Richard Tatum, George Mason University


Analysis of a reaction-diffusion system with local and nonlocal terms

Abstract: We analyze a mixed reaction-diffusion system containing an instability that results in nontrivial Turing structures. This system uses the homotopy parameter beta to vary the effect of both local ( beta = 1 ) and nonlocal ( beta = 0 ) diffusion. Furthermore, we consider epsilon-scaled kernels J such that the product of J and epsilon to the theta power is epsilon-independent for all real theta. For theta less than 1 and positive beta less than 1, we show that the generated Turing patterns are explained using only a finite number of eigenfunctions corresponding to the most unstable eigenvalues of the linearization. However, for theta = 1 and beta less than 1, we show how the nonlinearity is no longer bounded above by an epsilon-dependent bound that ensures the smallness of the nonlinearity as in the theta less than 1 case. The lack of this critical bound allows for a greater influence of the nonlinearity. Consequently, the unstable eigenfunctions of the linearization do not describe the solutions as well as they do for the solutions of the theta less than 1 case. The numerics provided show little agreement between the solutions and their linearized counterparts as a consequence of the greater influence of the nonlinearity.

Advisor: Evelyn Sander (George Mason University)