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On stationary patterns of a reaction-diffusion model with autocatalysis and saturation law. (English) Zbl 1148.35094
Summary: Understanding of spatial and temporal behaviour of interacting species or reactants in ecological or chemical systems has become a central issue, and rigorously determining the formation of patterns in models from various mechanisms is of particular interest to applied mathematicians. In this paper, we study a bimolecular autocatalytic reaction-diffusion model with saturation law and are mainly concerned with the corresponding steady-state problem subject to the homogeneous Neumann boundary condition. In particular, we derive some results for the existence and non-existence of non-constant stationary solutions when the diffusion rate of a certain reactant is large or small. The existence of non-constant stationary solutions implies the possibility of pattern formation in this system. Our theoretical analysis shows that the diffusion rate of this reactant and the size of the reactor play decisive roles in leading to the formation of stationary patterns.
MSC:
35Q80Appl. of PDE in areas other than physics (MSC2000)
35J55Systems of elliptic equations, boundary value problems (MSC2000)
35B45A priori estimates for solutions of PDE
92C15Developmental biology, pattern formation
92C40Biochemistry, molecular biology