A Quest Towards a Mathematical Theory of Living Systems

Scientific investigations in certain areas of applied mathematics, chemistry, and physics conducted during the period from the 1960s to the early 1990s produced new areas of interdisciplinary research along with keywords such as emergence, self-organization, chaos, complex adaptive systems, and swarm intelligence. During this period, we also saw the establishment of new research institutes, such as the Santa-Fe Institute, established in 1984 with the mission to study problems that are closely associated with the aforementioned keywords. Some of this research in science and mathematics had notable success at the time in capturing the interest of the general public, as evidenced by the positive reception of Chaos: Making a New Science by James Gleick and other similar books which provide a popular account of some results from nonlinear dynamical systems.

Rather quickly, mathematicians and scientists realized that burgeoning theories in nonlinear dynamics and complex systems theory had as much significance for the life sciences as for physics and chemistry, leading to the maturing of the field of mathematical biology. This is evidenced by the formation of institutes such as the Mathematical Biosciences Institute and the National Institute for Mathematical and Biological Synthesis, and the publication of books such as Mathematical Biology by J. D. Murray.

Initially, the greatest focus in mathematical biology was on population dynamics, epidemiology, the dynamics of excitable cells, pattern formation in biochemical systems, and mathematical modeling of certain biomedical phenomena. What all of the listed areas of mathematical biology have in common is that in each case the systems under study involved either very small populations, or populations so large that only the population density mattered and not the interaction of individuals.

As certain concepts in the study of complex adaptive systems (such as networks and interacting particle systems) were further developed, however, mathematicians and biologists alike realized the possibilities for the mathematical study of large populations of interacting individuals. This is evidenced by books such as Self-Organization in Biological Systems by Camazine et al., and Collective Animal Behavior by Sumpter. A Quest Towards a Mathematical Theory of Living Systems is aimed at presenting some of the more recent developments in the line of inquiry just traced out, mostly from a mathematical perspective. Before discussing the book in greater detail, let us point out that this work in inspired as much by recent discoveries in biology as it is by recent discoveries in applied mathematics.

In the first chapter of A Quest Towards a Mathematical Theory of Living Systems, the authors identify five key questions that the book attempts to address. These questions are:

1. Which are the most relevant common complexity features of living systems?
2. Can appropriate mathematical structures be derived to capture the main feature of living systems?
3. How can mathematical models be referred to the mathematical structures deemed to depict complexity features of living systems?
4. Models offer a predictive ability, but how can they be validated?
5. Which are the conceptual paths which might lead to a mathematical theory of living systems?

The type of living systems considered by the authors are characterized by such features as having a large number of interacting components that are adaptive to input from their local environment, interacting in time and over multiple scales. The authors continue their discussion elucidate what they consider to be the most relevant features of living systems in order to establish the basic assumptions upon which they will build their mathematical modeling framework. The remainder of the book then presents the mathematical framework the authors promote as a promising approach toward developing a mathematical theory of living systems.

The mathematical framework that the authors of this book promote largely consists of kinetic equations and Monte Carlo methods. In this, the book has some overlap with Interacting Multi-agent Systems: Kinetic equations and Monte Carlo methods by Pareschi and Toscani. This book is somewhat more accessible to the newcomer to the field, however, and focuses more on developing intuition and broad understanding than on the technical mathematical details. A Quest Towards a Mathematical Theory of Living Systems also presents some interesting applications to the social sciences. The book contains a substantial bibliography to which the reader is referred for more information and greater details.

In all, A Quest Towards a Mathematical Theory of Living Systems should prove to be an interesting book, especially for those with interests in collective behavior, emergent phenomenon, and mathematical approaches to studying such phenomena. The authors do an excellent job of orienting the reader with little if any prior exposure to the kinetic theory approach to modeling living systems. I recommend this book as a point of entry into this popular field of current research in the mathematical life sciences.

Jason M. Graham is an assistant professor in the department of mathematics at the University of Scranton, Scranton, Pennsylvania. His current professional interests are in teaching applied mathematics and mathematical biology, and collaborating with biologists specializing in the collective behavior of groups of organisms.