We carry out computer simulations of an excitable reaction-diffusion equation for an activator and an inhibitor both in one and two dimensions to study Various pattern formations such as propagating pulses, and concentric and spiral waves. By choosing a suitable nonlinearity, a stable limit cycle solution can coexist with an equilibrium uniform solution. In this situation, the excitability is still preserved in the sense that a propagating pulse is Stable. We have found that propagating pulses do not always annihilate upon collision but cause a domain that emits outgoing wave trains. In two dimensions a concentric wave (target pattern) is formed spontaneously without any pacemaker. The mechanism for these dynamical structures is qualitatively discussed.
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