Oscillatory and Stationary Patterns in a Diffusive Model with Delay Effect

In this paper, a reaction-diffusion model with delay effect and Dirichlet boundary condition is considered. Firstly, the existence, multiplicity, and patterns of spatially nonhomogeneous steady-state solution are obtained by using the Lyapunov-Schmidt reduction. Secondly, by means of space decomposition, we subtly discuss the distribution of eigenvalues of the infinitesimal generator associated with the linearized system at a spatially nonhomogeneous synchronous steady-state solution, and then we derive some sufficient conditions to ensure that the nontrivial synchronous steady-state solution is asymptotically stable. By using the symmetric bifurcation theory of differential equations together with the representation theory of standard dihedral groups, we not only investigate the effect of time delay on the pattern formation, but also obtain some important results on the spontaneous bifurcation of multiple branches of nonlinear wave solutions and their spatiotemporal patterns.

Automated summary

This paper investigates a reaction-diffusion model with delay effect and Dirichlet boundary condition, obtaining the existence and patterns of spatially nonhomogeneous steady-state solutions through Lyapunov-Schmidt reduction. Furthermore, the stability conditions of nontrivial synchronous steady-state solutions are discussed, along with the effect of time delay on pattern formation. The study also explores the spontaneous bifurcation of multiple branches of nonlinear wave solutions and their spatiotemporal patterns using symmetric bifurcation theory and representation theory of standard dihedral groups.