Modeling the interactions between prey and predators has become an important method to reveal their evolutionary patterns. This paper investigated a predator-prey model incorporating Holling II functional response, considering the dynamic effects of time delay and cross-diffusion on the model. First, the existence and local stability of a positive equilibrium were proven without time delay and diffusion. Next, we selected time delay as the bifurcation parameter to study the existence of Hopf bifurcation, and determined the critical value for Hopf bifurcation. Furthermore, by utilizing multi-scale analysis, the amplitude equation was derived, thereby obtaining the direction of Hopf bifurcation and the stability of bifurcation periodic solutions. Our results show that changes in time delay or diffusion parameters can lead to periodic oscillatory solutions in time and space, corresponding to supercritical (subcritical) Hopf bifurcation, and Turing instability. Finally, the theoretical results were validated through numerical simulations.
Citation: Ping Li, Haisong Cao, Can Chen, Jianfeng Jiao. Amplitude equation for the Hopf bifurcation of a reaction-diffusion predator-prey system incorporating time delay[J]. AIMS Mathematics, 2026, 11(1): 2595-2612. doi: 10.3934/math.2026105
Modeling the interactions between prey and predators has become an important method to reveal their evolutionary patterns. This paper investigated a predator-prey model incorporating Holling II functional response, considering the dynamic effects of time delay and cross-diffusion on the model. First, the existence and local stability of a positive equilibrium were proven without time delay and diffusion. Next, we selected time delay as the bifurcation parameter to study the existence of Hopf bifurcation, and determined the critical value for Hopf bifurcation. Furthermore, by utilizing multi-scale analysis, the amplitude equation was derived, thereby obtaining the direction of Hopf bifurcation and the stability of bifurcation periodic solutions. Our results show that changes in time delay or diffusion parameters can lead to periodic oscillatory solutions in time and space, corresponding to supercritical (subcritical) Hopf bifurcation, and Turing instability. Finally, the theoretical results were validated through numerical simulations.
| [1] |
B. Blasius, L. Rudolf, G. Weithoff, U. Gaedke, G. Fussmann, Long-term cyclic persistence in an experimental predator-prey system, Nature, 577 (2020), 226–230. https://doi.org/10.1038/s41586-019-1857-0 doi: 10.1038/s41586-019-1857-0
|
| [2] |
F. R. Zhang, Y. Li, C. P. Li, Hopf bifurcation in a delayed diffusive leslie-gower predator-prey model with herd behavior, Int. J. Bifurcat. Chaos, 29 (2019), 1950055. https://doi.org/10.1142/S021812741950055X doi: 10.1142/S021812741950055X
|
| [3] |
M. Song, S. Gao, C. Liu, Y. Bai, L. Zhang, B. Xie, et al., Cross-diffusion induced Turing patterns on multiplex networks of a predator-prey model, Chaos Soliton. Fract., 168 (2023), 113131. https://doi.org/10.1016/j.chaos.2023.113131 doi: 10.1016/j.chaos.2023.113131
|
| [4] |
A. J. Lotka, Analytical note on certain rhythmic relations in organic systems, Proc. Natl. Acad. Sci. U.S.A., 6 (1920), 410–415. https://doi.org/10.1073/pnas.6.7.410 doi: 10.1073/pnas.6.7.410
|
| [5] |
V. Volterra, Fluctuations in the abundance of a species considered mathematically$^{1}$, Nature, 118 (1926), 558–560. https://doi.org/10.1038/118558a0 doi: 10.1038/118558a0
|
| [6] |
C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Memoirs of the Entomological Society of Canada, 97 (1965), 5–60. https://doi.org/10.4039/entm9745fv doi: 10.4039/entm9745fv
|
| [7] |
C. S. Holling, Some characteristics of simple types of predation and parasitism, Can. Entomol., 91 (1959), 385–398. https://doi.org/10.4039/ent91385-7 doi: 10.4039/ent91385-7
|
| [8] |
F. Wu, Y. J. Jiao, Stability and Hopf bifurcation of a predator-prey model, Bound. Value Probl., 2019 (2019), 129. https://doi.org/10.1186/s13661-019-1242-9 doi: 10.1186/s13661-019-1242-9
