Mathematical Biosciences and Engineering, 2014, 11(4): 807-821. doi: 10.3934/mbe.2014.11.807.

Primary: 35K40, 35K57, 35B36; Secondary: 35Q92.

Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

On a diffusive predator-prey model with nonlinear harvesting

1. Department of Mathematics, Florida Gulf Coast University, 11501 FGCU Blvd. S., Fort Myers, FL 33965

In this paper, we study the dynamics of a diffusive Leslie-Gower model with a nonlinear harvesting term on the prey. We analyze the existence of positive equilibria and their dynamical behaviors. In particular, we consider the model with a weak harvesting term and find the conditions for the local and global asymptotic stability of the interior equilibrium. The global stability is established by considering a proper Lyapunov function. In contrast, the model with strong harvesting term has two interior equilibria and bi-stability may occur for this system. We also give the conditions of Turing instability and perform a series of numerical simulations and find that the model exhibits complex patterns.
  Figure/Table
  Supplementary
  Article Metrics

Keywords pattern formation; global stability.; turing instability; nonlinear harvesting; Leslie-Gower; Predator-prey

Citation: Peng Feng. On a diffusive predator-prey model with nonlinear harvesting. Mathematical Biosciences and Engineering, 2014, 11(4): 807-821. doi: 10.3934/mbe.2014.11.807

References

  • 1. Chaos Sol. and Fractals, 14 (2002), 1275-1293.
  • 2. Appl. Math. Lett., 16 (2003), 1069-1075.
  • 3. J. Math. Biol., 36 (1997), 149-168.
  • 4. J. Diff. Eqns., 246 (2009), 3932-3956.
  • 5. Springer, New-York, 1983.
  • 6. Nonlinear Analysis: Real World Applications, 12 (2011), 2385-2395.
  • 7. J. Math. Anal. Appl., 398 (2013), 278-295.
  • 8. Appl. Math. Lett., 14 (2011), 697-699.
  • 9. J. Comput. Appl Math., 185 (2006), 19-33.
  • 10. Disc. Cont. Dyn. Syst. S, 1 (2008), 303-315.
  • 11. Biometrika, 35 (1948), 213-245.
  • 12. Princeton University Press, Princeton, NJ, 1974.
  • 13. Nonlinear Anal. Real World Appl., 7 (2006), 1104-1118.
  • 14. Bull. Math. Biol., 50 (1988), 379-409.
  • 15. J. Math. Anal and Appl., 324 (2006), 14-29.
  • 16. Science Press, Beijing, 1990.
  • 17. Discrete Dyn. Nat. Soc., (2011), Art. ID 473949, 14 pp.
  • 18. J. Math. Anal. Appl., 384 (2011), 400-408.

 

This article has been cited by

  • 1. Xuebing Zhang, Hongyong Zhao, 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, 2016, 93, 9, 1447, 10.1080/00207160.2015.1056169
  • 2. Fengrong Zhang, Yan Li, Stability and Hopf bifurcation of a delayed-diffusive predator–prey model with hyperbolic mortality and nonlinear prey harvesting, Nonlinear Dynamics, 2017, 88, 2, 1397, 10.1007/s11071-016-3318-8
  • 3. Nayana Mukherjee, S Ghorai, Malay Banerjee, Detection of Turing patterns in a three species food chain model via amplitude equation, Communications in Nonlinear Science and Numerical Simulation, 2018, 10.1016/j.cnsns.2018.09.023

Reader Comments

your name: *   your email: *  

Copyright Info: 2014, Peng Feng, licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved