Phase reduction of reaction-diffusion systems with delay

We develop a phase reduction method for reaction-diffusion systems with a discrete delay. On the basis of the recent developments in the phase reduction theory for infinite-dimensional systems, we introduce a bilinear form tailored to spatially extended systems involving a discrete delay. By solving the adjoint equation associated with the bilinear form, we obtain the phase sensitivity function, which quantifies the shift of the phase in response to a given perturbation. The theory is verified numerically with the use of the Schnakenberg system with a discrete delay in one spatial dimension. We further demonstrate the utility of the theory by optimizing the interaction between a pair of the Schnakenberg systems, with the use of the phase equation, for maximizing the stability of in-phase synchronization. This study serves as a step toward establishing a theory for analyzing oscillatory systems that involve both spatial degrees of freedom and delay.