A food chain model with Allee effect: Analysis on the behaviors of equilibria

Huda Abdul Satar; Raid Kamel Naji; Mainul Haque; Huda Abdul Satar; Raid Kamel Naji; Mainul Haque

doi:10.3934/math.2025568

AIMS Mathematics

2025, Volume 10, Issue 5: 12598-12618. doi: 10.3934/math.2025568

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Research article

A food chain model with Allee effect: Analysis on the behaviors of equilibria

1.
Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq
2.
Scientific Research Commission, Baghdad, Iraq
3.
School of Mathematical Science, The University of Nottingham Ningbo China, Ningbo, China

Received: 16 March 2025 Revised: 19 May 2025 Accepted: 23 May 2025 Published: 29 May 2025
MSC : 92D25, 34L30, 92D40

The Allee effect is crucial to population ecology and conservation biology because it clarifies how difficult it can be for tiny populations to endure and expand. Low population densities make it more difficult for individuals to survive or reproduce in a phenomenon. Therefore, it is important to study the ecological system, including the Allee effect. Accordingly, our goal in this paper was to examine equilibria's behaviors in a three-species food chain model incorporating the Allee effect. This model includes a linear type of functional response. The points of equilibrium are categorized and depicted. The behaviors of these equilibrium points are then illustrated analytically through stability and bifurcation analyses. Moreover, the numerical simulation utilizes realistic hypothetical data to confirm the analytical results and detect the influence of varying the parameters on the system's dynamics. It is observed that the system undergoes bistable behavior; otherwise, the trivial equilibrium point is globally asymptotically stable. Under specific circumstances, the food chain system experiences a transcritical bifurcation around the axial and border equilibrium points. However, under some conditions, a Hopf bifurcation happens around the border equilibrium point.
- Allee effect,
- food chain,
- stability,
- bifurcation
Citation: Huda Abdul Satar, Raid Kamel Naji, Mainul Haque. A food chain model with Allee effect: Analysis on the behaviors of equilibria[J]. AIMS Mathematics, 2025, 10(5): 12598-12618. doi: 10.3934/math.2025568

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Abstract

The Allee effect is crucial to population ecology and conservation biology because it clarifies how difficult it can be for tiny populations to endure and expand. Low population densities make it more difficult for individuals to survive or reproduce in a phenomenon. Therefore, it is important to study the ecological system, including the Allee effect. Accordingly, our goal in this paper was to examine equilibria's behaviors in a three-species food chain model incorporating the Allee effect. This model includes a linear type of functional response. The points of equilibrium are categorized and depicted. The behaviors of these equilibrium points are then illustrated analytically through stability and bifurcation analyses. Moreover, the numerical simulation utilizes realistic hypothetical data to confirm the analytical results and detect the influence of varying the parameters on the system's dynamics. It is observed that the system undergoes bistable behavior; otherwise, the trivial equilibrium point is globally asymptotically stable. Under specific circumstances, the food chain system experiences a transcritical bifurcation around the axial and border equilibrium points. However, under some conditions, a Hopf bifurcation happens around the border equilibrium point.

