  In this paper we consider mathematical modeling of the dynamics of
self-organized patterning of spatially confined human embryonic stem cells
(hESCs) treated with BMP4 (gastruloids) described in recent experimental works.
In the first part of the paper we use the activator-inhibitor equations of
Gierer and Meinhardt to identify 3 reaction-diffusion regimes for each of the
three morphogenic proteins, BMP4, Wnt and Nodal, based on the characteristic
features of the dynamic patterning. We identify appropriate boundary conditions
which correspond to the experimental setup and perform numerical simulations of
the reaction-diffusion (RD) systems, using the finite element approximation, to
confirm that the RD systems in these regimes produce realistic dynamics of the
protein concentrations. In the second part of the paper we use analytic tools
to address the questions of the existence and stability of non-homogeneous
steady states for the reaction-diffusion systems of the type considered in the
first part of the paper. We find sufficient conditions on the data of the
problem under which the system has an universal attractor.
