[ 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

Spatiotemporal Patterns and Bifurcation Analysis of a Diffusive Predator-Prey Model with Hyperbolic Mortality

Journal of Applied Nonlinear Dynamics 14(4) (2025) 795--806 | DOI:10.5890/JAND.2025.12.003

M. Sivakumar$^1$, S. Dharani$^1$, K. Balachandran$^2$

$^1$ Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, Tamil Nadu, India

$^3$ Department of Mathematics, Bharathiar University, Coimbatore, India

Download Full Text PDF

 

Abstract

The dynamics of the predator-prey system with hyperbolic mortality subject to Neumann boundary conditions are investigated. Stability of the positive equilibrium have been discussed through distribution of the eigenvalues. With different initial values, rich spatial patterns in Turing-Hopf domain are obtained. Especially, the labyrinthine-like patterns are also discovered close to the codimension two Turing-Hopf bifurcation point under suitable conditions.

References

  1. [1]  Lotka, A.J. (1920), Analytical note on certain rhythmic relations in organic systems, Proceedings of the National Academy of Sciences, 6(7), 410-415.
  2. [2]  Volterra, V. (1926), Variazioni e fluttuazioni del numero d’individui in specie animali conviventi, Memoria della Reale Accademia Nazionale dei Lincei, 15(2), 31-113.
  3. [3]  Bazykin, A.D. (1976), Structural and Dynamic Stability of Model Predator-Prey Systems, IIASA, Laxenburg, Austria.
  4. [4]  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.
  5. [5]  Peng, R. and Wang, M. (2007), Global stability of the equilibrium of a diffusive Holling-Tanner prey-predator model, Applied Mathematics Letters, 20, 664-670.
  6. [6]  Sivakumar, M. and Balachandran, K. (2016), Phase portraits, Hopf bifurcations and limit cycles of the ratio-dependent Holling-Tanner models for predator-prey interactions, Journal of Applied Nonlinear Dynamics, 5, 283-304.
  7. [7]  Wang, Y.X. and Li, W.T. (2013), Spatial patterns of the Holling-Tanner predator-prey model with nonlinear diffusion effects, Applicable Analysis, 92, 2168-2181.
  8. [8]  Du, L. and Wang, M. (2010), Hopf bifurcation analysis in the 1-D Lengyel-Epstein reaction-diffusion model, Journal of Mathematical Analysis and Applications, 366, 473-485.
  9. [9]  Yi, F., Wei, J., and Shi, J. (2008), Diffusion-driven instability and bifurcation in the Lengyel-Epstein system, Nonlinear Analysis: Real World Applications, 9, 1038-1051.
  10. [10]  Zhang, X. and Zhao, H. (2014), Bifurcation and optimal harvesting of a diffusive predator-prey system with delays and interval biological parameters, Journal of Theoretical Biology, 363, 390-403.
  11. [11]  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.
  12. [12]  Zhang, X. and Zhao, H. (2016), Stability and bifurcation of a reaction-diffusion predator-prey model with non-local delay and Michaelis-Menten-type prey-harvesting, International Journal of Computer Mathematics, 93(9), 1447-1469.
  13. [13]  Taylor, R.J. (1984), Predation, Chapman and Hall, New York.
  14. [14]  Ghosh, J., Sahoo, B., and Poria, S. (2017), Prey-predator dynamics with prey refuge providing additional food to predator, Chaos, Solitons $\&$ Fractals, 96, 110-119.
  15. [15]  Ripple, W.J. and Beschta, R.L. (2004), Wolves and the ecology of fear: Can predation risk structure ecosystems?, BioScience, 54(8), 755-766.
  16. [16]  Turing, A.M. (1952), The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society of London. Series B, Biological Science, 237, 37-72.
  17. [17]  Sivasamy, R., Sivakumar, M., Balachandran, K., and Sathiyanathan, K. (2019), Spatial pattern of ratio-dependent predator-prey model with prey harvesting and cross-diffusion, International Journal of Bifurcation and Chaos, 29(1), 1950006, 1-16.
  18. [18]  Xiao, J. and Xia, Y. (2023), Global dynamics of a predator–prey model with simplified Holling type IV functional response and fear effect, International Journal of Bifurcation and Chaos, 33(2), 2350098.
  19. [19]  Wu, D., Lu, F., Shen, C., and Gao, J. (2023), Impact of spatial memory on a predator–prey system with Allee effect, International Journal of Bifurcation and Chaos, 33(2), 2350086.
  20. [20]  Liu, Y. and Wei, X. (2023), Turing–Hopf bifurcation analysis of the Sel’kov–Schnakenberg system, International Journal of Bifurcation and Chaos, 33(2), 2350012.
  21. [21]  Cai, J., Pinto, M., and Xia, Y. (2022), Stability and bifurcation analysis of a commensal model with Allee effect and herd behavior, International Journal of Bifurcation and Chaos, 32(2), 2250216.
  22. [22]  Sivakumar, M., Balachandran, K., and Karuppiah, K. (2017), Bifurcation and spatiotemporal patterns of a density-dependent predator-prey model with Crowley-Martin functional response, International Journal of Biomathematics, 10(1), 1750079, 1-25.
  23. [23]  Sambath, M., Balachandran, K., and Suvinthra, M. (2015), Stability and Hopf bifurcation of a diffusive predator-prey model with hyperbolic mortality, Complexity, 12(S1), 34-43.
  24. [24]  Kerner, E.H. (1957), A statistical mechanics of interacting biological species, Bulletin of Mathematical Biology, 19(2), 121-146.
  25. [25]  Murray, J.D. (2002), Mathematical Biology II: Spatial Models and Biomedical Applications, Springer-Verlag, Berlin.
  26. [26]  Murray, J.D. (2002), Mathematical Biology I: An Introduction, Springer-Verlag, Berlin.
  27. [27]  Fan, Y. (2014), Pattern formation of an epidemic model with cross diffusion, Applied Mathematics and Computation, 228, 311-319.

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