Hopf Bifurcation Analysis for the Pest-Predator Models Under Insecticide Use with Time Delay

Yanhui Zhai; Xiaona Ma; Ying Xiong

doi:https://doi.org/10.14445/22315373/IJMTT-V9P512

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 9 | Number 2 | Year 2014 | Article Id. IJMTT-V9P512 | DOI : https://doi.org/10.14445/22315373/IJMTT-V9P512

Hopf Bifurcation Analysis for the Pest-Predator Models Under Insecticide Use with Time Delay

Yanhui Zhai , Xiaona Ma , Ying Xiong

Abstract

We consider a delayed Pest-predator model under insecticide use. First, the paper considers the stability and local Hopf bifurcation for a modified Pest-predator model with time delay. In succession, using the normal form theory and center manifold argument, we obtain some explicit results which determine the stability, direction and other properties of bifurcation periodic solutions.

Keywords

Hopf bifurcation, Stability, time delay, Pest-predator model, Center manifold, Normal form 2010 Mathematics Subject Classi_cation Primary 34D35; Secondary 34K20

References

[1] B.D.Hassard, N.D.Kazarino_, and Y.H.Wan, Theory and Applications of Hopf Bifurcation, vol.41, Cambridge University Press, Cambridge, UK, 1981.
[2] Zhonghua Lu, Xuebin Chi, Lansun Chen. Impulsive control strategies in biological control of pesticide Theoretical Population Biology 64 (2003),39-47.
[3] Yongli Song, Yahong Peng, Stability and bifurcation analysis on a Logistic model with discrete and distribution delays, Applied Mathematics and Computation 181(2006),1745- 1757.
[4] Wenzhi Zhang, Zhichao jiang, Global Hopf Bifurcations in a Delayed Predator-prey System. ICINA.
[5] Junjie Wei and Chunbo Yu, Hopf bifurcation analysis in a model of oscillatory gene expression with delay, Proceedings of the Royal Society of Edinburgh,139A,879-895,2009.
[6] J. Hale, Theory of functional differential equations (Springer,1977)
[7] J.K. Hale, S.M. Verduyn Lunel, Introduction to functional differential equations. Springer-Verlag, 1995.
[8] G.Mircea, M.Neamtu, D.Opris, Dynamical systems from economy, mechanic and biology described by di_erential equations with time delay. Editura Mirton, 2003 (in Romanian).
[9] N. A. M. Monk. Oscillatory expression of HES1, p53, and NF-KB driven by transcriptional time delays. Curr. Biol. 13 (2003), 1409-1413.
[10] Yongli Song, JunjieWei, Bifurcation analysis for Chen's system with delayed feedback and its application to control of chaos. Chaos, Solutions and Fractals 22 (2004) 75-91.
[11] Y. Song, J. Wei and M. Han, Local and global Hopf bifurcation in a delayed hematopoiesis. Int. J. Bifur. Chaos 14 (2004), 3909-3919.
[12] S.Ruan and J.Wei, On the zeros of transcendental function with applications to stability of delayed differential equations with two delays.Dynam. Cont. Discrete Impuls.Syst.A10 (2003),863-874.
[13] J.Wei, Bifurcation analysis in a scalar delay differential equation. Nonlinearity 20 (2007),2483-2498.

Citation :

Yanhui Zhai , Xiaona Ma , Ying Xiong, "Hopf Bifurcation Analysis for the Pest-Predator Models Under Insecticide Use with Time Delay," International Journal of Mathematics Trends and Technology (IJMTT), vol. 9, no. 2, pp. 115-121, 2014. Crossref, https://doi.org/10.14445/22315373/IJMTT-V9P512