Pattern alternations induced by nonlocal interactions

TL;DR

This paper investigates pattern formation in nonlocal reaction-diffusion systems, deriving amplitude equations to predict Turing pattern selection, and validates the analytical results with numerical simulations.

Contribution

It derives the amplitude equations for nonlocal RD models near Turing bifurcation, extending analytical tools beyond local models and applying to various nonlocal systems.

Findings

Analytical amplitude equations predict pattern selection near bifurcation.

Numerical simulations confirm analytical predictions close to bifurcation.

Good agreement between theory and simulation for small nonlocal parameters.

Abstract

Pattern formation is a visual understanding of the dynamics of complex systems. Patterns arise in many ways, such as the segmentation of animals, bacterial colonies during growth, vegetation, chemical reactions, etc. In most cases, the long-range diffusion occurs, and the usual reaction-diffusion (RD) model can not capture such phenomena. The nonlocal RD model, on the other hand, can fill the gap. Analytical derivation of the amplitude equations (AE) for an RD system is a valuable tool to predict the pattern selections, in particular, the stationary Turing patterns when they occur. In this paper, we analyze the conditions for the Turing bifurcation for the nonlocal model and also derive the AE for the nonlocal RD model near the Turing bifurcation threshold to describe the reason behind the pattern selections. This derivation of the AE is not only limited to the nonlocal prey-predator model, as shown in our representative example but also can be applied to other nonlocal models near the Turing bifurcation threshold. The analytical prediction agrees with numerical simulation near the Turing bifurcation threshold. Moreover, the analytical and numerical results fit each other well even more remote from the Turing bifurcation threshold for the small values of the nonlocal parameter but not for the higher values.