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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Response of Prey-Predator-Scavenger Interactions with the Inclusion of Prey Harvesting

Journal of Applied Nonlinear Dynamics 13(2) (2024) 279--306 | DOI:10.5890/JAND.2024.06.008

Lakshmi Narayan Guin, Gourav Mandal, Pallab Pal, Santabrata Chakravarty

Department of Mathematics, Visva-Bharati, Santiniketan-731 235, West Bengal, India

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Abstract

The proposed analytical investigation deals with the interactions between prey, predator, and scavenger population in the event of prey harvesting through the use of an updated mathematical model. It includes the usage of the modified Previte and Hoffman tri-trophic food web model along with its issues governing the system of nonlinear differential equations. The prey population is assumed to follow a logistic growth with predation from both the predator and the scavenger population. The intervention of prey harvesting is another inclusion of the concerned study for the purpose of having a complete understanding of the influence of both the predator and scavenger species. The present attempt intends to explore the mechanism of harvesting scenario in a three-dimensional system of interacting species, namely one prey and two specialist predators. At the onset, the proposed model is sought for dimension-free so as to reduce the number of model parameters of the system successfully. The existence of feasible equilibria, together with their topological classification regarding coexistence positive equilibrium points of the system is thoroughly examined.~The stability analysis is carried out for all the feasible equilibria of the system in order to understand the dynamics of the interacting species. The sensitivity analysis of the model system is also performed numerically so as to estimate the harvesting rate culpable for balancing biomass. One may perceive that so long as the harvesting rate is uncontrolled, that is, beyond the threshold value, both the predator and scavenger species become extinct together with the prey population.

Acknowledgments

\bibitem{1} Baca{\"e}r, N. (2011), \textit{A Short History of Mathematical Population Dynamics}, Springer-Verlag, New York. \bibitem{27} Tanabe, K. and Namba, T. (2005), Omnivory creates chaos in simple food web models, \textit{Ecology}, \textbf{86}, 3411-3414. \bibitem{22} Previte, J. and Hoffman, K. (2013), Period doubling cascades in a predator-prey model with a scavenger, \textit{SIAM Review}, \textbf{55}, 523-546. \bibitem{holt1997theoretical} Holt, R.D. and Polis, G.A. (1997), A theoretical framework for intraguild predation, \textit{The American Naturalist}, \textbf{149}, 745-764. \bibitem{5} Clark, C.W. (1976), \textit{Mathematical Bioeconomics: Optimal Management of Renewable Resources}, Wiley, New York. \bibitem{6} Clark, C.W. (1985), \textit{Bioeconomic Modelling and Fisheries Management}, Wiley, New York.

