, Nondegenerate centers and limit cycles of cubic Kolmogorov systems, Nonlinear Dyn., 91 (2018), 487–496. https://doi.org/10.1007/s11071-017-3883-5 doi: 10.1007/s11071-017-3883-5
[2]	X. Chen, J. Llibre, Z. Zhang, Suffificient conditions for the existence of at least n or exactly $n$ limit cycles for the Liénard differential systems, J. Differ. Equations, 242 (2007), 11–23. https://doi.org/10.1016/j.jde.2007.07.004 doi: 10.1016/j.jde.2007.07.004
[3]	H. Chen, M. Han, Y. Xia, Limit cycles of a Liénard system with symmetry allowing for discontinuity, J. Math. Anal. Appl., 468 (2018), 799–816. https://doi.org/10.1016/j.jmaa.2018.08.050 doi: 10.1016/j.jmaa.2018.08.050
[4]	C. Du, W. Huang, Center-focus problem and limit cycles bifurcations for a class of cubic Kolmogorov model, Nonlinear Dyn., 72 (2013), 197–206. https://doi.org/10.1007/s11071-012-0703-9 doi: 10.1007/s11071-012-0703-9
[5]	C. Du, Y. Liu, W. Huang, Limit cycles bifurcations for a class of Kolmogorov model in symmetrical vector field, Int. J. Bifurcat. Chaos, 24 (2014), 1450040. https://doi.org/10.1142/S0218127414500400 doi: 10.1142/S0218127414500400
[6]	C. Du, Y. Liu, Q. Zhang, Limit cycles in a class of quartic Kolmogorov model with three positive equilibrium points, Int. J. Bifurcat. Chaos, 25 (2015), 1550080. https://doi.org/10.1142/S0218127415500807 doi: 10.1142/S0218127415500807
[7]	C. Du, Q. Wang, W. Huang, Three-Dimensional Hopf bifurcation for a class of cubic Kolmogorov model, Int. J. Bifurcat. Chaos, 24 (2014), 1450036. https://doi.org/10.1142/S0218127414500369 doi: 10.1142/S0218127414500369
[8]	J. Gu, A. Zegeling, W. Huang, Bifurcation of limit cycles and isochronous centers on center manifolds for a class of cubic Kolmogorov systems in R$^3$, Qual. Theory Dyn. Syst., 22 (2023), 42. https://doi.org/10.1007/s12346-023-00745-8 doi: 10.1007/s12346-023-00745-8
[9]	D. He, W. Huang, Q. Wang, Small amplitude limit cycles and local bifurcation of critical periods for a quartic Kolmogorov system, Qual. Theory Dyn. Syst., 19 (2020), 68. https://doi.org/10.1007/s12346-020-00401-5 doi: 10.1007/s12346-020-00401-5
[10]	X. Huang, L. Zhu, Limit cycles in a general kolmogorov model, Nonlinear Anal. Theor., 60 (2005), 1394–1414. https://doi.org/10.1016/j.na.2004.11.003 doi: 10.1016/j.na.2004.11.003
[11]	M. Han, Y. Lin, P. Yu, A study on the existence of limit cycles of a planar system with 3rd-degree polynomials, Int. J. Bifurcat. Chaos, 14 (2004), 41–60. https://doi.org/10.1142/S0218127404009247 doi: 10.1142/S0218127404009247
[12]	A. Kolmogorov, Sulla teoria di Volterra della lotta per lésistenza, Giornale dell'Istituto Italiano degli Attuari, 7 (1936), 74–80.
[13]	A. Q. Khan, S. A. H. Bukhari, M. B. Almatrafi, Global dynamics, Neimark-Sacker bifurcation and hybrid control in a Leslie's prey-predator model, Alex. Eng. J., 61 (2022), 11391–11404. https://doi.org/10.1016/j.aej.2022.04.042 doi: 10.1016/j.aej.2022.04.042
[14]	A. Q. Khan, F. Nazir, M. B. Almatrafi, Bifurcation analysis of a discrete Phytoplankton CZooplankton model with linear predational response function and toxic substance distribution, Int. J. Biomath., 16 (2023), 2250095. https://doi.org/10.1142/S1793524522500954 doi: 10.1142/S1793524522500954
[15]	A. Q. Khan, M. Tasneem, M. B. Almatrafi, Discrete-time COVID-19 epidemic model with bifurcation and control, Math. Biosci. Eng., 19 (2022), 1944–1969. https://doi.org/10.3934/mbe.2022092 doi: 10.3934/mbe.2022092
[16]	Y. Liu, Theory of center-focus for a class of higher-degree critical points and infinite points, Sci. China Ser. A-Math., 44 (2001), 365–377. https://doi.org/10.1007/BF02878718 doi: 10.1007/BF02878718
[17]	Y. Liu, H. Chen, Formulas of singular point quantities and the first 10 saddle quantities for a class of cubic system, Acta Math. Appl. Sin., 25 (2002), 295–302.
[18]	Y. Liu, J. Li, Theory of values of singular point in complex autonomous differential system, Sci. China Ser. A-Math., 3 (1990), 10–24.
[19]	J. Llibre, Y. Martínez, Dynamics of a family of Lotka-Volterra systems in R$^3$, Nonlinear Anal., 199 (2020), 111915. https://doi.org/10.1016/j.na.2020.111915 doi: 10.1016/j.na.2020.111915
[20]	J. Llibre, Y. Martínez, C. Valls, Limit cycles bifurcating of Kolmogorov systems in R$^2$ and in R$^3$, Commun. Nonlinear Sci., 91 (2020), 105401. https://doi.org/10.1016/j.cnsns.2020.105401 doi: 10.1016/j.cnsns.2020.105401
[21]	J. Llibre, X. Zhang, Limit cycles of the classical Liénard differential systems: A survey on the Lins Neto, de Melo and Pughs conjecture, Expo. Math., 35 (2017), 286–299. https://doi.org/10.1016/j.exmath.2016.12.001 doi: 10.1016/j.exmath.2016.12.001
[22]	N. G. Lloyd, J. M. Pearson, E. Saéz, I. Szántó, A cubic Kolmogorov system with six limit cycles, Comput. Math. Appl., 44 (2002), 445–455. https://doi.org/10.1016/S0898-1221(02)00161-X doi: 10.1016/S0898-1221(02)00161-X
[23]	Z. Lu, B. He, Multiple stable limit cycles for a cubic kolmogorov system, Chinese Journal of Engineering Mathematics, 4 (2001), 115–117.
[24]	N. G. Lloyd, J. M. Pearson, E. Saez, I. Szanto, Limit cycles of a cubic kolmogorov system, Appl. Math. Lett., 9 (1996), 15–18. https://doi.org/10.1016/0893-9659(95)00095-X doi: 10.1016/0893-9659(95)00095-X
[25]	Y. Wu, C. Zhang, Integrability and non-linearizability of weak saddles in a cubic Kolmogorov model, Chaos Soliton. Fract., 153 (2021), 111514. https://doi.org/10.1016/j.chaos.2021.111514 doi: 10.1016/j.chaos.2021.111514
[26]	Y. Yuan, H. Chen, C. Du, Y. Yuan, The limit cycles of a general Kolmogorov system, J. Math. Anal. Appl., 392 (2012), 225–237. https://doi.org/10.1016/j.jmaa.2012.02.065 doi: 10.1016/j.jmaa.2012.02.065

Reader Comments

Your name:*

Email:*
© 2023 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)