Nonequilibrium potential approach: Local and global stability of stationary patterns in an activator-inhibitor system with fast inhibition

We study the formation and global stability of stationary patterns in a finite one-dimensional reaction-diffusion model of the activator-inhibitor type. The analysis proceeds through the study of the nonequilibrium potential or Lyapunov functional for this system considering the fast inhibitor case and, in order to obtain analytical results, the adoption of a piecewise linear version of the model. We have studied the changes in relative stability among the different patterns as the ratio between the diffusion coefficients is varied and have discussed the meaning of the different contributions to the nonequilibrium potential.