Target patterns in reaction-diffusion systems

We consider the general reaction-diffusion system At = F(A) + ε D_M ²A + εg(→x, A), 0 < ε 1, where the small term εg(→x, A) represents the effects of localized impurities. We assume that the system A_t = F(A) has a stable time-periodic solution. Then we construct stable target pattern solutions of the full system. For typical initial conditions we find that these target patterns will arise only if g(→x, A) 0. Finally, we determine how target patterns interact and show that higher frequency target patterns eventually engulf neighboring lower frequency target patterns.