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

Abstract

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 problems of reaction-diffusion systems, we rigorously derive ordinary differential equations describing the motions of them on a metric star graph. As the application, 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.