This paper establishes a Filippov integrated pest management model with the fear effect and prey refuge, aiming to investigate economic-threshold-based intermittent control strategies and their dynamical behavior. First, the existence and local stability of the equilibria of the two subsystems are discussed, and the nonsmooth dynamics of the Filippov system are analyzed. Furthermore, sliding bifurcations, boundary equilibrium bifurcations, and global sliding bifurcations are investigated through numerical simulations. The results indicate that the control effect is highly sensitive to the threshold value, and as the threshold varies, the system exhibits a series of complex bifurcation phenomena. Furthermore, pest refuge behavior may weaken the effectiveness of control measures. Although the fear effect can suppress pest growth, an excessively strong fear effect may reduce prey availability for natural enemies, thereby limiting their population growth.
Citation: Wen Sun, Zhongyi Xiang. Dynamical analysis of a Filippov pest-natural enemy model incorporating fear and prey refuge[J]. Electronic Research Archive, 2026, 34(10): 7362-7391. doi: 10.3934/era.2026318
This paper establishes a Filippov integrated pest management model with the fear effect and prey refuge, aiming to investigate economic-threshold-based intermittent control strategies and their dynamical behavior. First, the existence and local stability of the equilibria of the two subsystems are discussed, and the nonsmooth dynamics of the Filippov system are analyzed. Furthermore, sliding bifurcations, boundary equilibrium bifurcations, and global sliding bifurcations are investigated through numerical simulations. The results indicate that the control effect is highly sensitive to the threshold value, and as the threshold varies, the system exhibits a series of complex bifurcation phenomena. Furthermore, pest refuge behavior may weaken the effectiveness of control measures. Although the fear effect can suppress pest growth, an excessively strong fear effect may reduce prey availability for natural enemies, thereby limiting their population growth.
| [1] |
N. Y. Su, R. H. Scheffrahn, A review of subterranean termite control practices and prospects for integrated pest management programmes, Integr. Pest Manag. Rev., 3 (1998), 1–13. https://doi.org/10.1023/A:1009684821954 doi: 10.1023/A:1009684821954
|
| [2] |
J. Pretty, Z. P. Bharucha, Integrated pest management for sustainable intensification of agriculture in Asia and Africa, Insects, 6 (2015), 152–182. https://doi.org/10.3390/insects6010152 doi: 10.3390/insects6010152
|
| [3] |
M. Kogan, Integrated pest management: Historical perspectives and contemporary developments, Annu. Rev. Entomol., 43 (1998), 243–270. https://doi.org/10.1146/annurev.ento.43.1.243 doi: 10.1146/annurev.ento.43.1.243
|
| [4] |
P. B. Angon, S. Mondal, I. Jahan, M. Datto, U. B. Antu, F. J. Ayshi, et al., Integrated pest management (ipm) in agriculture and its role in maintaining ecological balance and biodiversity, Adv. Agric., 2023 (2023), 5546373. https://doi.org/10.1155/2023/5546373 doi: 10.1155/2023/5546373
|
| [5] |
