[ Log In | Register]
! Skip Navigation Links
Skip Navigation Links
Image of Discontinuity, Nonlinearity and Complexity
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

Bifurcation and Stability Analysis of a Diffusive Predator-Prey Model with Ratio-Dependent Type III Functional Response

Discontinuity, Nonlinearity, and Complexity 6(3) (2017) 393--407 | DOI:10.5890/DNC.2017.09.008

M. Sivakumar; K. Balachandran

DRDO-BU-CLS, Bharathiar University, Coimbatore - 641046, India.

Department of Mathematics, Bharathiar University, Coimbatore - 641046, India.

Download Full Text PDF

 

Abstract

This paper is concerned with a diffusive Leslie-Gower predator-prey system with ratio dependent Holling type III functional response subject to Neumann boundary conditions. By linearizing the system at the positive constant steady-state solution and analyzing the associated characteristic equation in detail, local stability, existence of a Hopf bifurcation at the coexistence of the equilibrium and stability of bifurcating periodic solutions of the system in the absence of diffusion are studied. Furthermore, Turing instability and Hopf bifurcation analysis for the system with diffusion are studied. Finally, numerical simulations are provided in order to verify our theoretical results.

References

  1. [1]  Murdoch,W., Briggs, C. (2003), Consumer-Resource Dynamics, Princeton University Press, New York.
  2. [2]  May, R.M. (1973), Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton.
  3. [3]  Holling, C.S. (1965), The functional response of predator to prey density and its role in mimicry and population regulation, Memories of the Entomological Society of Canada, 45, 5-60.
  4. [4]  Tanner, J.T. (1975), The stability and the intrinistic growth rates of prey and predator populations, Ecology, 56, 855- 886.
  5. [5]  Wollkind, D.J., Collings, J.B., and Logan, J.A. (1988),Metastability in a temperature-dependedmodel system for trees, Bulletin of Mathematical Biology, 50, 379-409.
  6. [6]  Murray, J.D. (2002), Mathematical Biology I: An Introduction, Springer-Verlag, Berlin.
  7. [7]  Leslie, P.H. and Gower, J.C. (1960), The properties of a stochastic model for the predator-prey type of interactions between two species, Biometrica, 47, 219-234.
  8. [8]  Shi, H. B. and Li, Y. (2015), Global asymptotic stability os a dffusive predator prey model with ratio-dependent functional responses, Applied Mathematics and Computation, 250, 71-77.
  9. [9]  Turing, A.M. (1952), The chemical basis ofmorphogenesis, Philosophical Transactions of the Royal Society of Londan. Series B, Biological Science, 237, 37-72.
  10. [10]  Li, X., Jiang, W. and Shi, J. (2013), Hopf bifurcation and Turing instability in the reaction-diffusion Holling-Tanner predator-prey model, IMA Journal of Applied Mathematics, 78, 287-306.
  11. [11]  Liao, M., Tang, X., and Xu, C. (2011), Stability and instability analysis for a ratio-dependent predator-prey system with diffusion effects, Nonlinear Analysis RWA, 12, 1616-1626.
  12. [12]  Zhang, J.F., Li, W.T., and Yan, X.P. (2011), Hopf bifurcation and Turing instability in spatial homogeneous and inhomogeneous predator-prey models, Applied Mathematics and Computation, 218, 1883-1893.
  13. [13]  Allman, E.S. and Rhodes, J.A. (2004),Mathematical Models in Biology, An Introduction, Cambridge University Press, Cambridge.
  14. [14]  Garvie, M.R. and Trenchea, C. (2009), Spatiotemporal dynamics of two generic predator-prey models, Journal of Biological Dynamics, 4, 559-570.
  15. [15]  Hallam, T.G. and Deluna, J.T. (1984), Effects of toxicants on populations: a qualitative approach III, Journal of Theoretical Biology, 109, 411-429.
  16. [16]  Hassard, B.D., Kazarinoff, N.D., and Wan, Y.H. (1982), Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge.
  17. [17]  Ma, Z. and Li, W. (2013), Bifurcation analysis on a diffusive Holling-Tanner predator-prey model, Applied Mathematical Modelling, 37, 4731-4384.
  18. [18]  Murray, J.D. (2003), Mathematical Biology II: Spatial Models and Biomedical Applications, Springer-Verlag, Berlin.
  19. [19]  Pielou, E.C. (1969), An Introduction to Mathematical Ecology,Wiley, New York.
  20. [20]  Sivakumar, M., Sambath, M., and Balachandran, K. (2015), Stability and Hopf bifurcation analysis of a diffusive predator-prey model with Smith growth, International Journal of Biomathematics, 18, 1550013, 1-18.
  21. [21]  Sivakumar,M. and Balachandran, K. (2016), Phase Portraits, Hopf Bifurcations and Limit Cycles of the Ratio Dependent Holling-TannerModels for Predator-prey Interactions, Journal of Applied Nonlinear Dynamics, 5(3), 283-304.
  22. [22]  Wang, X. and Wei, J. (2011), Diffusion driven stability and bifurcation in a predator-prey system with Lvlev-type functional response, Applicable Analysis, 90, 1-24.
  23. [23]  Yi, F.,Wei, J., and Shi, J. (2008), Diffusion-driven instability and bifurcation in the Lendyel-Epstein system, Nonlinear Analysis RWA, 9, 1038-1051.
  24. [24]  Zhang, J.F., Li,W.T., andWang, Y.X. (2011), Turing patterns of a strongly coupled predator-prey system with diffusion effects, Nonlinear Analysis, 74, 847-858.
  25. [25]  Wang, M. X. (2008), Stability and Hopf bifurcation for a predator-prey models with prey-stage structure and diffusion, Mathematical Biosciences, 212, 149-160.

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