Dynamical behavior of a host–commensal system with multiplicative Allee effect and nonselective harvesting

Jinbo Zhang; Qin Yue; Jinbo Zhang; Qin Yue

doi:10.3934/era.2026280

Electronic Research Archive

2026, Volume 34, Issue 9: 6404-6431. doi: 10.3934/era.2026280

Previous Article Next Article

Research article Special Issues

Dynamical behavior of a host–commensal system with multiplicative Allee effect and nonselective harvesting

Jinbo Zhang ,
Qin Yue ^,

School of Finance and Mathematics, West Anhui University, Lu'an 237000, China

Received: 28 May 2026 Revised: 01 July 2026 Accepted: 13 July 2026 Published: 20 July 2026

This paper studies a host–commensal model with a multiplicative Allee effect on the host and nonselective harvesting on both species. Exploiting the upper-triangular structure, equilibrium and stability analyses reduce to one-dimensional problems. Three harvesting thresholds corresponding to a fold and two transcritical bifurcations yield a complete classification of at most six equilibria. Global dynamics is determined via asymptotically autonomous systems: The separatrix $ y = y_1 $ divides coexistence and extinction basins at low harvesting, and the system collapses to monostability beyond the fold. Numerical simulations verify the three qualitatively distinct dynamical regimes.
- host–commensal model,
- multiplicative Allee effect,
- nonselective harvesting,
- upper-triangular structure,
- global dynamics,
- bistability,
- fold bifurcation,
- transcritical bifurcation
Citation: Jinbo Zhang, Qin Yue. Dynamical behavior of a host–commensal system with multiplicative Allee effect and nonselective harvesting[J]. Electronic Research Archive, 2026, 34(9): 6404-6431. doi: 10.3934/era.2026280

Related Papers:

Abstract

This paper studies a host–commensal model with a multiplicative Allee effect on the host and nonselective harvesting on both species. Exploiting the upper-triangular structure, equilibrium and stability analyses reduce to one-dimensional problems. Three harvesting thresholds corresponding to a fold and two transcritical bifurcations yield a complete classification of at most six equilibria. Global dynamics is determined via asymptotically autonomous systems: The separatrix $ y = y_1 $ divides coexistence and extinction basins at low harvesting, and the system collapses to monostability beyond the fold. Numerical simulations verify the three qualitatively distinct dynamical regimes.