References

  1. [1]  Baca{\"e}r, N. (2011), A Short History of Mathematical Population Dynamics, Springer-Verlag, New York.
  2. [2]  Tanabe, K. and Namba, T. (2005), Omnivory creates chaos in simple food web models, Ecology, 86, 3411-3414.
  3. [3]  Previte, J. and Hoffman, K. (2013), Period doubling cascades in a predator-prey model with a scavenger, SIAM Review, 55, 523-546.
  4. [4]  Holt, R.D. and Polis, G.A. (1997), A theoretical framework for intraguild predation, The American Naturalist, 149, 745-764.
  5. [5]  Clark, C.W. (1976), Mathematical Bioeconomics: Optimal Management of Renewable Resources, Wiley, New York.
  6. [6]  Clark, C.W. (1985), Bioeconomic Modelling and Fisheries Management, Wiley, New York.
  7. [7]  Gupta, R. P. and D. K. Yadav, D. K. (2022), Role of Allee effect and harvesting of a food-web system in the presence of scavenger, Journal of Biological Systems, 30, 149-181.
  8. [8]  Jana, D.K. and Panja, P. (2022), Effects of supplying additional food for a scavenger species in a prey-predator-scavenger model with quadratic harvesting, International Journal of Modelling and Simulation, 43(3), 250-264.
  9. [9]  Satar, H.A. and Naji, R.K. (2019), Stability and bifurcation in a prey-predator-scavenger system with Michaelis-Menten type of harvesting function, Differential Equations and Dynamical Systems, 1-24, https://doi.org/10.1007/s12591-018-00449-5.
  10. [10]  Feng, P. (2014), Analysis of a delayed predator-prey model with ratio-dependent functional response and quadratic harvesting, Journal of Applied Mathematics and Computer Science, 44, 251-262.
  11. [11]  Gupta, R.P., Banerjee, M., and Chandra, P. (2014), Period doubling cascades of prey-predator model with nonlinear harvesting and control of over exploitation through taxation, Communications in Nonlinear Science and Numerical Simulation, 19, 2382-2405.
  12. [12]  Song, X. and Chen, L. (2001), Optimal harvesting and stability for a two-species competitive system with stage structure, Mathematical Biosciences, 170, 173-186.
  13. [13]  Kar, T.K. and Chaudhuri, K.S. (2004), Harvesting in a two-prey one predator fishery: a bioeconomic model, The ANZIAM Journal, 45, 443-456.
  14. [14]  Rai, V. and Sreenivasan, R. (1993), Period-doubling bifurcations leading to chaos in a model food chain, Ecological Complexity, 69, 63-77.
  15. [15]  Hale, J.K. (1976), Theory of Functional Differential Equations, Springer-Verlag, New York.
  16. [16]  Nagumo, M. (1942), {\"U}ber die lage der integralkurven gew{\"o}hnlicher differentialgleichungen, Proceedings of the Physico-Mathematical Society of Japan. 3rd Series, 24, 551-559.
  17. [17]  Birkhoff, G. and Rota, G.C. (1982), Ordinary Differential Equations, Ginn, Boston.
  18. [18]  Freedman, H. and Waltman, P. (1984), Persistence in models of three interacting predator-prey populations, Mathematical Biosciences, 68, 213-231.
  19. [19]  Freedman, H.I. and Ruan, S.G. (1995), Uniform persistence in functional differential equations, Journal of Differential Equations, 115, 173-192.
  20. [20]  LaSalle, J.P. (1976), The Stability of Dynamical Systems, SIAM, United States.
  21. [21]  Perko, L. (2001), Differential Equations and Dynamical Systems, Springer, New York.
  22. [22]  Guin, L.N., Mondal, B., and Chakravarty, S. (2018), Spatiotemporal patterns of a pursuit-evasion generalist predator-prey model with prey harvesting, Journal of Applied Nonlinear Dynamics, 7, 165-177.
  23. [23]  Guin, L.N. (2014), Existence of spatial patterns in a predator-prey model with self- and cross-diffusion, Applied Mathematics and Computation, 226, 320-335.
  24. [24]  Guin, L.N. and Mandal, P.K. (2014), Spatiotemporal dynamics of reaction-diffusion models of interacting populations, Applied Mathematical Modelling, 38, 4417-4427.
  25. [25]  Guin, L.N. and Acharya, S. (2017), Dynamic behaviour of a reaction-diffusion predator-prey model with both refuge and harvesting, Nonlinear Dynamics, 88, 1501-1533.
  26. [26]  Guin, L.N., Murmu, R., Baek, H., and Kim, K.H. (2020), Dynamical analysis of a Beddington-DeAngelis interacting species system with harvesting, Mathematical Problems in Engineering, 1-22, Article ID 7596394.
  27. [27]  Han, R., Guin, L.N., and Dai, B. (2021), Consequences of refuge and diffusion in a spatiotemporal predator-prey model, Nonlinear Analysis: Real World Applications, 60, 103311.
  28. [28]  Guin, L.N., Mondal, B., and Chakravarty, S. (2016), Existence of spatiotemporal patterns in the reaction-diffusion predator-prey model incorporating prey refuge, International Journal of Biomathematics, 9, 1650085.
  29. [29]  Sander, E. and Yorke, J.A. (2011), Period-doubling cascades galore, Ergodic Theory and Dynamical Systems, 31, 1249-1267.
  30. [30]  Guin, L.N. and Mandal, P.K. (2014), Effect of prey refuge on spatiotemporal dynamics of reaction-diffusion system, Computers and Mathematics with Applications, 68, 1325-1340.
  31. [31]  Guin, L.N., Mandal, G., Mondal, M., and Chakravarty, S. (2022), A chaotic tri-trophic food chain model supplemented by Allee effect, International Journal of Dynamics and Control, 11(2), 528-554.
  32. [32]  Souna, F., Lakmeche, A., and Djilali, S. (2020), Spatiotemporal patterns in a diffusive predator-prey model with protection zone and predator harvesting, Chaos, Solitons $\&$ Fractals, 140, 110180.
  33. [33]  Souna, F., Djilali, S., and Charif, F. (2020), Mathematical analysis of a diffusive predator-prey model with herd behavior and prey escaping, Mathematical Modelling of Natural Phenomena, 15, 23.
  34. [34]  Boudjema, I. and Djilali, S. (2018), Turing-Hopf bifurcation in Gauss-type model with cross diffusion and its application, Nonlinear Studies, 25, 665-687.
  35. [35]  Bentout, S., Djilali, S., and Kumar, S. (2021), Mathematical analysis of the influence of prey escaping from prey herd on three species fractional predator-prey interaction model, Physica A: Statistical Mechanics and its Applications, 572, 125840.
  36. [36]  Djilali, S., Benahmadi, L., Tridane, A., and Niri, K. (2020), Modeling the impact of unreported cases of the COVID-19 in the North African countries, Biology, 9, 373.
  37. [37]  Djilali, S., Benahmadi, L., Tridane, A., and Niri, K. (2021), Pattern formations of a delayed diffusive predator-prey model with predator harvesting and prey social behavior, Mathematical Methods in the Applied Sciences, 44, 9128-9142.
  38. [38]  Bentout, S., Djilali, S., and Ghanbari, B. (2021), Backward, Hopf bifurcation in a heroin epidemic model with treat age, International Journal of Modeling, Simulation, and Scientific Computing, 12, 2150018.
  39. [39]  Djilali, S., Bentout, S., Ghanbari, B. and Kumar, S. (2021), Spatial patterns in a vegetation model with internal competition and feedback regulation, The European Physical Journal Plus, 136, 1-24.
  40. [40]  Djilali, S. and Bentout, S. (2021), Global dynamics of SVIR epidemic model with distributed delay and imperfect vaccine, Results in Physics, 25, 104245.
  41. [41]  Han, R., Mandal, G., Guin, L.N., and Chakravarty, S. (2022), Dynamical response of a reaction-diffusion predator-prey system with cooperative hunting and prey refuge, Journal of Statistical Mechanics: Theory and Experiment, 2022(10), 103502.

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