[ 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

Consequence of Prey Refuge in a Tri-trophic Prey-dependent Food Chain Model with Intra-specific Competition

Journal of Applied Nonlinear Dynamics 5(2) (2016) 199--219 | DOI:10.5890/JAND.2016.06.007

Nijamuddin Ali$^{1}$; Santabrata Chakravarty$^{2}$

$^{1}$ Department of Education, Katwa College, Dist-Burdwan, West Bengal, India, 713130

$^{2}$ Department of Mathematics, Visva-Bharati, Dist-Birbhum, West Bengal, India, 731235

Download Full Text PDF

 

Abstract

The present article deals with the influence of a constant proportion of prey refuge in presence of intra-specific competition among predator population of a prey-dependent three species food chain model. The behaviour of the system near the biologically feasible equilibria is thoroughly analyzed. The preliminary results such as boundedness and dissipativeness of the system are established. Stability analysis including local and global stability of the equilibria has been carried out in order to examine the behaviour of the system. The present system experiences Hopf-Andronov bifurcation for suitable choice of the parameter values. The influences of the prey refuge parameters on the dynamical behaviour of the system are exhibited through several plots and discussed at some equilibrium positions. It is worth-noting that prey refuge has stabilization effect in some selected situations and bears the potential to control chaotic dynamics of the system. Hence, prey refuge may be of some use for biological control mechanism. Numerical simulations are performed to validate the applicability of the model under consideration.

Acknowledgments

This research work is supported by Minor Research Project of University Grants Commission, New Delhi, India vide Ref. No. F. No. PSW-021/14-15 (ERO), ID No. WBI-042 dated 03.02.2015.

