[ Log In | Register]
! Skip Navigation Links
Skip Navigation Links
Image of Journal of Applied Nonlinear Dynamics
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

Qualitative Analysis of a Modified Leslie-Gower Model with Gestation Delay

Journal of Applied Nonlinear Dynamics 10(3) (2021) 397--411 | DOI:10.5890/JAND.2021.09.004

R. Sivasamy , K. Nivethitha, S. Maheswari

Department of Science and Humanities, M.Kumarasamy College of Engineering, Tamil Nadu, India

Download Full Text PDF

 

Abstract

This paper explores the qualitative analysis of a modified Leslie-Gower prey-predator model where the consumption rate of prey is by per capita predator according to Beddington-DeAngelis functional response. Moreover, time-lag $(\tau)$ is established to exploit a gestation period of predations. The permanence analysis of the proposed system is investigated. We study the local stability of the non-delayed model at all possible equilibrium points. It is demonstrated that the given model experiences Hopf bifurcation about the interior equilibrium point with respect to delay $\tau$. Thereafter the stability and direction of Hopf bifurcation are formulated through normal and center manifold theorems. The derived criteria are justified with the help of numerical simulations.

References

  1. [1]  Lotka, A.J. (1956), Elements of Mathematical Biology, Cambridge University Press.
  2. [2]  Maynard-Smith, J. (1978), Models in Ecology, Cambridge University Press Archive.
  3. [3]  Cantrell, R.S. and Cosner, C. (2001), On the dynamics of predator-prey models with the Beddington-DeAngelis functional response, Journal of Mathematical Analysis and Applications, 257(1), 206-222.
  4. [4]  Zhu, H., Campbell, S.A., and Wolkowicz, G.S. (2003), Bifurcation analysis of a predator-prey system with nonmonotonic functional response, SIAM Journal on Applied Mathematics, 63(2), 636-682.
  5. [5]  Yuan, S. and Song, Y. (2009), Bifurcation and stability analysis for a delayed Leslie-Gower predator-prey system, IMA Journal of Applied Mathematics, 74(4), 574-603.
  6. [6]  Feng, P. (2014), Analysis of a delayed predator-prey model with ratio-dependent functional response and quadratic harvesting, Journal of Applied Mathematics and Computing, 44(1-2), 251-262.
  7. [7]  Celik, C. (2015), Stability and Hopf bifurcation in a delayed ratio dependent Holling-Tanner type model, Applied Mathematics and Computation, 255, 228-237.
  8. [8] Aziz-Alaoui, M.A. and Okiye, M.D. (2003), Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Applied Mathematics Letters, 16(7), 1069-1075.
  9. [9]  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.
  10. [10]  DeAngelis, D.L., Goldstein, R.A., and Oneill, R.V. (1975), A model for tropic interaction, Ecology, 56(4), 881-892.
  11. [11]  Pal, P.J. and Mandal, P.K. (2014), Bifurcation analysis of a modified Leslie-Gower predator-prey model with Beddington-Deangelis functional response and strong Allee effect, Mathematics and Computers in Simulation, 97, 123-146.
  12. [12]  Yu, S. (2014), Global stability of a modified Leslie-Gower model with Beddington-Deangelis functional response, Advances in Difference Equations, 2014(1), 84.
  13. [13]  Abid, W., Yafia, R., Aziz-Alaoui, M.A., Bouhafa, H., and Abichou, A. (2015), Global dynamics on a circular domain of a diffusion predator-prey model with modified Leslie-Gower and Beddington-DeAngelis functional type, Evolution Equations \& Control Theory, 4(2), 115.
  14. [14]  Indrajaya, D., Suryanto, A., and Alghofari, A. R. (2016), Dynamics of modified Leslie-Gower predator-prey model with Beddington-DeAngelis functional response and additive Allee effect, International Journal of Ecology and Development, 31(3), 60-71.
  15. [15]  Lahrouz, A., Settati, A., and Mandal, P.S. (2016), Dynamics of a switching diffusion modified Leslie-Gower predator-prey system with Beddington-Deangelis functional response, Nonlinear Dynamics, 85(2), 853-870.