C. Cortés-García, Population dynamics of a Filippov Gause predator-prey model with or without competition among predators with respect to prey density, Nonlinear Dyn., 113 (2025), 30161–30185. https://doi.org/10.1007/s11071-025-11600-7 doi: 10.1007/s11071-025-11600-7
|
| [6] |
C. Cortés-García, Dynamics in a Filippov-Gause predator-prey model with hunting cooperation or competition among predators for high or low prey densities, Int. J. Bifurcation Chaos, 35 (2025), 2550014. https://doi.org/10.1142/S0218127425500142 doi: 10.1142/S0218127425500142
|
| [7] |
C. Cortés-García, Population dynamics in a Leslie-Gower predator-prey model with predator harvesting at high densities, Math. Methods Appl. Sci., 48 (2025), 804–838. https://doi.org/10.1002/mma.10359 doi: 10.1002/mma.10359
|
| [8] |
W. Qin, S. Tang, C. Xiang, Y. Yang, Effects of limited medical resource on a Filippov infectious disease model induced by selection pressure, Appl. Math. Comput., 283 (2016), 339–354. https://doi.org/10.1016/j.amc.2016.02.042 doi: 10.1016/j.amc.2016.02.042
|
| [9] |
Y. Zhang, Y. Xiao, Global dynamics for a Filippov epidemic system with imperfect vaccination, Nonlinear Anal. Hybrid Syst., 38 (2020), 100932. https://doi.org/10.1016/j.nahs.2020.100932 doi: 10.1016/j.nahs.2020.100932
|
| [10] |
A. Wang, Y. Xiao, A Filippov system describing media effects on the spread of infectious diseases, Nonlinear Anal. Hybrid Syst., 11 (2014), 84–97. https://doi.org/10.1016/j.nahs.2013.06.005 doi: 10.1016/j.nahs.2013.06.005
|
| [11] |
X. Zhang, Z. Xiang, Impact of combined effects of vectors, protective measures, and vaccination under threshold policy control on the SISV model, Chaos, 35 (2025), 053117. https://doi.org/10.1063/5.0256966 doi: 10.1063/5.0256966
|
| [12] |
X. Jiao, X. Liu, Rich dynamics of a delayed Filippov avian-only influenza model with two-thresholds policy, Chaos Solitons Fractals, 182 (2024), 114710. https://doi.org/10.1016/j.chaos.2024.114710 doi: 10.1016/j.chaos.2024.114710
|
| [13] |
W. Li, Bifurcation analysis of a filippov predator-prey model with two thresholds, Nonlinear Dyn., 112 (2024), 9639–9656. https://doi.org/10.1007/s11071-024-09527-6 doi: 10.1007/s11071-024-09527-6
|
| [14] |
M. Hu, C. Xiang, B. Chu, Dynamic behaviors and bifurcation analysis of a three-dimensional filippov ecosystem with fear effect, Math. Methods Appl. Sci., 48 (2025), 6607–6623. https://doi.org/10.1002/mma.10699 doi: 10.1002/mma.10699
|
| [15] |
X. Zhang, X. Zhang, Complex dynamics analysis of a non-smooth predator-prey model with stage structure and threshold-dependent refuge mechanism, Nonlinear Dyn., 114 (2026), 533. https://doi.org/10.1007/s11071-026-12405-y doi: 10.1007/s11071-026-12405-y
|
| [16] |
Z. Xiang, X. Zhang, Application of a Filippov model incorporating the Allee effect and proportional release of sterile mosquitoes for dengue mosquito population control, Commun. Nonlinear Sci. Numer. Simul., 154 (2026), 109590. https://doi.org/10.1016/j.cnsns.2025.109590 doi: 10.1016/j.cnsns.2025.109590
|
| [17] |
D. M. Fawzy, A. Elsaid, W. Zahra, A. A. Arafa, Qualitative analysis of a Filippov wild-sterile mosquito population model with immigration, Chaos, 33 (2023), 113101. https://doi.org/10.1063/5.0167157 doi: 10.1063/5.0167157
|
| [18] |
