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We describe and analyze a bistable reaction-diffusion (RD) model for twointerconverting chemical species that exhibits a phenomenon of wave-pinning: awave 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 thisprocess. This behavior arises in a model for chemical polarization of a cell byRho GTPases in response to stimulation. The initially spatially homogeneousconcentration profile (representative of a resting cell) develops into anasymmetric stationary front profile (typical of a polarized cell). Wave-pinninghere 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. Usingmatched asymptotic analysis, we explain the mathematical basis of wave-pinning,and predict the speed and pinned position of the wave. An analysis of thebifurcation of the pinned front solution reveals how the wave-pinning regimedepends 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 thedetails of the reaction kinetics: a saddle-node or a pitchfork bifurcation.

Asymptotic and bifurcation analysis of wave-pinning in a reaction-diffusion model for cell polarization