The dynamics of pulse solutions for reaction diffusion systems on a star shaped metric graph with the kirchhoff?s boundary condition

In this paper, we consider motions of localized patterns for reaction-diffusion systems of general types on a metric star graph which consists of several half-lines with a common end point called "the junction point", where the Kirchhoff boundary condition is imposed. Assuming the existence and the stability of pulse and front like patterns for corresponding 1dimensional prob-lems of reaction-diffusion systems, we rigorously derive ordinary differential equations describing the motions of them on a metric star graph. As the ap-plication, the attractive motion of a single pulse solution for the Gray-Scott model toward the junction point is shown. It is also shown that a single front solution of Allen-Cahn equation is repulsive against the junction point. The motion of multi pulse solutions and front solutions are also treated.