The Characteristics of Patterns in Simple Discrete Reaction-Diffusion Systems of Different Dimensionality and Number of Species

Abstract: The formation of smooth quasi-periodic patterns in static discrete reaction-diffusion systems of various kinds is here studied from a mathematical point of view. The regularity of the patterns can be attributed to the integrability of the system, so that, when searching for such patterns in a system, it is important to investigate whether or not there exist any conserved quantities. The work presented herein is an attempt to generalize a known method, which finds a mathematical representation of the reaction mechanisms allowing for a conserved quantity in the one-dimensional one-species case. It is shown that although the method does not generalize to higher spatial dimensionality, it is possible to generalize to an arbitrary number of species in one dimension, where the reaction mechanism is a generalization of that found in the known method. A fixed-point relation for the system is found and analyzed for an arbitrary number of species, whereas the stabilities and other characteristics of the fixed-points are studied for the two-species case. Numerical simulations show that quasi-periodic patterns indeed can exist along the spatial dimension of the two-species system, and have characteristics that agree with the developed theory. The stabilities in time of these patterns are unclear, since a static case of the reaction-diffusion dynamics is studied. It should be noted that the static case investigated is not uniquely defined by reaction-diffusion dynamics, so that the patterns, if stable, might be found so under some dynamics other than reaction-diffusion.