References

[1]	P. Quansah, M. Ibrahim, Global dynamics of a ratio-dependent commensalism model with nonselective harvesting and partial closure for the populations, J. Math. Comput. Sci., 16 (2026), 2. https://doi.org/10.28919/jmcs/9762 doi: 10.28919/jmcs/9762
[2]	R. Fawthrop, J. Cerca, G. Pacheco, G. Satre, E. S. C. Scordato, M. Ravinet, et al., Understanding human-commensalism through an ecological and evolutionary framework, Trends Ecol. Evol., 40 (2025), 159–169. https://doi.org/10.1016/j.tree.2024.10.006 doi: 10.1016/j.tree.2024.10.006
[3]	A. Mougi, The roles of amensalistic and commensalistic interactions in large ecological network stability, Sci. Rep., 6 (2016), 29929. https://doi.org/10.1038/srep29929 doi: 10.1038/srep29929
[4]	T. Li, Q. Wang, Bifurcation analysis for two-species commensalism (amensalism) systems with distributed delays, Int. J. Bifurcation Chaos, 32 (2022), 2250133. https://doi.org/10.1142/S0218127422501334 doi: 10.1142/S0218127422501334
[5]	Y. Chong, A. J. Kashyap, S. Chen, F. Chen, Dynamics analysis of a discrete-time commensalism model with additive Allee effect for the host species, Axioms, 12 (2023), 1031. https://doi.org/10.3390/axioms12111031 doi: 10.3390/axioms12111031
[6]	E. C. D. Stewart, N. Jimi, C. Moreau, T. G. Dahlgren, A. G. Glover, Novel deep-sea commensalism: a new genus and two new species of Myzostomida from the abyssal Pacific Ocean, Invertebr. Syst., 40 (2026), IS25078. https://doi.org/10.1071/IS25078 doi: 10.1071/IS25078
[7]	K. A. Mathis, J. L. Bronstein, Our current understanding of commensalism, Annu. Rev. Ecol. Evol. Syst., 51 (2020), 167–189. https://doi.org/10.1146/annurev-ecolsys-011720-040844 doi: 10.1146/annurev-ecolsys-011720-040844
[8]	C. R. Dickman, Body size, prey size, and community structure in insectivorous mammals, Ecology, 69 (1988), 569–580. https://doi.org/10.2307/1941006 doi: 10.2307/1941006
[9]	X. He, Z. Zhu, J. Chen, F. Chen, Dynamical analysis of a Lotka–Volterra commensalism model with additive Allee effect, Open Math., 20 (2022), 646–665. https://doi.org/10.1515/math-2022-0055 doi: 10.1515/math-2022-0055
[10]	L. Xu, Y. Xue, Q. Lin, C. Lei, Global attractivity of symbiotic model of commensalism in four populations with Michaelis–Menten type harvesting in the first commensal populations, Axioms, 11 (2022), 337. https://doi.org/10.3390/axioms11070337 doi: 10.3390/axioms11070337
[11]	P. Lemes, F. G. Barbosa, B. Naimi, M. B. Araújo, Dispersal abilities favor commensalism in animal–plant interactions under climate change, Sci. Total Environ., 835 (2022), 155157. https://doi.org/10.1016/j.scitotenv.2022.155157 doi: 10.1016/j.scitotenv.2022.155157
[12]	R. Wu, L. Li, Dynamic behaviors of a commensal symbiosis model with ratio-dependent functional response and one party cannot survive independently, J. Math. Comput. Sci., 16 (2016), 495–506. https://doi.org/10.22436/jmcs.016.04.03 doi: 10.22436/jmcs.016.04.03
[13]	W. C. Allee, Animal aggregations, Q. Rev. Biol., 2 (1927), 367–398. https://doi.org/10.1086/394281 doi: 10.1086/394281
[14]	F. Courchamp, T. Clutton-Brock, B. Grenfell, Inverse density dependence and the Allee effect, Trends Ecol. Evol., 14 (1999), 405–410. https://doi.org/10.1016/S0169-5347(99)01683-3 doi: 10.1016/S0169-5347(99)01683-3
[15]	P. A. Stephens, W. J. Sutherland, Consequences of the Allee effect for behaviour, ecology and conservation, Trends Ecol. Evol., 14 (1999), 401–405. https://doi.org/10.1016/S0169-5347(99)01684-5 doi: 10.1016/S0169-5347(99)01684-5
[16]	K. Fang, Y. Wang, F. Chen, X. Chen, Dynamics of a modified Lotka–Volterra commensalism system incorporating Allee effect and symmetric nonselective harvest, Symmetry, 17 (2025), 852. https://doi.org/10.3390/sym17060852 doi: 10.3390/sym17060852
[17]	J. Zhong, L. Chen, F. Chen, Stability and bifurcation in a two-patch commensal symbiosis model with nonlinear dispersal and additive Allee effect, Int. J. Biomath., 19 (2026), 2450099. https://doi.org/10.1142/S1793524524500992 doi: 10.1142/S1793524524500992
[18]	M. B. Almatrafi, Stability and period-doubling bifurcation of fractional-order commensal symbiosis model with Allee effect, Fractal Fract., 10 (2026), 226. https://doi.org/10.3390/fractalfract10040226 doi: 10.3390/fractalfract10040226
[19]	S. Işık, Stability and period-doubling bifurcation in a modified commensal symbiosis model with Allee effect, Erzincan Univ. J. Sci. Technol., 15 (2022), 310–324. https://doi.org/10.18185/erzifbed.879963 doi: 10.18185/erzifbed.879963
[20]	A. Tassaddiq, R. Ahmed, A. Ditta, Exploring stability and bifurcation in a discretized commensalism model, Nonlinear Dyn., 113 (2025), 28463–28475. https://doi.org/10.1007/s11071-025-11530-4 doi: 10.1007/s11071-025-11530-4
