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ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com

Dynamics of a Predator-Prey Model with group defense and Exponential Fading Memory

Discontinuity, Nonlinearity, and Complexity 12(3) (2023) 497--510 | DOI:10.5890/DNC.2023.09.003

Prabir Panja

Department of Applied Science, Haldia Institute of Technology, Haldia-721657, West Bengal, India

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Abstract

In this paper, a predator-prey model with Monod-Haldane type functional response is developed. Here, it is assumed that prey grows logistically in the absence of predator. Also, predator population is divided into two subpopulations such as juvenile predator and mature predator respectively. To incorporate the group defense behavior in the model, Monod-Haldane type functional response is considered. It is considered that a portion of juvenile predator becomes mature predator. It is assumed that the growth rate of predator at an instant is not depends only the density of prey at the present time, but also depends on the density of the prey on the previous instant of time. Different possible equilibrium points are determined. Also, the stability of the model around these equilibrium points is studied. Hopf bifurcation analysis of the model is done with respect to some important parameters. It is observed that exponential fading memory has a big role in the stability of the model. Finally, some numerical simulation results have been presented for better understanding of the dynamics of the model.

References

  1. [1]  Wu, C.C. (2019), The spreading speed for a predator-prey model with one predator and two preys, Applied Mathematics Letters, 91, 9-14.
  2. [2]  Liu, Q., Jiang, D., Hayat, T., and Alsaedi, A. (2019), Stationary distribution of a regime-switching predator-prey model with anti-predator behaviour and higher-order perturbations, Physica A: Statistical Mechanics and its Applications, 515, 199-210.
  3. [3]  Jia, Y., Luo, B., Wu, J., and Xu, H. (2018), Analysis on the existence of the steady-states for an ecological-mathematical model with predator-prey-dependent functional response, Computers $\&$ Mathematics with Applications, 76, 1767-1778.
  4. [4]  Wu, D. and Zhao, M. (2019), Qualitative analysis for a diffusive predator-prey model with hunting cooperative, Physica A: Statistical Mechanics and its Applications, 515, 299-309.
  5. [5]  Kumar, S. and Kharbanda, H. (2019), Chaotic behavior of predator-prey model with group defense and non-linear harvesting in prey, Chaos, Solitons $\&$ Fractals, 119, 19-28.
  6. [6]  Ginoux, J., Naeck, R., Ruhomally, Y.B., Dauhoo, M.Z. and Perc, M. (2019), Chaos in a predator-prey-based mathematical model for illicit drug consumption, Applied Mathematics and Computation, 347, 502-513.
  7. [7]  Wang, Y., Lu, M., and Liu, J. (2020), Global stability of a delayed virus model with latent infection and Beddington-DeAngelis infection function, Applied Mathematics Letters, 107, 106463.
  8. [8]  Wang, K. (2011), Periodic solutions to a delayed predator-prey model with Hassell-Varley type functional response, Nonlinear Analysis: Real World Applications, 12, 137-145.
  9. [9]  Sokol, W. and Howell, J.A. (1980), Kinetics of phenol oxidation by washed cells, Biotechnology and Bioengineering, 23, 2039-2049.
  10. [10]  Khajanchi, S. (2017), Modeling the dynamics of stage-structure predator-prey system with Monod-Haldane type response function, Applied Mathematics and Computation, 302, 122-143.
  11. [11]  Liu, Z. and Tan, R. (2007), Impulsive harvesting and stocking in a Monod-Haldane functional response predator-prey system, Chaos, Solitons \& Fractals, 34, 454-464.
  12. [12]  Pal, R., Basu, D. and Banerjee, M. (2009), Modelling of phytoplankton allelopathy with Monod-Haldane-type functional response-A mathematical study, Biosystems, 95, 243-253.
  13. [13]  Chen, Y. and Liu, Z. (2009), Modelling and analysis of an impulsive SI model with Monod-Haldane functional response, Chaos, Solitons \& Fractals, 39, 1698-1714.
  14. [14]  Panja, P. (2020), Prey-predator-scavenger model with Monod-Haldane type functional response, Rendiconti del Circolo Matematico di Palermo Series 2, 69, 1205-1219.
  15. [15]  Yang, P. and Wang, Y. (2020), Hopf-Zero bifurcation in an age-dependent predator-prey system with Monod-Haldane functional response comprising strong Allee effect, Journal of Differential Equations, 269, 9583-9618.
  16. [16]  Alsakaji, H.J., Kundu, S., and Rihan, F.A. (2021), Delay differential model of one-predator two-prey system with Monod-Haldane and holling type II functional responses, Applied Mathematics and Computation, 397, 125919.
  17. [17]  Sahoo, B. and Poria, S. (2019), Dynamics of predator-prey system with fading memory, Applied Mathematics and Computation, 347, 319-333.
  18. [18]  Al-Jumaily, A.M. and Du, Y. (2011), Fading memory model for airway smooth muscle dynamic response, Journal of Theoretical Biology, 283, 10-13.
  19. [19]  Castro, R., Sierra, W., and Stange, E. (2017), Bifurcations in a predator-prey model with general logistic growth and exponential fading memory, Applied Mathematical Modelling, 45, 134-147.
  20. [20]  Mondal, J. (2018), A mathematical model about predator-prey holling type II effect of fading memory-a mathematical approach, International Journal of Engineering Science Invention, 7, 43-54.
  21. [21]  Cavalli, F. and Naimzada, A. (2015), A tatonnement process with fading memory, stabilization and optimal speed of convergence, Chaos, Solitons \& Fractals, 79, 116-129.
  22. [22]  Wang, N. and Han, M. (2016), Slow-fast dynamics of Hopfield spruce-budworm model with memory effects, Advances in Difference Equations, 2016, 73.
  23. [23]  Okada, M. and Kawashima, S. (2017), Global solutions to the equation of thermoelasticity with fading memory, Journal of Differential Equations, 263, 338-364.
  24. [24]  Tarasova, V.V. and Tarasov, V.E. (2018), Dynamic intersectoral models with power-law memory, Communications in Nonlinear Science and Numerical Simulation, 54, 100-117.
  25. [25]  Tarasov, V.E. and Tarasova, V.V. (2018), Macroeconomic models with long dynamic memory: Fractional calculus approach, Applied Mathematics and Computation, 338, 466-486.
  26. [26]  MacDonald, N. (1976), Time delay in prey-predator models, Mathematical Biosciences, 28, 321-330.
  27. [27]  MacDonald, N. (1977), Time delay in prey-predator models II. Bifurcation theory, Mathematical Biosciences, 33, 227-234.
  28. [28]  Farkas, M. (1988), Stable oscillations in a predator-prey model with time lag, Journal of Mathematical Analysis and Applications, 130, 191-199.
  29. [29]  Farkas, M., Farkas, A., and Szabo, G. (1988), Multiparameter bifurcation diagrams in predator-prey models with time lag, Journal of Mathematical Biology, 26, 93-103.
  30. [30]  Nagumo, M. (1926), Eine hinreichende Bedingung fur die Unitat der Losung von Dierentialgleichungen erster Ordnung, Japan Journal of Mathematics, 3, 107-112.
  31. [31]  Birkhoff, G. and Rota, G.C. (1982), Ordinary differential equations, Ginn, Boston.
  32. [32]  Hassard, B.D., Kazarinoff, N.D., and Wan, Y.H. (1981), Theory and application of hopf bifurcation, London Mathematical Society Lecture Note Series, vol. 41, Cambridge University Press.

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