References

[1]	M. Agarwal, V. Singh, Rich dynamics of a food chain model with ratio-dependent type Ⅲ functional responses, Int. J. Eng. Sci. Technol., 5 (2013), 124–141. http://dx.doi.org/10.4314/ijest.v5i3.10 doi: 10.4314/ijest.v5i3.10
[2]	N. Ali, M. Haque, E. Venturino, S. Chakravarty, Dynamics of a three-species ratio-dependent food chain model with intra-specific competition within the top predator, Comput. Biol. Med., 85 (2017), 63–74. https://doi.org/10.1016/j.compbiomed.2017.04.007 doi: 10.1016/j.compbiomed.2017.04.007
[3]	S. A. Momen, R. K. Naji, The dynamics of Sokol-Howell prey-predator model involving strong Allee effect, Iraqi J. Sci., 62 (2021), 3114–3127. https://doi.org/10.24996/ijs.2021.62.9.27 doi: 10.24996/ijs.2021.62.9.27
[4]	S. A. Momen, R. K. Naji, The dynamics of modified Leslie-Gower predator-prey model under the influence of nonlinear harvesting and fear effect, Iraqi J. Sci., 63 (2022), 259–282. https://doi.org/10.24996/ijs.2022.63.1.27 doi: 10.24996/ijs.2022.63.1.27
[5]	W. M. Alwan, H. A. Satar, The effects of media coverage on the dynamics of disease in prey-predator model, Iraqi J. Sci., 62 (2021), 981–996. https://doi.org/10.24996/ijs.2021.62.3.28 doi: 10.24996/ijs.2021.62.3.28
[6]	P. Cong, M. Fan, X. Zou, Dynamics of a three-species food chain model with fear effect, Commun. Nonlinear Sci., 99 (2021), 105809. https://doi.org/10.1016/j.cnsns.2021.105809 doi: 10.1016/j.cnsns.2021.105809
[7]	S. Debnath, U. Ghosh, S. Sarkar, Global dynamics of a tritrophic food chain model subject to the Allee effects in the prey population with sexually reproductive generalised type top predator, Comput. Math. Methods, 2 (2020), e1079. https://doi.org/10.1002/cmm4.1079 doi: 10.1002/cmm4.1079
[8]	S. Debnath, P. Majumdar, S. Sarkar, U. Ghosh, Chaotic dynamics of a tri-topic food chain model with Beddington-DeAngelis functional response in presence of fear effect, Nonlinear Dynam., 106 (2021), 2621–2653. https://doi.org/10.1007/s11071-021-06896-0 doi: 10.1007/s11071-021-06896-0
[9]	S. Gakkhar, R. K. Naji, Order and chaos in predator to prey ratio-dependent food chain, Chaos Soliton. Fract., 18 (2003), 229–239. https://doi.org/10.1016/S0960-0779(02)00642-2 doi: 10.1016/S0960-0779(02)00642-2
[10]	Z. M. Hadi, D. K. Bahlool, The effect of alternative resource and refuge on the dynamical behavior of food chain model, Malays. J. Math. Sci., 17 (2023), 731–754. https://doi.org/10.47836/MJMS.17.4.13 doi: 10.47836/MJMS.17.4.13
[11]	J. Hale, Ordinary differential equations, Wiley-Interscience, 1969.
[12]	M. Haque, N. Ali, S. Chakravarty, Study of a tri-trophic prey-dependent food chain model of interacting populations, Math. Biosci., 246 (2013), 55–71. https://doi.org/10.1016/j.mbs.2013.07.021 doi: 10.1016/j.mbs.2013.07.021
[13]	S. Hsu, T. Hwang, Y. Kuang, Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system, J. Math. Biol., 42 (2001), 489–506. https://doi.org/10.1007/s002850100079 doi: 10.1007/s002850100079
[14]	S. Hsu, T. Hwang, Y. Kuang, A ratio-dependent food chain model and its applications to biological control, Math. Biosci., 181 (2003), 55–83. https://doi.org/10.1016/s0025-5564(02)00127-x doi: 10.1016/s0025-5564(02)00127-x
[15]	H. A. Ibrahim, R. K. Naji, Chaos in Beddington-DeAngelis food chain model with fear, J. Phys. Conf. Ser., 1591 (2020), 012082. https://doi.org/10.1088/1742-6596/1591/1/012082 doi: 10.1088/1742-6596/1591/1/012082
[16]	B. Kumar, R. K. Sinha, Dynamics of an eco-epidemic model with Allee effect in prey and disease in predator, Comput. Math. Biophys., 11 (2023), 16. https://doi.org/10.1515/cmb-2023-0108 doi: 10.1515/cmb-2023-0108
[17]	V. Kumar, N. Kumari, Controlling chaos in three species food chain model with fear effect, AIMS Math., 5 (2020), 828–842. https://doi.org/10.3934/math.2020056 doi: 10.3934/math.2020056
[18]	R. Lavanya, S. Vinoth, K. Sathiyanathan, Z. N. Tabekoueng, P. Hammachukiattikul, R. Vadivel, Dynamical behavior of a delayed holling Type-Ⅱ predator-prey model with predator cannibalism, J. Math., 2022 (2022), 4071375. https://doi.org/10.1155/2022/4071375 doi: 10.1155/2022/4071375
