Gierer-Meinhardt system is a mathematical model to describe biological pattern formation due to activator and inhibitor. Turing pattern is
expected in the presence of local self-enhancement and long-range inhibition.
The long-time behavior of the solution, however, has not yet been clarified mathematically. In this paper, we study the case when its ODE part takes
periodic-in-time solutions, that is, Ï„=s+1. Under some additional assumptions on parameters, we show that the solution exists global-in-time and absorbed into one of these ODE orbits. Thus spatial patterns eventually dis- appear if those parameters are in a region without local self-enhancement or long-range inhibition
