  Many systems of physical and biological interest are characterized by
assemblies of phase oscillators whose interaction is mediated by a diffusing
chemical. The coupling effect results from the fact that the local
concentration of the mediating chemical affects both its production and
absorption by each oscillator. Since the chemical diffuses through the medium
in which the oscillators are embedded, the coupling among oscillators is
non-local: it considers all the oscillators depending on their relative spatial
distances. We considered a mathematical model for this coupling, when the
diffusion time is arbitrary with respect to the characteristic oscillator
periods, yielding a system of coupled nonlinear integro-differential equations
which can be solved using Green functions for appropriate boundary conditions.
In this paper we show numerical solutions of these equations for three finite
domains: a linear one-dimensional interval, a rectangular, and a circular
region, with absorbing boundary conditions. From the numerical solutions we
investigate phase and frequency synchronization of the oscillators, with
respect to changes in the coupling parameters for the three considered
geometries.
