Bifurcation and stability analysis of steady states to a Brusselator model

In this paper we consider the structure of the nonnegative steady state solutions of a Brusselator model. We prove the existence and boundedness of continua of the steady state solutions by applying the theorems of global bifurcation theory. The concentration of one intermediary reactant is treated as a bifurcation parameter. Through the asymptotic bifurcation analysis, we derive an explicit formula for the non-constant steady states (i.e., stationary patterns). Using it as a base, we establish the stability criteria and find a selection mechanism of the wave modes for the stable patterns by computing the leading term of the principal eigenvalue. Our results demonstrate that all of the bifurcations other than the one at the first bifurcation location are unstable, and if the pattern is stable, then its mode must be a positive integer which minimizes the bifurcation parameter. A specific example is presented to illustrate our analytical results.