Stability and Hopf bifurcation in a delayed competition system

Abstract In this paper, Hopf bifurcation for two-species Lotka–Volterra competition systems with delay dependence is investigated. By choosing the delay as a bifurcation parameter, we prove that the system is stable over a range of the delay and beyond that it is unstable in the limit cycle form, i.e., there are periodic solutions bifurcating out from the positive equilibrium. Our results show that a stable competition system can be destabilized by the introduction of a maturation delay parameter. Further, an explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived by using the theory of normal forms and center manifolds, and numerical simulations supporting the theoretical analysis are also given.

[1]  Peixuan Weng,et al.  Global attractivity in a competition system with feedback controls , 2003 .

[2]  Xue-Zhong He,et al.  Oscillations and Convergence in an Almost Periodic Competition System , 1997 .

[3]  Yasuhiro Takeuchi,et al.  Permanence and global attractivity for competitive Lotka-Volterra systems with delay , 1994 .

[4]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[5]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[6]  M. Zhien,et al.  Harmless delays for uniform persistence , 1991 .

[7]  Jinde Cao,et al.  Stability and Hopf bifurcation in a delayed competitive web sites model , 2006 .

[8]  Jianhong Wu.  SYMMETRIC FUNCTIONAL DIFFERENTIAL EQUATIONS AND NEURAL NETWORKS WITH MEMORY , 1998 .

[9]  Xianhua Tang,et al.  Global attractivity of non-autonomous Lotka-Volterra competition system without instantaneous negative feedback , 2003 .

[10]  B. Hassard,et al.  Theory and applications of Hopf bifurcation , 1981 .

[11]  Yasuhisa Saito,et al.  The Necessary and Sufficient Condition for Global Stability of a Lotka–Volterra Cooperative or Competition System with Delays , 2002 .

[12]  Shawgy Hussein,et al.  Stability and Hopf bifurcation for a delay competition diffusion system , 2002 .

[13]  M. Zhien,et al.  Stability switches in a class of characteristic equations with delay-dependent parameters , 2004 .

[14]  J. Hale.  Theory of Functional Differential Equations , 1977 .

[15]  Maoan Han,et al.  Stability and Hopf bifurcations in a competitive Lotka-Volterra system with two delays , 2004 .

[16]  Zhien Ma,et al.  Stability for a competitive Lotka-Volterra system with delays , 2002 .