References

  1. [1]  Sih, A. (1987), Prey refuges and predator-prey stability, Theoretical Population Biology, 31, 1-12.
  2. [2]  Sih, A., Petranka, J. W. and Kats, L. B. (1988), The dynamics of prey refuge use: a model and tests with sunfish and salamander larvae, American Naturalist, 132, 463-483.
  3. [3]  Ko, W. and Ryu, K. (2006), Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a prey refuge, Journal of Differential Equations, 231, 534-550.
  4. [4]  Kar, T. K. (2005), Stability analysis of a prey-predator model incorporating a prey refuge, Communications in Nonlinear Science and Numerical Simulation, 10, 681-691.
  5. [5]  Collings, J. B. (1995), Bifurcation and stability analysis of a temperature-dependent mite predator-prey interaction model incorporating a prey refuge, Bulletin of mathematical biology, 57, 63-76.
  6. [6]  Huang, Y., Chen, F. and Zhong, L. (2006), Stability analysis of a prey-predator model with Holling type III response function incorporating a prey refuge, Applied Mathematics and Computation, 182, 672-683.
  7. [7]  Chen, F., Chen, L. and Xie, X. (2009), On a Leslie-Gower predator-prey model incorporating a prey refuge, Nonlinear Analysis: Real World Applications, 10, 2905-2908.
  8. [8]  Olivares, E. G. and Jiliberto, R. R. (2003), Dynamic consequences of prey refuges in a simple model system: more prey, fewer predators and enhanced stability, Ecological Modelling, 166, 135-146.
  9. [9]  Chen, L. and Chen, F. (2010), Global analysis of a harvested predator-prey model incorporating a constant prey refuge, International Journal of Biomathematics, 3, 205-223.
  10. [10]  Tao, Y., Wang, X. and Song, X. (2011), Effect of prey refuge on a harvested predator-prey model with generalized functional response, Communications in Nonlinear Science and Numerical Simulation, 16, 1052- 1059.
  11. [11]  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, 2285-2295.
  12. [12]  Liu, X. and Han, M. (2011), Chaos and Hopf bifurcation analysis for a two species predator-prey system with prey refuge and diffusion, Nonlinear Analysis: Real World Applications, 12, 1047-1061.
  13. [13]  Wang, Y. and Wang, J. (2012), Influence of prey refuge on predator-prey dynamics, Nonlinear Dynamics, 67, 191-201.
  14. [14]  Guan, X., Wang, W. and Cai, Y. (2011), Spatiotemporal dynamics of a Leslie-Gower predator-prey model incorporating a prey refuge, Nonlinear Analysis: Real World Applications, 12, 2385-2395.
  15. [15]  Jana, S., Chakraborty, M., Chakraborty, K. and Kar, T. K. (2012), Global stability and bifurcation of time delayed prey-predator system incorporating prey refuge, Mathematics and Computers in Simulation, 85, 57-77.
  16. [16]  Kuang, Y., Fagan, W. F. and Loladze, I. (2003), Biodiversity, habitat area, resource growth rate and interference competition, Bulletin of Mathematical Biology, 65, 497-518.
  17. [17]  Ruan, S. and He, X. Z. (1998), Global stability in chemostat-type competition models with nutrient recycling, SIAM Journal on Applied Mathematics, 58, 170-192.
  18. [18]  Lobry, C. and Harmand, J. (2006), A new hypothesis to explain the coexistence of n species in the presence of a single resource, Comptes Rendus Biologies, 329, 40-46.
  19. [19]  Lobry, C., Rapaport, A. and Mazenc, F. (2006), Sur un modele densite-dependant de competition pour une ressource, Comptes Rendus Biologies, 329, 63-70.
  20. [20]  Grognard, F., Mazenc, F. and Rapaport, A. (2007), Polytopic Lyapunov functions for persistence analysis of competing species, Discrete and Continuous Dynamical Systems B, 8,73-93.
  21. [21]  Hastings, A. and Powell, T. (1991), Chaos in a three-species food chain, Ecology, 72, 896-903.
  22. [22]  Abrams, P.A. and Roth, J.D. (1994), The effects of enrichment of three-species food chains with nonlinear functional responses, Ecology, 1118-1130.
  23. [23]  Feo, O. D. and Rinaldi, S. (1998), Singular homoclinic bifurcations in tritrophic food chains, Mathematical Biosciences, 148, 7-20.
  24. [24]  Eisenberg, J. N. and Maszle, D. R. (1995), The structural stability of a three-species food chain model, Journal of Theoretical Biology, 176, 501-510.
  25. [25]  Klebanoff, A. and Hastings, A. (1994), Chaos in three species food chains, Journal of Mathematical Biology, 32, 427-451.
  26. [26]  Mccann, K. and Yodzis, P. (1995), Bifurcation structure of a three-species food-chain model, Theoretical Population Biology, 48, 93-125.
  27. [27]  Kuznetsov, Y.A. and Rinaldi, S. (1996), A Remarks on food chain dynamics, Mathematical Biosciences, 134, 1-33.
  28. [28]  Chiu, C.H. and Hsu, S.B. (1998), Extinction of top-predator in a three-level food-chain model, Journal of Mathematical Biology, 37, 372-380.
  29. [29]  Haque, M. and Ali, N. and Chakravarty, S. (2013), Study of a tri-trophic prey-dependent food chain model of interacting populations, Mathematical Biosciences, 246, 55-71.
  30. [30]  Birkhoff, G. and Rota, G. C. (1989), Ordinary Differential Equations, Ginn, Boston.
  31. [31]  Hale, J.K. (1969), Ordinary Differential Equations, Wiley-Interscience, New York.
  32. [32]  Cressman, R. and Garay, J. (2009), A predator-prey refuge system: evolutionary stability in ecological systems, Theoretical Population Biology, 76, 248-257.
  33. [33]  Freedman, I. H. (1980), Deterministic mathematical models in population ecology, Marcel dekker, inc.
  34. [34]  Sun, Y. G. and Saker, S. H. (2006), Positive periodic solutions of discrete three-level food-chain model of Holling type II, Applied Mathematics and Computation, 180, 353-365.
  35. [35]  Hsu, S.B., Hwang, T.W. and Kuang, Y. (2003), A ratio-dependent food chain model and its applications to biological control, Mathematical Biosciences, 181, 55-83.
  36. [36]  Zhao, M. and Lv, S. (2009), Chaos in a three-species food chain model with a Beddington-DeAngelis functional response, Chaos, Solitons & Fractals, 40, 2305-2316.

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