Fully illustrated mathematical guide to pattern formation. Includes instructive exercises and examples.

An account of how complex patterns form in sustained nonequilibrium systems; for graduate students in biology, chemistry, engineering, mathematics, and physics.

Granular materials are an integral part of our everyday life. They are also the base material for most industrial processing techniques. The highly dissipative nature of the particle collisions means energy input is needed in order to mobilize the grains. This interplay of dissipation and excitation leads to a wide variety of pattern formation processes, which are addressed in this book. The reader is introduced to this wide field by, first, a description of the material properties of granular materials under different experimental conditions that are important in connection with the pattern formation dynamics and, second, by further details given later on in the description of the specific system.

This book addresses topics of mobile multi-agent systems, pattern formation, biological modelling, artificial life, unconventional computation, and robotics. The behaviour of a simple organism which is capable of remarkable biological and computational feats that seem to transcend its simple component parts is examined and modelled. In this book the following question is asked: How can something as simple as Physarum polycephalum - a giant amoeboid single-celled organism which does not possess any neural tissue, fixed skeleton or organised musculature - can approximate complex computational behaviour during its foraging, growth and adaptation of its amorphous body plan, and with such limited resources? To answer this question the same apparent limitations as faced by the organism are applied: using only simple components with local interactions. A synthesis approach is adopted and a mobile multi-agent system with very simple individual behaviours is employed. It is shown their interactions yield emergent behaviour showing complex self-organised pattern formation with material-like evolution. The presented model reproduces the biological behaviour of Physarum; the formation, growth and minimisation of transport networks. In its conclusion the book moves beyond Physarum and provides results of scoping experiments approximating other complex systems using the multi-agent approach. The results of this book demonstrate the power and range of harnessing emergent phenomena arising in simple multi-agent systems for biological modelling, computation and soft-robotics applications. It methodically describes the necessary components and their interactions, showing how deceptively simple components can create powerful mechanisms, aided by abundant illustrations, supplementary recordings and interactive models. It will be of interest to those in biological sciences, physics, computer science and robotics who wish to understand how simple components can result in complex and useful behaviours and who wish explore the potential of guided pattern formation themselves.

We present examples of familiar phenomena found in nonequilibrium systems, including oscillatory phenomena, order-formation processes, and pattern formation. In particular, we introduce commonly used mathematical methods to analyze their characteristics. First, we present oscillations described by the Lotka–Volterra and van der Pol equations, the Brusselator, the Oregonator, and relaxation oscillations as examples of oscillatory phenomena. Second, we investigate the order-formation process in colloidal crystals and present an experimental observation of 2D array formation. Third, we demonstrate pattern formation in crystals on the basis of the Mullins–Sekerka instability, and in chemical and biological systems on the basis of the Turing instability. In particular, we describe the optical properties and development of sophisticated structural patterns that directly interact with light. Finally, we briefly describe a theoretical phase-transition analogy that might clarify the concept of order formation in nonequilibrium systems.

This volume collects several in-depth articles giving lucid discussions on new developments in statistical and condensed matter physics. Many, though not all, contributors had been in touch with the late S-K Ma. Written by some of the world's experts and originators of new ideas in the field, this book is a must for all researchers in theoretical physics. Most of the articles should be accessible to diligent graduate students and experienced readers will gain from the wealth of materials contained herein.

Pattern Formation in Morphogenesis is a rich source of interesting and challenging mathematical problems. The volume aims at showing how a combination of new discoveries in developmental biology and associated modelling and computational techniques has stimulated or may stimulate relevant advances in the field. Finally it aims at facilitating the process of unfolding a mutual recognition between Biologists and Mathematicians of their complementary skills, to the point where the resulting synergy generates new and novel discoveries. It offers an interdisciplinary interaction space between biologists from embryology, genetics and molecular biology who present their own work in the perspective of the advancement of their specific fields, and mathematicians who propose solutions based on the knowledge grasped from biologists.

Spatio-temporal patterns appear almost everywhere in nature, and their description and understanding still raise important and basic questions. However, if one looks back 20 or 30 years, definite progress has been made in the modeling of insta bilities, analysis of the dynamics in their vicinity, pattern formation and stability, quantitative experimental and numerical analysis of patterns, and so on. Universal behaviors of complex systems close to instabilities have been determined, leading to the wide interdisciplinarity of a field that is now referred to as nonlinear science or science of complexity, and in which initial concepts of dissipative structures or synergetics are deeply rooted. In pioneering domains related to hydrodynamics or chemical instabilities, the interactions between experimentalists and theoreticians, sometimes on a daily basis, have been a key to progress. Everyone in the field praises the role played by the interactions and permanent feedbacks between ex perimental, numerical, and analytical studies in the achievements obtained during these years. Many aspects of convective patterns in normal fluids, binary mixtures or liquid crystals are now understood and described in this framework. The generic pres ence of defects in extended systems is now well established and has induced new developments in the physics of laser with large Fresnel numbers. Last but not least, almost 40 years after his celebrated paper, Turing structures have finally been ob tained in real-life chemical reactors, triggering anew intense activity in the field of reaction-diffusion systems.

This text explores the use of cellular automata in modeling pattern formation in biological systems. It describes several mathematical modeling approaches utilizing cellular automata that can be used to study the dynamics of interacting cell systems both in simulation and in practice. New in this edition are chapters covering cell migration, tissue development, and cancer dynamics, as well as updated references and new research topic suggestions that reflect the rapid development of the field. The book begins with an introduction to pattern-forming principles in biology and the various mathematical modeling techniques that can be used to analyze them. Cellular automaton models are then discussed in detail for different types of cellular processes and interactions, including random movement, cell migration, adhesive cell interaction, alignment and cellular swarming, growth processes, pigment cell pattern formation, tissue development, tumor growth and invasion, and Turing-type patterns and excitable media. In the final chapter, the authors critically discuss possibilities and limitations of the cellular automaton approach in modeling various biological applications, along with future research directions. Suggestions for research projects are provided throughout the book to encourage additional engagement with the material, and an accompanying simulator is available for readers to perform their own simulations on several of the models covered in the text. QR codes are included within the text for easy access to the simulator. With its accessible presentation and interdisciplinary approach, Cellular Automaton Modeling of Biological Pattern Formation is suitable for graduate and advanced undergraduate students in mathematical biology, biological modeling, and biological computing. It will also be a valuable resource for researchers and practitioners in applied mathematics, mathematical biology, computational physics, bioengineering, and computer science. PRAISE FOR THE FIRST EDITION “An ideal guide for someone with a mathematical or physical background to start exploring biological modelling. Importantly, it will also serve as an excellent guide for experienced modellers to innovate and improve their methodologies for analysing simulation results.” —Mathematical Reviews

This volume bridges two topics of considerable current interest: pattern formation in nonequilibrium phenomena and physics of liquid crystals, both active and diverse areas of research. Because liquid crystals form large-scale and regular patterns under the influence of a variety of applied fields they are fruitful materials to study the spontaneous formation and evolution of ordered and disordered patterns. The chapters, each by a noted researcher in the field, briefly summarize the fundamental work done in the 1960s but concentrate on reviewing results from the recent resurgence of interest in the field as well as indicating the direction of current work.