Hopf bifurcation analysis of the delayed diffusive predator-prey model with harvesting terms

Honglu Yu; Yu Sui; Dan Jin; Ruizhi Yang; Honglu Yu; Yu Sui; Dan Jin; Ruizhi Yang

doi:10.3934/era.2025227

Electronic Research Archive

2025, Volume 33, Issue 8: 5064-5084. doi: 10.3934/era.2025227

Previous Article Next Article

Research article

Hopf bifurcation analysis of the delayed diffusive predator-prey model with harvesting terms

1.
Department of Mathematics, Northeast Forestry University, Harbin 150040, China
2.
School of Ecology, Northeast Forestry University, Harbin 150040, China

Received: 01 July 2025 Revised: 03 August 2025 Accepted: 20 August 2025 Published: 29 August 2025

This paper studied a delayed diffusive predator-prey model with predator and prey harvesting terms. The existence of constant steady-state solutions and Hopf bifurcation were analyzed. Then by applying the central manifold theorem and the normal form method, the direction of the Hopf bifurcation and stability of the bifurcating period solution were studied. Numerical simulations were conducted to confirm the accuracy of the proposed theory. In addition, taking the predator-prey relationship between sharks and tuna as an example, this study investigated the impact of predator harvesting coefficients on the constant steady-state solutions of the system and the time required for the system to reach stability.
- predator-prey model,
- delay,
- harvesting term,
- Hopf bifurcation
Citation: Honglu Yu, Yu Sui, Dan Jin, Ruizhi Yang. Hopf bifurcation analysis of the delayed diffusive predator-prey model with harvesting terms[J]. Electronic Research Archive, 2025, 33(8): 5064-5084. doi: 10.3934/era.2025227

Related Papers:

Abstract

This paper studied a delayed diffusive predator-prey model with predator and prey harvesting terms. The existence of constant steady-state solutions and Hopf bifurcation were analyzed. Then by applying the central manifold theorem and the normal form method, the direction of the Hopf bifurcation and stability of the bifurcating period solution were studied. Numerical simulations were conducted to confirm the accuracy of the proposed theory. In addition, taking the predator-prey relationship between sharks and tuna as an example, this study investigated the impact of predator harvesting coefficients on the constant steady-state solutions of the system and the time required for the system to reach stability.

