The Mathematics And Mechanics Of Biological Growth

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The Mathematics and Mechanics of Biological Growth

This monograph presents a general mathematical theory for biological growth. It provides both a conceptual and a technical foundation for the understanding and analysis of problems arising in biology and physiology. The theory and methods are illustrated on a wide range of examples and applications. A process of extreme complexity, growth plays a fundamental role in many biological processes and is considered to be the hallmark of life itself. Its description has been one of the fundamental problems of life sciences, but until recently, it has not attracted much attention from mathematicians, physicists, and engineers. The author herein presents the first major technical monograph on the problem of growth since D’Arcy Wentworth Thompson’s 1917 book On Growth and Form. The emphasis of the book is on the proper mathematical formulation of growth kinematics and mechanics. Accordingly, the discussion proceeds in order of complexity and the book is divided into five parts. First, a general introduction on the problem of growth from a historical perspective is given. Then, basic concepts are introduced within the context of growth in filamentary structures. These ideas are then generalized to surfaces and membranes and eventually to the general case of volumetric growth. The book concludes with a discussion of open problems and outstanding challenges. Thoughtfully written and richly illustrated to be accessible to readers of varying interests and background, the text will appeal to life scientists, biophysicists, biomedical engineers, and applied mathematicians alike.
Morpho-elasticity

Growth plays a fundamental role in many biological processes and is considered to be the hallmark of life itself. It is a process of extreme complexity and its description has been one of the fundamental problems of life sciences. However, until recently, it has not attracted much attention from mathematicians, physicists, and mechanicians. The goal of this monograph is to present the state of knowledge in the mechanics of growth, to provide a rigorous foundation and, to offer a set of mathematical tools for the analysis of specific problems arising in biology accessible to mathematicians, physicists, and biologists. Simple examples, applications and building discussions are included. The emphasis of the book is on the kinematics and mechanics of growth. Accordingly, the three first parts of the book are: Growth of curves and lamentary structure, Growth of surfaces, membranes, and shells, and Volumetric growth.
Parabolic Equations in Biology

This book presents several fundamental questions in mathematical biology such as Turing instability, pattern formation, reaction-diffusion systems, invasion waves and Fokker-Planck equations. These are classical modeling tools for mathematical biology with applications to ecology and population dynamics, the neurosciences, enzymatic reactions, chemotaxis, invasion waves etc. The book presents these aspects from a mathematical perspective, with the aim of identifying those qualitative properties of the models that are relevant for biological applications. To do so, it uncovers the mechanisms at work behind Turing instability, pattern formation and invasion waves. This involves several mathematical tools, such as stability and instability analysis, blow-up in finite time, asymptotic methods and relative entropy properties. Given the content presented, the book is well suited as a textbook for master-level coursework.