Dynamical analysis of a Filippov pest-natural enemy model incorporating fear and prey refuge

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Abstract

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.

Keywords

threshold strategy / Filippov system / sliding mode / fear effect / prey refuge

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References

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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