  16. [16]  Roy, B., Roy, S.K., and Gurung, B.D. (2017), Holling-Tanner model with Beddington-Deangelis functional response and time delay introducing harvesting, Mathematics and Computers in Simulation, 142, 1-14.
  17. [17]  Song, Y. and Wei, J. (2005), Local Hopf bifurcation and global periodic solutions in a delayed predator-prey system, Journal of Mathematical Analysis and Applications, 301(1), 1-21.
  18. [18]  Saha, T. and Chakrabarti, C. (2009), Dynamical analysis of a delayed ratio-dependent Holling-Tanner predator-prey model, Journal of Mathematical Analysis and Applications, 358(2), 389-402.
  19. [19]  Karaoglu, E. and Merdan, H. (2014), Hopf bifurcations of a ratio-dependent predator-prey model involving two discrete maturation time delays, Chaos, Solitons $\&$ Fractals, 68, 159-168.
  20. [20]  El Foutayeni, Y. and Khaladi, M. (2016), Equilibrium Points and Their Stability Properties of a Multiple Delays Model, Differential Equations and Dynamical Systems, 1-18.
  21. [21]  Huang, J., Gong, Y., and Chen, J. (2013), Multiple bifurcations in a predator-prey system of Holling and Leslie type with constant-yield prey harvesting, International Journal of Bifurcation and Chaos, 23(10), 1350164.
  22. [22]  Huang, J., Liu, S., Ruan, S., and Zhang, X. (2016), Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting, Communication in Pure and Applied Analysis, 15(3), 1041-1055.
  23. [23]  Chakraborty, K., Jana, S., and Kar, T.K. (2012), Global dynamics and bifurcation in a stage structured prey-predator fishery model with harvesting, Applied Mathematics and Computation, 218(18), 9271-9290.
  24. [24]  Gupta, R.P. and Chandra, P. (2013), Bifurcation analysis of modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting, Journal of Mathematical Analysis and Applications, 398(1), 278-295.
  25. [25]  Yuan, R., Jiang, W., and Wang, Y. (2015), Saddle-node-Hopf bifurcation in a modified Leslie-Gower predator-prey model with time-delay and prey harvesting, Journal of Mathematical Analysis and Applications, 422(2), 1072-1090.
  26. [26]  Hu, D. and Cao, H. (2017), Stability and bifurcation analysis in a predator-prey system with Michaelis-Menten type predator harvesting, Nonlinear Analysis: Real World Applications, 33, 58-82.
  27. [27]  Yang, R. and Zhang, C. (2017), Dynamics in a diffusive modified Leslie-Gower predator-prey model with time delay and prey harvesting, Nonlinear Dynamics, 87(2), 863-878.
  28. [28]  Hassard, B.D., Kazarinoff, N.D., and Wan, Y.W. (1981), Theory and applications of Hopf bifurcation, CUP Archive.
  29. [29]  Saifuddin, M., Biswas, S., Samanta, S., Sarkar, S., and Chattopadhyay, J. (2016), Complex dynamics of an eco-epidemiological model with different competition coefficients and weak Allee in the predator, Chaos, Solitons \& Fractals, 91, 270-285.
  30. [30]  Singh, M.K., Bhadauria, B.S., and Singh, B.K. (2018), Bifurcation analysis of modified Leslie-Gower predator-prey model with double Allee effect, Ain Shams Engineering Journal, 9(4), 1263-1277.
  31. [31]  Peng, Y. and Zhang, T. (2016), Turing instability and pattern induced by cross-diffusion in a predator-prey system with Allee effect, Applied Mathematics and Computation, 275, 1-12.
  32. [32]  Slobodkin, L.B. (1980), Growth and regulation of animal populations, Dover Publications.
  33. [33]  Dennis, B. (1989), Allee effects: population growth, critical density, and the chance of extinction, Natural Resource Modeling, 3(4), 481-538.
  34. [34]  Sivasamy, R., Sathiyanathan, K., and Balachandran, K. (2019), Dynamics of a Modified Leslie-Gower Model with Crowley-Martin Functional Response and Prey Harvesting, Journal of Applied Nonlinear Dynamics, 8(4), 621-636.

Copyright © 2011-2024 L & H Scientific Publishing. All rights reserved. Wildcard SSL Certificates