Stability And Hopf Branch of A Predator-prey Model with Two Time Delays And Refuges Effect

  IJMTT-book-cover
 
International Journal of Mathematics Trends and Technology (IJMTT)
 
© 2020 by IJMTT Journal
Volume-66 Issue-5
Year of Publication : 2020
Authors : SU Xiao-ya, ZHAIYan-hui
  10.14445/22315373/IJMTT-V66I5P503

MLA

MLA Style:SU Xiao-ya, ZHAIYan-hui "Stability And Hopf Branch of A Predator-prey Model with Two Time Delays And Refuges Effect" International Journal of Mathematics Trends and Technology 66.4 (2020):12-31. 

APA Style: SU Xiao-ya, ZHAIYan-hui(2020). Stability And Hopf Branch of A Predator-prey Model with Two Time Delays And Refuges Effect International Journal of Mathematics Trends and Technology, 12-31.

Abstract
This paper mainly investigated a Predator-prey Model with two time delays and refuges effect .By analyzing the characteristic equations,we discussed the local stability of equilibrium point of the system and the sufficient condition for the existence of Hopf branch.By choosing the delay as a bifurcation parameter,we can determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions by using the centermanifold theorem and normal form theory.At last, some numerical simulation results are confirmed that the feasibility of the theoretical analysis.

Reference
[1] FARIA T. Stability and Bifurcation for a Delayed Predator-prey Model and the Effect of Diffusion[J]. Journal of Mathematical Analysis and Applications,2001,254(4):433-643.
[2] CELIK C. The Stability and Hopf Bifurcation for a Predator-prey System with Time Delay[J]. Chaos,Solitons and Fractals,2008,37(1):87-99.
[3] CELIK C. Hopf Bifurcation of a Ratio-dependent Predator-prey System with Time Delay[J]. Chaos,Solitons and Fractals,2009,42(3):1474-1484.
[4] YUAN S. SONG Y. Stability and Hopf Bifurcation in a Delayed Leslie-Gower Predator-prey System[J]. Journal of Mathematical Analysis and Applications.2009.355(1):82-100.
[5] MA Y. Global Hopf Bifurcation in the Leslie-Gower Predator-preyModel with Two Delays[J]. Nonlinear Analysis:Real World Application,2012,13(1):370-375.
[6] CHEN Y. ZHANG F. Dynamics of a Delayed Predator-preyModel with Predator Migration[J]. Applied Mathematical Modelling,2013,37(3):1400-1412.
[7] B.D.Hassard,N.D.Kazarinoff ,Y.H.Wan,Theory and Applications of Hopf Bifurcations,Cambridge University Press,Cambridge.1981.

Keywords
Hopf bifurcation,Stability,Two time delays ,Center manifold theorem,Predator-prey Model