[ Log In | Register]
! Skip Navigation Links
Skip Navigation Links
Image of Journal of Applied Nonlinear Dynamics
ISSN:2325-6192 (print)
ISSN:2325-6206 (online)
Journal of Environmental Accounting and Management
António Mendes Lopes (editor), Jiazhong Zhang(editor)
António Mendes Lopes (editor)

University of Porto, Portugal

Email: aml@fe.up.pt

Jiazhong Zhang (editor)

School of Energy and Power Engineering, Xi'an Jiaotong University, Xi'an, Shaanxi Province 710049, China

Fax: +86 29 82668723 Email: jzzhang@mail.xjtu.edu.cn

Fear Effect in a Tri-Trophic Food Chain Model with Holling Type IV Functional Response

Journal of Environmental Accounting and Management 9(3) (2021) 235--253 | DOI:10.5890/JEAM.2021.09.004

Krishnendu Sarkar$^{1}$, Nijamuddin Ali$^{2}$, Lakshmi Narayan Guin$^{1}$

$^{1}$ Department of Mathematics, Visva-Bharati, Santiniketan 731235, India

$^{2}$ Department of Mathematics, Vivekananda Mahavidyalaya, Purba Bardhaman, West Bengal, 713103, India

Download Full Text PDF

 

Abstract

Of concern the present study deals with an updated food chain model comprising one prey and two predator species in a natural environment with the inclusion of fear effect in both the prey and the median predator population through Holling type IV functional response. The present model is affluent with intra-specific competition among the hunter species having specific mortality. The model system emphasizes its characteristics in the proximity of the probable equilibrium position in the realm of biological dynamics. The response of the system is explored further for its stability analysis based on prerequisites and Hopf-bifurcation phenomena as well with respect to some significant model parameters. A quantitative analysis based on numerical simulation is also carried out in order to validate the analytical results and thereby the applicability of the model is established.

Acknowledgments

The authors would like to express thank the anonymous referees and the editors for making productive remarks and suggestions in order to improve the quality of the work done. The first and third authors gratefully acknowledge the financial support in part from Special Assistance Programme (SAP-III) sponsored by the University Grants Commission (UGC), New Delhi, India (Grant No. F.510 / 3 / DRS-III / 2015 (SAP-I)). We are grateful to Prof. Santabrata Chakravarty, Department of Mathematics, Visva-Bharati, Santiniketan-731235, West Bengal, India for comments on an earlier version of the manuscript.

