Bifurcation Behaviors of Steady-State Solution to a Discrete General Brusselator Model

We study the local and global bifurcation of nonnegative nonconstant solutions of a discrete general Brusselator model. We generalize the linear in the standard Brusselator model to the nonlinear . Assume that is a strictly increasing function, and . Taking as the bifurcation parameter, we obtain that the solution set of the problem constitutes a constant solution curve and a nonconstant solution curve in a small neighborhood of the bifurcation point . Moreover, via the Rabinowitz bifurcation theorem, we obtain the global structure of the set of nonconstant solutions under the condition that is nonincreasing in . In this process, we also make a priori estimation for the nonnegative nonconstant solutions of the problem.