Pattern formation : exploring patterns through the convergence of mathematics and biology

Abstract

A pattern, according to Oxford Dictionary, is formally defined as "a repeated decorative design". Patterns in nature are exquisite manifestations of order and repetition, transcending mere decoration to reveal the underlying principles of the universe. From the fractal intricacies of snowflakes to the stripes of a zebra, nature鈥檚 patterns captivate our senses and ignite our curiosity. Patterns in nature are not merely ornamental; apart from serving practical purposes, they also illustrate the interconnectedness of all living things. This dissertation delves into the intricate models for pattern formation, presenting a thorough examination of these phenomena and their significance in various fields. A selection of different modeling approaches such as bifurcation theory, limit cycles, symmetry, reaction-diffusion systems, Turing patterns, and travelling plane waves shall be covered. This project also involves extensive use of programming and numerical implementation in Mathematica.