GLOBAL STABILITY AND HOPF BIFURCATION OF A DELAYED PREDATOR-PREY MODEL INCORPORATING ALLEE EFFECT AND FEAR EFFECT
DOI:
https://doi.org/10.25271/sjuoz.2025.13.1.1406Keywords:
Fear, Time delay, Allee effect, Stability analysis, Hopf bifurcationAbstract
This paper aims to discover the impact of the fear of predators in prey, Allee effect for predator reproduction and time delay corresponding to the gestation period on the dynamics of a predator- prey model. Existence, non-negativity, and boundedness of the model solutions are guaranteed. The criteria for asymptotically stability of all the biologically feasible steady state points are determined. It is also determined a critical value for time delay, where the model under goes Hopf -bifurcation near coexistence steady state point. Finally, with the help of the MATLAB program, to confirm the analytical results and discover the impact of fear, the Allee effect, and time delay, the model was solved numerically.it is observed that fear affect negatively on both prey and predator species and the time delay may system may induce a transition of the dynamics of system from the a stability situation to the state where the populations oscillate periodically or vice versa
References
Xiaoying, W., Liana Z., Xinfu Z.(2016).Modeling the fear effect in predator-prey interactions, Math. Bio, 73
(2), 1–38. https://doi.org/10.1007/s00285-016-0989-1
Pal S., Pal N, Samanta S.(2019) Chattopadhyay J, Effect of hunting cooperation and fear in a predator-prey
model, Ecol.Complex39. https://doi.org/10.1016/j.ecocom.2019.100770
Huisen Z., Yangli C., Shengmao F., Weiming W.(2019) Impact of the fear effect in a prey-predator model
Incorporating aprey refuge,Appl.Math.Comput,356,328–337.
http://dx.doi.org/10.1016/j.amc.2019.03.034
Pingping G., Meng F., Ingfu Z. (2021)Dynamics of a three-species food chain model with fear effect,
Commun. Nonlinear Sci. Number. Simulat 99. https://doi.org/10.1016/j.cnsns.2021.105809
Yiping T., Yongli C., Ruoxia Y., Maolin H., Weiming. W.(2022) Complex dynamics in an eco-Epidemiological
model with the cost of anti-predator behaviors.NonlinearDyn,107,3127–3141.
https://link.springer.com/article/10.1007%2Fs11071-021-07133-4
Jimil AR., Naji RK. (2023).Modeling and Analyzing the Influence of Fear on the Harvested Modified Leslie
Gower Model. Baghdad Sci. J, 20(5),1701-1712. https://dx.doi.org/10.21123/bsj.2023.ID
Xiaoqin W., Yipng T, Yongli C, Weimng W.(2020). Impact of the fear effect on the stability and Bifurcation of
a Leslie–Gower predator-prey model, Int.J.Bifur.Chaos30.doi:10.1142/S0218127420502107.
Mengxin He., zhong Li.(2022) stability of fear effect predator-prey model mutual interference or group
defense, J. Bio.Dyn, 16,480-498. https://doi.org/10.1080/17513758.2022.2091800
Soumitra P., Pankaj .KT., Arvind KM., Hao W.(2023). Fear effect in a three-species food chain model with
generalist predator, Math. Bio and Engineering, 21,1-33. https://doi.org/10.3934/mbe.2024001.
Yaseen, R.M., Helal, M.M., Dehingia, K., Mohsen, A.A.(2024). Effect of the Fear Factor and Prey Refuge in an
Asymmetric Predator–Prey Model. Braz J Phys 54, 214. http://dx.doi.org/10.1007/s13538-024-
-9.
Haque M., Sarwardi S., Preston S. and Venturino E.(2011) Effect of delay in a LotkaVolterra type predator-
prey model with a transmissible disease in the predator species”, Mathematical Biosciences,
(1).234,47-57. https://doi.org/10.1016/j.mbs.2011.06.009
liu J., lv P., Li,, B., Zhang T.(2021). Dynamics of a Predator-Prey Model with Fear Effect and Time Delay,
Complexity,16pages.https://doi.org/10.1155/2021/9184193.
Dehingia, K., Hosseini, K., Salahshour, S., Baleanu, D .(2022) A Detailed Study on a Tumor Model with Delayed Growth of Pro-Tumor Macrophages. Int. J. Appl. Comput. Math 8, 245 (2022). https://doi.org/10.1007/s40819-022-01433-y
Naji RK., Majeed SJ.(2020). the dynamical analysis of a delayed prey-predator model with arefuge-stage
structure prey population, Iranian.J.Math.Sci and inf, 1(2020), 135-159.
https://ijmsi.ir/article-1-1116-en.pdf
Lavanya R, Vinth S, Sathiyanathan, K, Zeric NT, Hammachukiatkul P, Vadivel R(2022) Dynamical behavior of
a delayed Holling type-II predator prey model with predator Cannibalism, J. Mathematics,15 pages.
https://doi.org/10.1155/2022/4071375.
Rihan FA, Alsakaji HJ, Rajivganthi C.(2020) Stability and Hopf Bifurcation of Three-Species Prey-Predator
System with Time Delays and Allee Effect ,Complexity,(2020), 15pages.
https://doi.org/10.1155/2020/7306412.
Alan J. Terry.(2015). Predator–prey models with component Allee effect for predator reproduction. J. Math.
Biol,71,1325–1352. https://doi.org/10.1007/s00285-015-0856-5
Soura KS.(2018) Population dynamics with multiple allee effects induced by fear factors-A Mathematical
study on prey-predator interactions, Appl.Math.Mode,64, 1–14.
https://doi.org/10.1016/j.apm.2018.07.021
Yining X., Zhao J., Yang R.(1996).Stability Analysis and Hopf Bifurcation of a Delayed Diffusive Predator–Prey
Model with a Strong Allee Effect on the Prey and the Effect of Fear on the Predator,
, Mathematics 11,(9). https://doi.org/10.3390/math11091996.
Dehingia ,K., Das ,P., Upadhyay ,R., Misra ,A., Rihan, F. , Kamyar Hosseini .(2023)Modeling and analysis of
delayed tumour–immune system with hunting T-cells, Mathematics,203,669-684.
http://dx.doi.org/10.1016/j.matcom.2022.07.009.
Das, A., Dehingia, K ., Hinçal, E .,Özköse, F., Hosseini, K. (2024)A study on the dynamics of a breast cancer
model with discrete-time delay, physica scripta, 99(3). http://dx.doi.org/10.1088/1402-4896/ad2753.
Dehingia, K ., Das, A ., Hinçal, E., Hosseini, K.(2024) The Effect of Time Delay on the Dynamics of a Plankton-Nutrient System with Refuge, Brazilian Journal of Physics, 55 (1). doi:10.1007/s13538-024-01670-0
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