[ 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

Dynamics of a Modified Leslie-Gower Model with Crowley-Martin Functional Response and Prey Harvesting

Journal of Applied Nonlinear Dynamics 8(4) (2019) 621--636 | DOI:10.5890/JAND.2019.12.008

R. Sivasamy$^{1}$, K. Sathiyanathan$^{1}$, K. Balachandran$^{2}$

$^{1}$ Department of Mathematics, SRMV College of Arts and Science, Coimbatore - 641020, India

$^{2}$ Department of Mathematics, Bharathiar University, Coimbatore - 641046, India

Download Full Text PDF

 

Abstract

In this paper, dynamics of modified Leslie-Gower predator-prey model with Crowley-Martin functional response and nonlinear prey harvesting is discussed. We discuss the permanence analysis and local stability of all possible equilibrium points. Also we derive the conditions for occurrence of Hopf bifurcation around positive equilibrium point and its stability. The global stability of positive equilibrium point is derived by using proper Lyapunov functional. Further the Hutchinson’s delay is introduced to utilize the fact that prey takes some time lag to convert the food into its growth. It is noted that the Hopf bifurcation occurs when the time lag parameter τ crosses its critical value. The proposed theoretical results are verified with the help of numerical simulations.

References

  1. [1]  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.
  2. [2]  DeAngelis, D.L., Goldstein, R.A., and O’neill, R.V. (1975), A model for tropic interaction, Ecology, 56(4), 881-892.
  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]  Bie, Q., Wang, Q., and Yao, Z. (2014), Cross-diffusion induced instability and pattern formation for a Holling type-II predator-prey model, Applied Mathematics and Computation, 247, 1-12.
  5. [5]  Cheng, L. and Cao, H. (2016), Bifurcation analysis of a discrete-time ratio-dependent predator-prey model with allee effect, Communications in Nonlinear Science and Numerical Simulation, 38, 288-302.
  6. [6]  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.
  7. [7]  Leslie, P.H. and Gower, J.C. (1960), The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47(3/4), 219-234.
  8. [8]  Crowley, P.H. and Martin, E.K. (1989), Functional responses and interference within and between year classes of a dragonfly population, Journal of the North American Benthological Society, 8(3), 211-221.
  9. [9]  Ali, N. and Jazar, M. (2013), Global dynamics of a modified Leslie-Gower predator-prey model with Crowley- Martin functional responses, Journal of Applied Mathematics and Computing, 43(1-2), 271-293.
  10. [10]  Zhou, J. (2015), Qualitative analysis of a modified Leslie-Gower predator-prey model with Crowley-Martin functional responses, Communications on Pure and Applied Analysis, 14(3), 1127-1145.
  11. [11]  Yin, H., Xiao, X., Wen, X., and Liu, K. (2014), Pattern analysis of a modified Leslie-Gower predator-prey model with Crowley-Martin functional response and diffusion, Computers and Mathematics with Applications, 67(8), 1607-1621.
  12. [12]  Ren, J., Yu, L., and Siegmund, S. (2017), Bifurcations and chaos in a discrete predator-prey model with Crowley-Martin functional response, Nonlinear Dynamics, 90, 19-41.
  13. [13]  Tripathi, J.P., Meghwani, S.S., Thakur, M., and Abbas, S. (2018), A modified Leslie-Gower predator-prey interaction model and parameter identifiability, Communications in Nonlinear Science and Numerical Simulation, 54, 331-346.
  14. [14]  Ji, L. and Wu, C. (2010), Qualitative analysis of a predator-prey model with constant-rate prey harvesting incorporating a constant prey refuge, Nonlinear Analysis: Real World Applications, 11(4), 2285-2295.
  15. [15]  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.
  16. [16]  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, Communications on Pure and AppliedAnalysis, 15(3), 1041-1055.
  17. [17]  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.
  18. [18]  Jana, D., Pathak, R., and Agarwal, M. (2016), On the stability and hopf bifurcation of a prey generalist predator system with independent age-selective harvesting, Chaos, Solitons and Fractals, 83, 252-273.
  19. [19]  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.
  20. [20]  Gupta, R.P., Chandra, P., and Banerjee, M. (2015), Dynamical complexity of a prey-predator model with nonlinear predator harvesting, Discrete and Continuous Dynamical Systems-Series B, 20(2), 423-443.
  21. [21]  Yuan, R., Jiang, W., andWang, Y. (2015), Saddle-node-hopf bifurcation in a modified Leslie-Gower predatorprey model with time-delay and prey harvesting, Journal of Mathematical Analysis and Applications, 422(2), 1072-1090.
  22. [22]  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.
  23. [23]  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.
  24. [24]  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.
  25. [25]  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.
  26. [26]  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.
  27. [27]  Li, Y. and Li, C. (2013), Stability and Hopf bifurcation analysis on a delayed Leslie-Gower predator-prey system incorporating a prey refuge, Applied mathematics and computation, 219(9), 4576-4589.
  28. [28]  Liu, J. and Zhang, L. (2016), Bifurcation analysis in a prey-predator model with nonlinear predator harvesting, Journal of the Franklin Institute, 353(17), 4701-4714.
  29. [29]  Pal, P.J., Mandal, P.K., and Lahiri, K.K. (2014), A delayed ratio-dependent predator- prey model of interacting populations with Holling type III functional response, Nonlinear Dynamics, 76(1), 201-220.
  30. [30]  Hutchinson, G.E. (1948), Circular causal systems in ecology. Annals of the New York Academy of Sciences, 50(1), 221-246.

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