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A reaction-diffusion system exhibiting Turing's diffusion driven instabilityis considered. The equation for an activator is supplemented by unilateralterms of the type $s_{-}(x)u^{-}$, $s_{+}(x)u^{+}$ describing sources and sinksactive only if the concentration decreases below and increases above,respectively, the value of the basic spatially constant solution which isshifted to zero. We show that the domain of diffusion parameters in whichspatially non-homogeneous stationary solutions can bifurcate from that constantsolution is smaller than in the classical case without unilateral terms. It isa dual information to previous results stating that analogous terms in theequation for an inhibitor imply the existence of bifurcation points even indiffusion parameters for which bifurcation is excluded without unilateralsources. The case of mixed (Dirichlet-Neumann) boundary conditions as well asthat of pure Neumann conditions is described.

Unilateral sources and sinks of an activator in reaction-diffusion systems exhibiting diffusion-driven instability