Convolution Representation of Traveling Pulses in Reaction-Diffusion Systems

Convolution representation manifests itself as an important tool in the reduction of partial differential equations. In this study, we consider the convolution representation of traveling pulses in reaction-diffusion systems. Under the adiabatic approximation of inhibitor, a two-component reaction-diffusion system is reduced to a one-component reaction-diffusion equation with a convolution term. To find the traveling speed in a reaction-diffusion system with a global coupling term, the stability of the standing pulse and the relation between traveling speed and bifurcation parameter are examined. Additionally, we consider the traveling pulses in the kernel-based Turing model. The stability of the spatially homogeneous state and most unstable wave number are examined. The practical utilities of the convolution representation of reaction-diffusion systems are discussed.

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