Speed of traveling waves for monotone reaction–diffusion systems as a function of diffusion coefficients

Highlights

• Partial derivatives of the speed with respect to diffusions satisfy a simple identity.

• Bigger diffusion of buffer molecules can slow the calcium wave for exogenous buffers.

• Laplacian of the speed can be negative/positive for endogenous/exogenous buffers.

Abstract

Traveling waves form a basic class of solutions to reaction–diffusion equations, which can describe a large number of phenomena. The basic property characterizing traveling wave solutions is their speed of propagation. In this study we analyze its dependence on the diffusivities of the interacting agents. We show that this dependence is subject to some relations, which can be derived by simple scaling properties. We augment our findings by an investigation of reaction–diffusion systems describing intercellular calcium dynamics in the presence of buffer molecules. We establish some mathematical results concerning the behavior of the velocity of traveling waves in the case of fast buffer kinetics, (paying special attention to the vicinity of zero speed), and present outcomes of numerical simulations showing how complicated the interplay between the diffusion coefficients of calcium and buffering molecules can be, especially in models with more than one kind of buffer molecules.