Scientific Computing, Applied and Industrial Mathematics (SCAIM) Seminar: Lawrence Ward

A lattice-indexed family of stochastic processes has quasi-cycle oscillations if its otherwise-damped oscillations are sustained by noise. Such a family performs the reaction part of a stochastic reaction-diffusion system when we insert a local Mexican Hat-type, difference of Gaussians, coupling on a one-dimensional and on a two-dimensional lattice. In one dimension we find that the phases of the quasi-cycles synchronize (establish a relatively constant relationship, or phase locking) rapidly at coupling strengths lower than those required to produce spatial patterns of their amplitudes. The patterns of phase locking persist and evolve but do not induce patterns in the amplitudes. In two dimensions the amplitude patterns form more quickly, but there remain parameter regimes in which phase patterns form without being accompanied by clear amplitude patterns. At higher coupling strengths we find patterns both of phase synchronization and of amplitude (resembling Turing patterns) corresponding to the patterns of phase synchronization. Specific properties of these patterns are controlled by the parameters of the reaction and of the Mexican Hat coupling.