H. Zhou, Q. Zhang, S. Tang, Qualitative analysis of the sliding vector field in a Filippov food chain model with integrated pest management strategy, Appl. Math. Comput., 490 (2025), 129188. https://doi.org/10.1016/j.amc.2024.129188 doi: 10.1016/j.amc.2024.129188
|
| [19] |
Y. Tian, X. Tian, X. Yan, J. Zheng, K. Sun, Complex dynamics of non-smooth pest-natural enemy Gomportz models with a variable searching rate based on threshold control, Electron. Res. Arch., 33 (2025), 26–49. https://doi.org/10.3934/era.2025002 doi: 10.3934/era.2025002
|
| [20] |
Y. Tian, H. Guo, W. Shen, X. Yan, J. Zheng, K. Sun, Dynamic analysis and validation of a prey-predator model based on fish harvesting and discontinuous prey refuge effect in uncertain environments, Electron. Res. Arch., 33 (2025), 973–994. https://doi.org/10.3934/era.2025044 doi: 10.3934/era.2025044
|
| [21] |
H. Zhou, X. Wang, S. Tang, Global dynamics of non-smooth filippov pest-natural enemy system with constant releasing rate, Math. Biosci. Eng., 16 (2019), 7327–7361. https://doi.org/10.3934/mbe.2019366 doi: 10.3934/mbe.2019366
|
| [22] |
W. Qin, X. Tan, M. Tosato, X. Liu Threshold control strategy for a non-smooth Filippov ecosystem with group defense, Appl. Math. Comput., 362 (2019), 124532. https://doi.org/10.1016/j.amc.2019.06.046 doi: 10.1016/j.amc.2019.06.046
|
| [23] |
B. Kang, X. Hou, B. Liu, Threshold control strategy for a Filippov model with group defense of pests and a constant-rate release of natural enemies, Math. Biosci. Eng., 20 (2023), 12076–12092. https://doi.org/10.3934/mbe.2023537 doi: 10.3934/mbe.2023537
|
| [24] |
A. A. Arafa, S. A. Hamdallah, S. Tang, Y. Xu, G. M. Mahmoud, Dynamics analysis of a filippov pest control model with time delay, Commun. Nonlinear Sci. Numer. Simul., 101 (2021), 105865. https://doi.org/10.1016/j.cnsns.2021.105865 doi: 10.1016/j.cnsns.2021.105865
|
| [25] |
L. Y. Zanette, A. F. White, M. C. Allen, M. Clinchy, Perceived predation risk reduces the number of offspring songbirds produce per year, Science, 334 (2011), 1398–1401. https://doi.org/10.1126/science.1210908 doi: 10.1126/science.1210908
|
| [26] |
X. Wang, L. Zanette, X. Zou, Modelling the fear effect in predator-prey interactions, J. Math. Biol., 73 (2016), 1179–1204. https://doi.org/10.1007/s00285-016-0989-1 doi: 10.1007/s00285-016-0989-1
|
| [27] |
M. A. Abbasi, M. Samreen, Analyzing multi-parameter bifurcation on a prey-predator model with the allee effect and fear effect, Chaos Solitons Fractals, 180 (2024), 114498. https://doi.org/10.1016/j.chaos.2024.114498 doi: 10.1016/j.chaos.2024.114498
|
| [28] |
Y. Li, M. He, Z. Li, Dynamics of a ratio-dependent Leslie-Gower predator-prey model with allee effect and fear effect, Math. Comput. Simul., 201 (2022), 417–439. https://doi.org/10.1016/j.matcom.2022.05.017 doi: 10.1016/j.matcom.2022.05.017
|
| [29] |
H. Qi, B. Liu, Stationary distribution of a stochastic reaction-diffusion predator-prey model with additional food and fear effect, Appl. Math. Lett., 150 (2024), 108978. https://doi.org/10.1016/j.aml.2023.108978 doi: 10.1016/j.aml.2023.108978
|
| [30] |
H. Wang, Y. Zhang, L. Ma, Bifurcation and stability of a diffusive predator-prey model with the fear effect and time delay, Chaos, 33 (2023), 073137. https://doi.org/10.1063/5.0157410 doi: 10.1063/5.0157410
|
| [31] |