[21]	P. Georgescu, D. Maxin, H. Zhang, Global stability results for models of commensalism, Int. J. Biomath., 10 (2017), 1750037. https://doi.org/10.1142/S1793524517500371 doi: 10.1142/S1793524517500371
[22]	F. Courchamp, L. Berec, J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, 2008. https://doi.org/10.1093/acprof: oso/9780198570301.001.0001
[23]	L. Berec, E. Angulo, F. Courchamp, Multiple Allee effects and population management, Trends Ecol. Evol., 22 (2007), 185–191. https://doi.org/10.1016/j.tree.2006.12.002 doi: 10.1016/j.tree.2006.12.002
[24]	L. Zhao, J. Shen, Canards and homoclinic orbits in a slow-fast modified May–Holling–Tanner predator-prey model with weak multiple Allee effect, Discrete Contin. Dyn. Syst. - Ser. B, 27 (2022), 6745–6769. https://doi.org/10.3934/dcdsb.2022018 doi: 10.3934/dcdsb.2022018
[25]	F. Kangalgil, M. Altunkaynak, Chaos control and bifurcations in a modified discrete prey–predator model with weak multiple Allee effect, Int. J. Bifurcation Chaos, 35 (2025), 2550124. https://doi.org/10.1142/S021812742550124X doi: 10.1142/S021812742550124X
[26]	J. Xiao, Y. Xia, Spatiotemporal dynamics in a diffusive predator-prey model with multiple Allee effect and herd behavior, J. Math. Anal. Appl., 529 (2024), 127569. https://doi.org/10.1016/j.jmaa.2023.127569 doi: 10.1016/j.jmaa.2023.127569
[27]	C. W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources, Wiley, 1990.
[28]	A. Ditta, P. A. Naik, R. Ahmed, Z. Huang, Exploring periodic behavior and dynamical analysis in a harvested discrete-time commensalism system, Int. J. Dyn. Control, 13 (2025), 63. https://doi.org/10.1007/s40435-024-01551-z doi: 10.1007/s40435-024-01551-z
[29]	S. Jawad, Study the dynamics of commensalism interaction with Michaelis–Menten type prey harvesting, Al-Nahrain J. Sci., 25 (2022), 45–50. https://doi.org/10.22401/ANJS.25.1.08 doi: 10.22401/ANJS.25.1.08
[30]	C. Lu, X. Liu, Z. Li, The dynamics and harvesting strategies of a predator-prey system with Allee effect on prey, AIMS Math., 8 (2023), 28897–28925. https://doi.org/10.3934/math.20231481 doi: 10.3934/math.20231481
[31]	H. Liu, H. Yu, C. Dai, Z. Ma, Q. Wang, M. Zhao, Dynamical analysis of an aquatic amensalism model with nonselective harvesting and Allee effect, Math. Biosci. Eng., 18 (2021), 8857–8882. https://doi.org/10.3934/mbe.2021437 doi: 10.3934/mbe.2021437
[32]	N. Min, P. Xing, H. Zhang, Role of Allee effect, hunting cooperation, and prey harvesting in a Leslie–Gower predator–prey model, Adv. Contin. Discrete Models, 2025 (2025), 96. https://doi.org/10.1186/s13662-025-03951-7 doi: 10.1186/s13662-025-03951-7
[33]	P. Sardar, S. Biswas, K. P. Das, S. Rana, B. Pal, F. Ali, et al., Exploring multiple bifurcation behaviors and chaos control via weak Allee effect and harvesting in a predator–prey interaction model, Nonlinear Sci., 4 (2025), 100033. https://doi.org/10.1016/j.nls.2025.100033 doi: 10.1016/j.nls.2025.100033
[34]	H. R. Thieme, Convergence results and a Poincaré–Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755–763. https://doi.org/10.1007/BF00173267 doi: 10.1007/BF00173267
[35]	K. Mischaikow, H. L. Smith, H. R. Thieme, Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions, Trans. Am. Math. Soc., 347 (1995), 1669–1685. https://doi.org/10.1090/S0002-9947-1995-1290727-7 doi: 10.1090/S0002-9947-1995-1290727-7
[36]	F. Chen, Y. Chen, Z. Li, L. Chen, Note on the persistence and stability property of a commensalism model with Michaelis–Menten harvesting and Holling type Ⅱ commensalistic benefit, Appl. Math. Lett., 134 (2022), 108381. https://doi.org/10.1016/j.aml.2022.108381 doi: 10.1016/j.aml.2022.108381
[37]	M. W. Hirsch, Systems of differential equations which are competitive or cooperative. Ⅰ: Limit sets, SIAM J. Math. Anal., 13 (1982), 167–179. https://doi.org/10.1137/0513013 doi: 10.1137/0513013
[38]	H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, 1995.
[39]	J. Sotomayor, Generic bifurcations of dynamical systems, in Dynamical Systems, (1973), 561–582. https://doi.org/10.1016/B978-0-12-550350-1.50047-3
[40]	G. Zhang, Y. Shen, B. Chen, Positive periodic solutions in a nonselective harvesting predator–prey model with multiple delays, J. Math. Anal. Appl., 395 (2012), 298–306. https://doi.org/10.1016/j.jmaa.2012.05.045 doi: 10.1016/j.jmaa.2012.05.045
[41]	G. Zhang, H. Guo, L. Wang, Exploring bifurcations in a differential-algebraic model of predator–prey interactions, Nonlinear Dyn., 112 (2024), 20549–20571. https://doi.org/10.1007/s11071-024-10098-9 doi: 10.1007/s11071-024-10098-9

Reader Comments

Your name:*

Email:*
© 2026 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)