Analysis of a Reaction-diffusion System with Local and Nonlocal Diffusion Terms

Abstract

Reaction-diffusion describes the process in which multiple participating chemicals or agents react with each other, while simultaneously diffusing or spreading through a liquid or gaseous medium. Typically, these processes are studied for their ability to produce nontrivial patterns that evolve over time. These patterns, often referred to as Turing structures or Turing patterns, are diffusion driven. In the presence of diffusion, the Turing patterns are observable, but are not present in the absence of diffusion. It is important for reactiondiffusion models to replicate the behavior that is experimentally observed. That is to say that the models must be able to produce solutions with traits, such as pattern type, that are similar to experimentally observed traits. Mathematically, we seek to explain certain aspects of the models such as pattern selection in the hope of broadening our understanding of the underlying process for which the model represents. I analyze a mixed reaction-diffusion system containing an instability that results in nontrivial Turing structures. This system uses a homotopy parameter β to vary the effect of both local (β = 1) and nonlocal (β = 0) diffusion. Furthermore, I consider −scaled kernels J such that θJ is −independent for θ ∈ R. For θ < 1 and 0 < β ≤ 1, I show that the generated Turing patterns are explained using only finite number of eigenfunctions corresponding to the most unstable eigenvalues of the linearization. However, for θ = 1 and β < 1, I show how the nonlinearity is no longer bounded above by an −dependent bound that ensures the smallness of the nonlinearity as in the θ < 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 θ < 1 case. The numerics provided show little agreement between the solutions and their linearized counterparts as a consequence of greater influence of the nonlinearity. The thesis is concluded with numerical pattern studies of the local and nonlocal reactiondiffusion systems. The patterns are studied as the values of various parameters of the reaction-diffusion system are changed. These numerical experiments reveal typical patterns such as stripes and spots, as well as irregular snakelike patterns. Furthermore, solutions for the local system subject to homogeneous Neumann boundary conditions are compared to the solutions of the local system subject to periodic boundary conditions. For some cases, the solutions for both systems are quite similar.