|
| [9] |
C. Wang, S. Yuan, H. Wang, Spatiotemporal patterns of a diffusive prey-predator model with spatial memory and pregnancy period in an intimidatory environment, J. Math. Biol., 84 (2022), 12. https://doi.org/10.1007/s00285-022-01716-4 doi: 10.1007/s00285-022-01716-4
|
| [10] |
R. J. Han, B. X. Dai, Spatiotemporal pattern formation and selection induced by nonlinear cross-diffusion in a toxic-phytoplankton-zooplankton model with Allee effect, Nonlinear Anal.-Real, 45 (2019), 822–853. https://doi.org/10.1016/j.nonrwa.2018.05.018 doi: 10.1016/j.nonrwa.2018.05.018
|
| [11] |
X. Y. Meng, L. Xiao, Hopf bifurcation and Turing instability of a delayed diffusive Zooplankton-Phytoplankton model with hunting cooperation, Int. J. Bifurcat. Chaos, 34 (2024), 2540090. https://doi.org/10.1142/S0218127424500901 doi: 10.1142/S0218127424500901
|
| [12] |
X. Tao, L. Zhu, Study of periodic diffusion and time delay induced spatiotemporal patterns in a predator-prey system, Chaos Soliton. Fract., 150 (2021), 111101. https://doi.org/10.1016/j.chaos.2021.111101 doi: 10.1016/j.chaos.2021.111101
|
| [13] |
Z. W. Liang, X. Y. Meng, Stability and Hopf bifurcation of a multiple delayed predator-prey system with fear effect, prey refuge and Crowley-Martin function, Chaos Soliton. Fract., 175 (2023), 113955. https://doi.org/10.1016/j.chaos.2023.113955 doi: 10.1016/j.chaos.2023.113955
|
| [14] |
G. Sun, Z. Jin, L. Li, M. Haque, B. Li, Spatial patterns of a predator-prey model with cross diffusion, Nonlinear Dyn., 69 (2012), 1631–1638. https://doi.org/10.1007/s11071-012-0374-6 doi: 10.1007/s11071-012-0374-6
|
| [15] |
G. Sun, Z. Jin, Y. Zhao, Q. Liu, L. Li, Spatial pattern in a predator-prey system with both self- and cross-diffusion, Int. J. Mod. Phys. C, 20 (2009), 71–84. https://doi.org/10.1142/S0129183109013467 doi: 10.1142/S0129183109013467
|
| [16] |
H. Yang, Z. M. Zhong, Turing instability of the periodic solutions for a predator-prey model with Allee effect and cross-diffusion, Adv. Cont. Discr. Mod., 2025 (2025), 53. https://doi.org/10.1186/s13662-025-03893-0 doi: 10.1186/s13662-025-03893-0
|
| [17] |
C. Y. Wang, S. Y. Qi, Spatial dynamics of a predator-prey system with cross diffusion, Chaos Soliton. Fract., 107 (2018), 55–60. https://doi.org/10.1016/j.chaos.2017.12.020 doi: 10.1016/j.chaos.2017.12.020
|
| [18] |
D. X. Song, Y. L. Song, C. Li, Stability and Turing patterns in a predator-prey model with hunting cooperation and Allee effect in prey population, Int. J. Bifurcat. Chaos, 30 (2020), 2050137. https://doi.org/10.1142/S0218127420501370 doi: 10.1142/S0218127420501370
|
| [19] |
Y. H. Peng, H. Y. Ling, Pattern formation in a ratio-dependent predator-prey model with cross-diffusion, Appl. Math. Comput., 331 (2018), 307–318. https://doi.org/10.1016/j.amc.2018.03.033 doi: 10.1016/j.amc.2018.03.033
|
| [20] |
D. X. Jia, T. H. Zhang, S. L. Yuan, Pattern dynamics of a diffusive toxin producing phytoplankton-zooplankton model with three-dimensional patch, Int. J. Bifurcat. Chaos, 29 (2019), 1930011. https://doi.org/10.1142/S0218127419300118 doi: 10.1142/S0218127419300118
|
| [21] |
X. Z. Lian, S. L. Yan, H. L. Wang, Pattern formation in predator-prey model with delay and cross diffusion, Abstr. Appl. Anal., 2013 (2013), 147232. https://doi.org/10.1155/2013/147232 doi: 10.1155/2013/147232
|
| [22] |
W. J. Li, L. T. Zhang, J. D. Cao, A note on Turing-Hopf bifurcation in a diffusive Leslie-Gower model with weak Allee effect on prey and fear effect on predator, Appl. Math. Lett., 172 (2026), 109741. https://doi.org/10.1016/j.am1.2025.109741 doi: 10.1016/j.am1.2025.109741
|
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Ping Li, Haisong Cao, Can Chen, Jianfeng Jiao. Amplitude equation for the Hopf bifurcation of a reaction-diffusion predator-prey system incorporating time delay[J]. AIMS Mathematics, 2026, 11(1): 2595-2612. doi: 10.3934/math.2026105
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