Back
 JAMP  Vol.3 No.7 , July 2015
Hopf Bifurcation Analysis for a Modified Time-Delay Predator-Prey System with Harvesting
Abstract: In this paper, we consider the direction and stability of time-delay induced Hopf bifurcation in a delayed predator-prey system with harvesting. We show that the positive equilibrium point is asymptotically stable in the absence of time delay, but loses its stability via the Hopf bifurcation when the time delay increases beyond a threshold. Furthermore, using the norm form and the center manifold theory, we investigate the stability and direction of the Hopf bifurcation.
Cite this paper: Ni, Y. , Meng, Y. and Ding, Y. (2015) Hopf Bifurcation Analysis for a Modified Time-Delay Predator-Prey System with Harvesting. Journal of Applied Mathematics and Physics, 3, 771-780. doi: 10.4236/jamp.2015.37094.
References

[1]   Lotka, A.J. (1925) Elements of Physical Biology. Nature, 116, 461. http://dx.doi.org/10.1038/116461b0

[2]   Volterra, V. (1926) Fluctuations in The Abundance of A Species Considered Mathematically. Nature, 118, 558-560. http://dx.doi.org/10.1038/118558a0

[3]   Yan, X. and Li, W. (2007) Bifurcation and Global Periodic Solutions in A Delayed Facultative Mutualism System. Physica D: Nonlinear Phenomena, 227, 51-69. http://dx.doi.org/10.1016/j.physd.2006.12.007

[4]   Tian, C. and Zhang, L. (2013) Hopf Bifurcation Analysis in a Diffusive Food-Chain Model with Time Delay. Computers & Mathematics with Applications, 66, 2139-2153. http://dx.doi.org/10.1016/j.camwa.2013.09.002

[5]   Kar, T.K. and Jana, S. (2012) Stability and Bifurcation Analysis of a Stage Structured Predator Prey Model with Time Delay. Applied Mathematics & Computation, 219, 3779-3792. http://dx.doi.org/10.1016/j.amc.2012.10.007

[6]   Zhang, J., Li, W. and Yan, X. (2011) Hopf Bifurcation and Turing Instability in Spatial Homogeneous and Inhomogeneous Predator-Prey Models. Applied Mathematics & Computation, 218, 1883-1893. http://dx.doi.org/10.1016/j.amc.2011.06.071

 
 
Top