  We describe and analyze a bistable reaction-diffusion (RD) model for two
interconverting chemical species that exhibits a phenomenon of wave-pinning: a
wave of activation of one of the species is initiated at one end of the domain,
moves into the domain, decelerates, and eventually stops inside the domain,
forming a stationary front. The second ("inactive") species is depleted in this
process. This behavior arises in a model for chemical polarization of a cell by
Rho GTPases in response to stimulation. The initially spatially homogeneous
concentration profile (representative of a resting cell) develops into an
asymmetric stationary front profile (typical of a polarized cell). Wave-pinning
here is based on three properties: (1) mass conservation in a finite domain,
(2) nonlinear reaction kinetics allowing for multiple stable steady states, and
(3) a sufficiently large difference in diffusion of the two species. Using
matched asymptotic analysis, we explain the mathematical basis of wave-pinning,
and predict the speed and pinned position of the wave. An analysis of the
bifurcation of the pinned front solution reveals how the wave-pinning regime
depends on parameters such as rates of diffusion and total mass of the species.
We describe two ways in which the pinned solution can be lost depending on the
details of the reaction kinetics: a saddle-node or a pitchfork bifurcation.
