Stability and Hopf bifurcation for a delayed cooperative system with diffusion effects. (English) Zbl 1162.35319

Yan, Xiang-Ping; Li, Wan-Tong

Int. J. Bifurcation Chaos Appl. Sci. Eng. 18, No. 2, 441-453 (2008).

Summary: The main purpose of this paper is to investigate the stability and Hopf bifurcation for a delayed two-species cooperative diffusion system with Neumann boundary conditions. By linearizing the system at the positive equilibrium and analyzing the corresponding characteristic equation, the asymptotic stability of positive equilibrium and the existence of Hopf oscillations are demonstrated. It is shown that, under certain conditions, the system undergoes only a spatially homogeneous Hopf bifurcation at the positive equilibrium when the delay crosses through a sequence of critical values; under the other conditions, except for the previous spatially homogeneous Hopf bifurcations, the system also undergoes a spatially inhomogeneous Hopf bifurcation at the positive equilibrium when the delay crosses through another sequence of critical values. In particular, in order to determine the direction and stability of periodic solutions bifurcating from spatially homogeneous Hopf bifurcations, the explicit formulas are given by using the normal form theory and the center manifold reduction for partial functional differential equations (PFDEs). Finally, to verify our theoretical predictions, some numerical simulations are also included.

MSC:

35B32	Bifurcations in context of PDEs
35K50	Systems of parabolic equations, boundary value problems (MSC2000)
35B35	Stability in context of PDEs
35K57	Reaction-diffusion equations
37N25	Dynamical systems in biology
92D25	Population dynamics (general)

Keywords:

Neumann boundary conditions; positive equilibrium; spatially homogeneous Hopf bifurcation; spatially inhomogeneous Hopf bifurcation; normal form theory; center manifold reduction; numerical simulations

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