Self-organization in an excitable reaction-diffusion system. III. Motionless localized versus propagating-pulse solutions

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Abstract
In a series of papers, we have studied pattern dynamics in the Bonhoeffer–van der Pol type reaction-diffusion system which is a coupled set of equations for an activator and an inhibitor and exhibits excitability. We have been concerned mainly with localized motionless solutions in one dimension, which have been shown to be stable when the diffusion constant of the inhibitor is sufficiently large. In this paper, we shall explore how the properties of the system change when we decrease the diffusion constant. It is shown that a motionless localized solution turns out to be unstable in such a situation while a propagating-pulse solution can exist stably. This crossover from the motionless to the pulse solutions does not occur as a clear bifurcation. There is a parameter regime where the two solutions can coexist, and rich variety of dynamical patterns is expected.