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.
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