[19]	F. H. Maghool, R. K. Naji, The dynamics of a Tritrophic Leslie-Gower food-web system with the effect of fear, J. Appl. Math., 2021 (2021), 2112814. https://doi.org/10.1155/2021/2112814 doi: 10.1155/2021/2112814
[20]	F. H. Maghool, R. K. Naji, Chaos in the three-species Sokol-Howell food chain system with fear, Commun. Math. Biol. Neu., 2022 (2022), 14. https://doi.org/10.28919/cmbn/7056 doi: 10.28919/cmbn/7056
[21]	S. Mandal, N. Sk, P. K. Tiwari, R. K. Upadhyay, Chaos and extinction risks of sexually reproductive generalist top predator in a seasonally forced food chain system with Allee effect, Chaos, 34 (2024), 063142. https://doi.org/10.1063/5.0212961 doi: 10.1063/5.0212961
[22]	N. Mukherjee, S. Ghorai, M. Banerjee, Detection of turing patterns in a three species food chain model via amplitude equation, Commun. Nonlinear Sci., 69 (2019), 219–236. https://doi.org/10.1016/j.cnsns.2018.09.023 doi: 10.1016/j.cnsns.2018.09.023
[23]	P. Panday, N. Pal, S. Samanta, J. Chattopadhyay, Stability and bifurcation analysis of a three-species food chain model with fear, Int. J. Bifurcat. Chaos, 28 (2018), 1850009. https://doi.org/10.1142/s0218127418500098 doi: 10.1142/s0218127418500098
[24]	S. Pathak, A. Maiti, G. P. Samanta, Rich dynamics of a food chain model with Hassell-Varley type functional responses, Appl. Math. Comput., 208 (2009), 303–317. https://doi.org/10.1016/j.amc.2008.12.015 doi: 10.1016/j.amc.2008.12.015
[25]	D. Pattanayak, A. Mishra, S. Dana, N. Bairagi, Bistability in a tri-trophic food chain model: Basin stability perspective, Chaos, 31 (2021), 073124. https://doi.org/10.1063/5.0054347 doi: 10.1063/5.0054347
[26]	L. Perko, Differential equations and dynamical systems, Springer Science & Business Media, 7 (2013).
[27]	S. Rana, S. Bhattacharya, S. Samanta, Complex dynamics of a three-species food chain model with fear and Allee effect, Int. J. Bifurcat. Chaos, 32 (2022). https://doi.org/10.1142/s0218127422500845
[28]	H. A. Satar, H. A. Ibrahim, D. K. Bahlool, On the dynamics of an eco-epidemiological system incorporating a vertically transmitted infectious disease, Iraqi J. Sci., 62 (2021), 1642–1658. https://doi.org/10.24996/ijs.2021.62.5.27 doi: 10.24996/ijs.2021.62.5.27
[29]	H. A. Satar, R. K. Naji, Stability and bifurcation in a prey-predator-scavenger system with Michaelis-Menten type of harvesting function, Differ. Equat. Dyn. Syst., 30 (2022), 933–956. https://doi.org/10.1007/s12591-018-00449-5 doi: 10.1007/s12591-018-00449-5
[30]	J. T. Tanner, The stability and the intrinsic growth rates of prey and predator populations, Ecology, 56 (1975), 855–867. https://doi.org/10.2307/1936296 doi: 10.2307/1936296
[31]	R. K. Upadhyay, R. K. Naji, Dynamics of a three species food chain model with Crowley-martin type functional response, Chaos Soliton. Fract., 42 (2009), 1337–1346. https://doi.org/10.1016/j.chaos.2009.03.020 doi: 10.1016/j.chaos.2009.03.020
[32]	S. Vinoth, R. Sivasamy, K. Sathiyanathan, G. Rajchakit, P. Hammachukiattikul, R. Vadivel, et al., Dynamical analysis of a delayed food chain model with additive Allee effect, Adv. Differ. Equ., 2021 (2021). https://doi.org/10.1186/s13662-021-03216-z
[33]	S. Vinoth, R. Vadivel, N. T. Hu, C. S. Chen, N. Gunasekaran, Bifurcation analysis in a harvested modified Leslie-Gower model incorporated with the fear factor and prey refuge, Mathematics, 11 (2023), 3118. https://doi.org/10.3390/math11143118 doi: 10.3390/math11143118
[34]	S. Vinoth, R. Vadivel, C. Dimplekumar, G. Nallappan, Dynamical complexities and chaos control in a Ricker type predator-prey model with additive Allee effect, AIMS Math., 8 (2023) 22896–22923. https://doi.org/10.3934/math.20231165
[35]	F. Zhu, R. Yang, Bifurcation in a modified Leslie-Gower model with nonlocal competition and fear effect, Discrete Cont. Dyn.-B, 30 (2025), 2865–2893. https://doi.org/10.3934/dcdsb.2024195 doi: 10.3934/dcdsb.2024195

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