References

  1. [1]  Arditi, R. and Ginzburg, L. (1989), Coupling in predator-prey dynamics: ratiodependence, Journal of Theoretical Biology, 139(3), 311-326.
  2. [2]  Birkhoff, G. and Rota, G.C. (1962), Ordinary Differential Equations, Ginn, Boston.
  3. [3]  Beddington, J.R. (1975), Mutual interference between parasites or predators and its effect on searching efficiency, The Journal of Animal Ecology, 44(1), 331-340.
  4. [4]  Castellano, C., Fortunato, S., and Loreto, V. (2009) Statistical physics of social dynamics, Reviews of Modern Physics, 81(2), 591.
  5. [5]  Creel, S. and Creel, N.M. (1995), Communal hunting and pack size in African wild dogs, Lycaonpictus, Animal Behaviour, 50(5), 1325-1339.
  6. [6]  Dejean, A., Leroy, C., Corbara, B., Roux, O., Cereghino, R., Orivel, J., and Boulay, R. (2010), Arboreal ants use the ``velcro principle" to capture very large prey, PLoS ONE 5(6), e11331.
  7. [7]  Gakkhar, S. and Naji, R. (2003), Seasonally perturbed prey-predator system with predator-dependent functional response, Chaos, Solitons and Fractals, 18(5), 1075-1083.
  8. [8]  Haque, M. and Venturino, E. (2007), An eco-epidemiological model with disease in predator: the ratio-dependent case, Mathematical Methods in the Applied Sciences, 30(4), 1791-1809.
  9. [9]  Hastings, A. and Powell, T. (1991), Chaos in three-species food chain, Ecology, 72(3), 896-903.
  10. [10]  Holling, C.S. (1959), The components of predation as revealed by a study of small-mammal predation of the European pine sawfly, Canadian Entomologist, 91(5), 293-329.
  11. [11]  Holmes, E., Lewis, M., Banks, J., and Veit, R. (1994), Partial differential equations in ecology: Spatial interactions and population dynamics, Ecology, 75(1), 17-29.
  12. [12]  Kot, M. (2001), Elements of Mathematical Ecology, Cambridge University Press, Cambridge.
  13. [13]  Lima, S. and Dill, L.M. (1990), Behavioral decisions made under the risk of predation: A review and prospectus, Canadian Journal of Zoology, 68(4), 619-640.
  14. [14]  Lotka, A.J. (1925), Elements of Physical Biology, Williams \& Wilkins Company, Baltiore, Maryland.
  15. [15]  Lotka, A.J. (1956), Elements of Mathematical Biology, Dover Publications, Mineola, New York.
  16. [16]  Meng, X., Liu, R., and Zhang, T. (2014), Adaptive dynamics for a non-autonomous Lotka-Volterra model with size-selective disturbance, Nonlinear Analysis: Real World Applications, 16(1), 202-213.
  17. [17]  Murray, J.D. (2001), Mathematical Biology II: Spatial Models and Biomedical Applications, Springer-Verlag, New York.
  18. [18]  Pal, S., Pal, N., Samanta, S., and Chattopadhyay, J. (2019), Effect of hunting cooperation and fear in a predator-prey model, Ecological Complexity, 39, 100770.
  19. [19]  Pandey, P., Pal, N., Samanta, S., and Chattopadhyay, J. (2018), Stability and bifurcation analysis of a three-species food chain model with fear, International Journal of Bifurcation and Chaos, 28(1), 1850009.
  20. [20]  Rosenzweig, M.L. and MacArthur, R.H. (1963), Graphical representation and stability conditions of predator-prey interactions, American Naturalist, 97(895), 209-223.
  21. [21]  Sasmal, S.K. (2018), Population dynamics with multiple Allee effects induced by fear factors-A mathematical study on prey-predator interactions, Applied Mathematical Modeling, 64(1), 1-14. 65, Springer-Verlag: New York.
  22. [22]  Seo, G. and DeAngelis D.L., (2011), A predator-prey model with a Holling type I functional response including a predator mutual interference, Journal of Nonlinear Sciences, 21(6), 811-833.
  23. [23]  Shigesada, N., Kawasaki, K., and Teramoto, E. (1979), Spatial segregation of interacting species, Journal of Theoretical Biology, 79(1), 83-99.
  24. [24]  Szolnoki, A., Mobilia, M., Jiang, L.-L., Szczesny, B., Rucklidge, A.M., and Perc, M. (2014), Cyclic dominance in evolutionary games: A review, Journal of the Royal Society Interface, 11(100), 20140735.
  25. [25]  Taylor, R.J. (1984), Predation, Chapman \& Hall .
  26. [26]  Volterra, V. (1926), Variazione e fluttuazioni del numerod individui in specie animaliconviventi, Memorie della Reale Accademia Nazionale dei Lincei, 6, 31-113.
  27. [27]  Wang, X., Zanette, L., and Zou, X. (2016), Modelling the fear effect in predator-prey interactions, Journal of Mathematical Biology, 73(5), 1179-1204.
  28. [28]  Xiao, Y. and Chen, L. (2001), Modeling and analysis of a predator-prey model with disease in the prey, Mathematical Biosciences, 171(1), 59-82.
  29. [29]  Xiao, D. and Ruan, S. (2001), Global analysis in a predator-prey system with nonmonotonic functional response, SIAM Journal of Applied Mathematics, 61(4), 1445-1472.

Copyright © 2011-2023 L & H Scientific Publishing. All rights reserved. Wildcard SSL Certificates