O. Akman, T. Comar, M. Henderson, An analysis of an impulsive stage structured integrated pest management model with refuge effect, Chaos Solitons Fractals, 111 (2018), 44–54. https://doi.org/10.1016/j.chaos.2018.03.039 doi: 10.1016/j.chaos.2018.03.039
|
| [32] |
Y. Cai, Q. Chen, Z. Teng, G. Zhang, R. Rifhat, Bifurcations in a discrete-time Beddington-DeAngelis prey-predator model with fear effect, prey refuge and harvesting, Nonlinear Dyn., 113 (2025), 931–969. https://doi.org/10.1007/s11071-024-10232-7 doi: 10.1007/s11071-024-10232-7
|
| [33] |
T. K. Kar, Stability analysis of a prey-predator model incorporating a prey refuge, Commun. Nonlinear Sci. Numer. Simul., 10 (2005), 681–691. https://doi.org/10.1016/j.cnsns.2003.08.006 doi: 10.1016/j.cnsns.2003.08.006
|
| [34] |
W. Ko, K. Ryu, Qualitative analysis of a predator-prey model with Holling type ii functional response incorporating a prey refuge, J. Differ. Equations, 231 (2006), 534–550. https://doi.org/10.1016/j.jde.2006.08.001 doi: 10.1016/j.jde.2006.08.001
|
| [35] |
M. Venzon, A. Janssen, A. Pallini, M. W. Sabelis, Diet of a polyphagous arthropod predator affects refuge seeking of its thrips prey, Anim. Behav., 60 (2000), 369–375. https://doi.org/10.1006/anbe.2000.1483 doi: 10.1006/anbe.2000.1483
|
| [36] |
S. Magalhaes, P. C. Van Rijn, M. Montserrat, A. Pallini, M. W. Sabelis, Population dynamics of thrips prey and their mite predators in a refuge, Oecologia, 150 (2007), 557–568. https://doi.org/10.1007/s00442-006-0548-3 doi: 10.1007/s00442-006-0548-3
|
| [37] |
K. Mokni, M. Ch-Chaoui, Strong Allee effect and evolutionary dynamics in a single-species Ricker population model, J. Biol. Syst., 31 (2023), 1341–1370. https://doi.org/10.1142/S0218339023500456 doi: 10.1142/S0218339023500456
|
| [38] |
K. Mokni, M. Ch-Chaoui, Exploring persistence, stability, and bifurcations: A Darwinian Ricker-Cushing model, Int. J. Dyn. Control, 13 (2025), 34. https://doi.org/10.1007/s40435-024-01539-9 doi: 10.1007/s40435-024-01539-9
|
| [39] |
M. Ahmidi, K. Mokni, M. Ch-Chaoui, Influence of prey harvesting on the dynamics of a prey-predator system: Multi-parameter bifurcation analysis, Nonlinear Dyn., 113 (2025), 31841–31870. https://doi.org/10.1007/s11071-025-11701-3 doi: 10.1007/s11071-025-11701-3
|
| [40] |
H. Benali, K. Mokni, H. Mouhsine, M. Ch-Chaoui, Unveiling complexity: A discrete-time prey-predator model with immigration effects, Iran J. Sci., 49 (2025), 449–462. https://doi.org/10.1007/s40995-024-01742-5 doi: 10.1007/s40995-024-01742-5
|
| [41] |
H. Zhang, Y. Cai, S. Fu, W. Wang, Impact of the fear effect in a prey-predator model incorporating a prey refuge, Appl. Math. Comput., 356 (2019), 328–337. https://doi.org/10.1016/j.amc.2019.03.034 doi: 10.1016/j.amc.2019.03.034
|
| [42] |
S. A. Hamdallah, A. A. Arafa, Stability analysis of Filippov prey-predator model with fear effect and prey refuge, J. Appl. Math. Comput., 70 (2024), 73–102. https://doi.org/10.1007/s12190-023-01934-z doi: 10.1007/s12190-023-01934-z
|
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Wen Sun, Zhongyi Xiang. Dynamical analysis of a Filippov pest-natural enemy model incorporating fear and prey refuge[J]. Electronic Research Archive, 2026, 34(10): 7362-7391. doi: 10.3934/era.2026318
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