Bifurcation Analysis Of Three Species Prey-Predator Delayed Food Chain Model With Holling Type-II Functional Response And Fear Effect
Main Article Content
Abstract
In this study, we establish and investigate a delayed three-species food chain model with Holling type-II functional response and fear impact in the prey population. We assume that the fear of predators is restricting the growth rate of the prey population. The middle predator is thought to be both a prey and a predator, whereas the top predator consumes both the prey and the middle predator. We investigate the presence of all positive equilibrium points as well as their local stability. In addition, for the non-delayed model, we perform the Hopf-bifurcation analysis around the interior equilibrium point based on the fear parameter. Furthermore, for the related delayed model, we demonstrate the lo-cal stability and presence of Hopf-bifurcation. Finally, some numerical results have been provided to validate our analytical conclusions.
Downloads
Article Details
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
W. B. Cannon. Bodily changes in pain, hunger, fear and rage: An account of recent researches into the function of emotional excitement. d appleton & company, 1915.
B. K. Das, D. Sahoo, and G. Samanta. Impact of fear in a delay-induced predator– prey system with intraspecific competition within predator species. Mathematics and Computers in Simulation, 191:134–156, 2022.
H. I. Freedman. Deterministic mathematical models in population ecology, vol- ume 57. Marcel Dekker Incorporated, 1980.
A. Kumar and B. Dubey. Modeling the effect of fear in a prey–predator system with prey refuge and gestation delay. International Journal of Bifurcation and Chaos, 29(14):1950195, 2019.
V. Kumar and N. Kumari. Controlling chaos in three species food chain model with fear effect. AIMS Mathematics, 5(2):828–842, 2020.
S. L. Lima and L. M. Dill. Behavioral decisions made under the risk of predation: a review and prospectus. Canadian journal of zoology, 68(4):619–640, 1990.
S. Magudeeswaran, K. Sathiyanathan, R. Sivasamy, S. Vinoth, M. Sivabalan, et al. Analysis on dynamics of delayed intraguild predation model with ratio-dependent functional response. Discontinuity, Nonlinearity, and Complexity, 10(03):381–396, 2021.
S. Magudeeswaran, S. Vinoth, K. Sathiyanathan, and M. Sivabalan. Impact of fear on delayed three species food-web model with holling type-ii functional response. International Journal of Biomathematics, page 2250014, 2021.
N. Pal, S. Samanta, S. Biswas, M. Alquran, K. Al-Khaled, and J. Chattopadhyay. Stability and bifurcation analysis of a three-species food chain model with delay. International Journal of Bifurcation and Chaos, 25(09):1550123, 2015.
S. Pal, N. Pal, S. Samanta, and J. Chattopadhyay. Fear effect in prey and hunting co- operation among predators in a leslie-gower model. Math. Biosci. Eng, 16(5):5146– 5179, 2019.
P. Panday, N. Pal, S. Samanta, and J. Chattopadhyay. Stability and bifurcation anal- ysis of a three-species food chain model with fear. International Journal of Bifurca- tion and Chaos, 28(01):1850009, 2018.
P. Panday, S. Samanta, N. Pal, and J. Chattopadhyay. Delay induced multiple sta- bility switch and chaos in a predator–prey model with fear effect. Mathematics and Computers in Simulation, 172:134–158, 2020.
W. J. Ripple and R. L. Beschta. Wolves and the ecology of fear: can predation risk structure ecosystems? BioScience, 54(8):755–766, 2004.
R. Taylor and C. Predation. Hall ltd. New York, 1984.
S. Vinoth, R. Sivasamy, K. Sathiyanathan, G. Rajchakit, P. Hammachukiattikul,
R. Vadivel, and N. Gunasekaran. Dynamical analysis of a delayed food chain model with additive allee effect. Advances in Difference Equations, 2021(1):1–20, 2021.
J. Wang, Y. Cai, S. Fu, and W. Wang. The effect of the fear factor on the dynamics of a predator-prey model incorporating the prey refuge. Chaos: An Interdisciplinary Journal of Nonlinear Science, 29(8):083109, 2019.
X. Wang, L. Zanette, and X. Zou. Modelling the fear effect in predator–prey inter- actions. Journal of mathematical biology, 73(5):1179–1204, 2016.
A. J. Wirsing, M. R. Heithaus, and L. M. Dill. Living on the edge: dugongs prefer to forage in microhabitats that allow escape from rather than avoidance of predators. Animal Behaviour, 74(1):93–101, 2007.
C. Xu and M. Liao. Bifurcation behaviours in a delayed three-species food-chain model with holling type-ii functional response. Applicable Analysis, 92(12):2468– 2486, 2013.
H. Zhang, Y. Cai, S. Fu, and W. Wang. Impact of the fear effect in a prey- predator model incorporating a prey refuge. Applied Mathematics and Computation, 356:328–337, 2019.