Chebyshev collocation methods for resolving spike dynamics in the GM reaction-diffusion model

In this work the Gierer-Meinhardt model is analyzed using Chebyshev collocation methods. This model is classified as a reaction-diffusion system and arises from biological pattern formations. Reaction-diffusion systems are characterized by having one substance, the activator, which steadily grows in concentration, while another substance, the inhibitor, absorbs the activator and lowers the activator's concentration. This work analyzes a variety of parameters for the Gierer-Meinhardt system which influence how quickly the inhibitor responds to activator growth. Typically when the inhibitor reacts quickly to increases in activator concentration a steady state will be reached. This work analyzes the system in one dimension and then extends the analysis into two dimensions; numerical results are presented for both cases. We have explored the computational and time complexity of the algorithms developed, and compared these results to published results obtained by using a finite difference method and a moving finite element method. Finally, observations are made concerning when to use the proposed spectral method as opposed to the established moving mesh method.