References

[1]	D. F. Duan, B. Niu, J. Wei, Y. Yuan, The dynamical analysis of a nonlocal predator-prey model with cannibalism, Eur. J. Appl. Math., 35 (2025), 707–731. https://doi.org/10.1017/S0956792524000019 doi: 10.1017/S0956792524000019
[2]	M. Chen, S. Ham, Y. Choi, H. Kim, J. Kim, Pattern dynamics of a harvested predator-prey model, Chaos, Solitons Fractals, 176 (2023), 114153. https://doi.org/10.1016/j.chaos.2023.114153 doi: 10.1016/j.chaos.2023.114153
[3]	W. Yang, M. Chen, The impact of predator-dependent prey refuge on the dynamics of a leslie-Gower predator-prey model, Asian Res. J. Math., 19 (2023), 203–211. https://doi.org/10.9734/arjom/2023/v19i11766 doi: 10.9734/arjom/2023/v19i11766
[4]	G. Guo, X. Yang, C. Zhang, S. Li, Pattern formation with jump discontinuity in a predator-prey model with Holling-Ⅱ functional response, Eur. J. Appl. Math., (2025), 1–23. https://doi.org/10.1017/S0956792525000063
[5]	F. Zhu, R. Yang, Bifurcation in a modified Leslie-Gower model with nonlocal competition and fear effect, Discrete Contin. Dyn. Syst. - Ser. B, 30 (2025), 2865–2893. https://doi.org/10.3934/dcdsb.2024195 doi: 10.3934/dcdsb.2024195
[6]	F. Wang, R. Yang, X. Zhang, Turing patterns in a predator-prey model with double Allee effect, Math. Comput. Simul., 220 (2024), 170–191. https://doi.org/10.1016/j.matcom.2024.01.015 doi: 10.1016/j.matcom.2024.01.015
[7]	R. Yang, F. Wang, D. Jin, Spatially inhomogeneous bifurcating periodic solutions induced by nonlocal competition in a predator-prey system with additional food, Math. Methods Appl. Sci., 45 (2022), 9967–9978. https://doi.org/10.1002/mma.8349 doi: 10.1002/mma.8349
[8]	P. A. Schmidt, L. D. Mech, Wolf pack size and food acquisition, Am. Nat., 150 (1997), 513–517. https://doi.org/10.1086/286079 doi: 10.1086/286079
[9]	F. Courchamp, D. Macdonald, Crucial importance of pack size in the African wild dog Lycaon pictus, Anim. Conserv., 4 (2010), 169–174. https://doi.org/10.1017/S1367943001001196 doi: 10.1017/S1367943001001196
[10]	D. Scheel, C. Packer, Group hunting behaviour of lions: a search for cooperation, Anim. Behav., 41 (1991), 697–709. https://doi.org/10.1016/S0003-3472(05)80907-8 doi: 10.1016/S0003-3472(05)80907-8
[11]	Y. Ma, R. Yang, Bifurcation analysis in a modified Leslie-Gower with nonlocal competition and Beddington-DeAngelis functional response, J. Appl. Anal. Comput., 15 (2025), 2152–2184. https://doi.org/10.11948/20240415 doi: 10.11948/20240415
[12]	C. Cosner, D. L. Deangelis, J. S. Ault, D. B. Olson, Effects of spatial grouping on the functional response of predators, Theor. Popul. Biol., 56 (1999), 65–75. https://doi.org/10.1006/tpbi.1999.1414 doi: 10.1006/tpbi.1999.1414
[13]	C. S. Holling, The functional response of predators to prey size and its role in mimicry and population regulation, Mem. Entomol. Soc. Can., 97 (1965), 5–60. https://doi.org/10.4039/entm9745fv doi: 10.4039/entm9745fv
[14]	K. Ryu, W. Ko, M. Haque, Bifurcation analysis in a predator-prey system with a functional response increasing in both predator and prey densities, Nonlinear Dyn., 94 (2018), 1639–1656. https://doi.org/10.1007/s11071-018-4446-0 doi: 10.1007/s11071-018-4446-0
[15]	R. Yang, C. Nie, D. Jin, Spatiotemporal dynamics induced by nonlocal competition in a diffusive predator-prey system with habitat complexity, Nonlinear Dyn., 110 (2022), 879–900. https://doi.org/10.1007/s11071-022-07625-x doi: 10.1007/s11071-022-07625-x
[16]	X. Zhang, J. Liu, G. Wang, Impacts of nonlocal fear effect and delay on a reaction-diffusion predator-prey model, Int. J. Biomath., 18 (2025), 2350089. https://doi.org/10.1142/S1793524523500894 doi: 10.1142/S1793524523500894
[17]	X. Zhang, S. Li, Y. Yuan, Q. An, Effect of discontinuous harvesting on a diffusive predator-prey model, Nonlinearity, 37 (2024), 115016. https://doi.org/10.1088/1361-6544/ad7fc3 doi: 10.1088/1361-6544/ad7fc3
[18]	Y. Wang, Q. Tu, Multiple positive periodic solutions for a delayed predator-prey system with Beddington-DeAngelis functional response and harvesting terms, in Annals of the University of Craiova, Mathematics and Computer Science Series, 42 (2015), 330–338. https://doi.org/10.52846/ami.v42i2.644
[19]	X. Chang, J. Wei, Hopf bifurcation and optimal control in a diffusive predator-prey system with time delay and prey harvesting, Nonlinear Anal.-Model. Control, 17 (2012), 379–409. https://doi.org/10.15388/NA.17.4.14046 doi: 10.15388/NA.17.4.14046
[20]	J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer Berlin, 1996.
[21]	B. D. Hassard, N. D. Kazarinoff, Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge-New York, 1981.

Reader Comments

Your name:*

Email:*
© 2025 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)