Propagation limits and velocity of reaction-diffusion fronts in a system of discrete random sources

Abstract

The effect of spatially randomizing a system of pointlike sources on the propagation of reaction-diffusion fronts is investigated in multidimensions. The dynamics of the reactive front are modeled by superimposing the solutions for diffusion from a single point source. A nondimensional parameter is introduced to quantify the discreteness of the system, based on the characteristic reaction time of sources compared to the diffusion time between sources. The limits to propagation and the average velocity of propagation are expressed as probabilistic quantities to account for the influence of the randomly distributed sources. In random systems, two- and three-dimensional fronts are able to propagate beyond a limit previously found for systems with regularly distributed sources, while a propagation limit in one dimension that is independent of domain size cannot be defined. The dimensionality of the system is seen to have a strong influence on the front propagation velocity, with higher dimensional systems propagating faster than lower dimensional systems. In a three-dimensional system, both the limit to propagation and average front velocity revert to a solution that assumes a spatially continuous source function as the discreteness parameter is increased to the continuum limit. The results indicate that reactive systems are able to exploit local fluctuations in source concentration to extend propagation limits and increase the velocity in comparison to regularly spaced systems.