Stability of delay-coupled reaction-diffusion systems with Turing or wave instability

A quenching phenomenon that emerges in coupled oscillators, namely, amplitude death (AD), has been widely accepted. The present paper analytically deals with AD in two identical reaction-diffusion systems with periodic boundary or zero-flux boundary conditions that are coupled by a diffusive connection, namely, the static connection or the delay connection. The main results of the present paper are to guarantee the following facts analytically. For reaction-diffusion systems with Turing instability, neither the static connection nor the delay connection induces AD. For reaction-diffusion systems with wave instability, the static connection cannot induce AD, but the delay connection can induce AD. These facts do not depend on the number of state variables in the reaction-diffusion system. Some numerical examples with the extended Brusselator model having three state variables demonstrate these facts.