Table Of ContentModeling and Simulation in Science,
Engineering and Technology
Nicola Bellomo
Abdelghani Bellouquid
Livio Gibelli
Nisrine Outada
A Quest Towards
a Mathematical
Theory of Living
Systems
Modeling and Simulation in Science, Engineering and Technology
Series Editors
Nicola Bellomo Tayfun E. Tezduyar
Department of Mathematics Department of Mechanical Engineering
Faculty of Sciences Rice University
King Abdulaziz University Houston, TX, USA
Jeddah, Saudi Arabia
Editorial Advisory Board
K. Aoki M.A. Herrero
National Taiwan University Departamento de Matematica Aplicada
Taipei, Taiwan Universidad Complutense de Madrid
Madrid, Spain
K.J. Bathe
Department of Mechanical Engineering P. Koumoutsakos
Massachusetts Institute of Technology Computational Science & Engineering
Cambridge, MA, USA Laboratory, ETH Zürich
Zürich, Switzerland
Y. Bazilevs
Department of Structural Engineering H.G. Othmer
University of California, San Diego Department of Mathematics
La Jolla, CA, USA University of Minnesota
Minneapolis, MN, USA
M. Chaplain
Division of Mathematics K.R. Rajagopal
University of Dundee Department of Mechanical Engineering
Dundee, Scotland, UK Texas A&M University
College Station, TX, USA
P. Degond
Department of Mathematics K. Takizawa
Imperial College London Department of Modern Mechanical
London, UK Engineering
Waseda University
A. Deutsch Tokyo, Japan
Center for Information Services
and High-Performance Computing Y. Tao
Technische Universität Dresden Dong Hua University
Dresden, Germany Shanghai, China
More information about this series at http://www.springer.com/series/4960
Nicola Bellomo • Abdelghani Bellouquid
Livio Gibelli • Nisrine Outada
A Quest Towards
a Mathematical Theory
of Living Systems
Nicola Bellomo Livio Gibelli
Department of Mathematics, Faculty of School of Engineering
Sciences University of Warwick
King Abdulaziz University Coventry
Jeddah UK
Saudi Arabia
Nisrine Outada
and Mathematics and Population Dynamics
Laboratory, UMMISCO, Faculty of
Department of Mathematical Sciences Sciences of Semlalia of Marrakech
(DISMA) Cadi Ayyad University
Politecnico di Torino Marrakech
Turin Morocco
Italy
and
Abdelghani Bellouquid
Jacques-Louis Lions Laboratory
Ecole Nationale des Sciences Appliquées de
Pierre et Marie Curie University
Marrakech
Paris 6
Académie Hassan II Des Sciences Et
France
Techniques, Cadi Ayyad Unviersity
Marrakech
Morocco
ISSN 2164-3679 ISSN 2164-3725 (electronic)
Modeling and Simulation in Science, Engineering and Technology
ISBN 978-3-319-57435-6 ISBN 978-3-319-57436-3 (eBook)
DOI 10.1007/978-3-319-57436-3
Library of Congress Control Number: 2017941474
Mathematics Subject Classification (2010): 35Q20, 35Q82, 35Q91, 35Q92, 91C99
© Springer International Publishing AG 2017
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar
methodology now known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publication does not imply, even in the absence of a specific statement, that such names are exempt from
the relevant protective laws and regulations and therefore free for general use.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authors or the editors give a warranty, express or implied, with respect to the material contained herein or
for any errors or omissions that may have been made. The publisher remains neutral with regard to
jurisdictional claims in published maps and institutional affiliations.
Printed on acid-free paper
This book is published under the trade name Birkhäuser
The registered company is Springer International Publishing AG
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
(www.birkhauser-science.com)
To the memory of Abdelghani Bellouquid
Preface
This book is devoted to the design of a unified mathematical approach to the
modeling and analysis of large systems constituted by several interacting living
entities. It is a challenging objective that needs new ideas and mathematical tools
based on a deep understanding of the interplay between mathematical and life
sciences. The authors do not naively claim that this is fully achieved, but simply that
a useful insight and some significant results are obtained toward the said objective.
The source of the contents of this book is the research activity developed in the
last 20 years, which involved several young and experienced researchers. This story
started with a book edited, at the beginning of this century, by N.B. with Mario
Pulvirenti [51], where the chapters of the book presented a variety of models of life
science systems which were derived by kinetic theory methods and theoretical tools
of probability theory. The contents of [51] were motivated by the belief that an
important new research frontier of applied mathematics had to be launched. The
basic idea was that methods of the mathematical kinetic theory and statistical
mechanics ought to be developed toward the modeling of large systems in life
science differently from the traditional application to the fluid dynamics of large
systems of classical particles.
Here, particles are living entities, from genes, cells, up to human beings. These
entities are called, within the framework of mathematics, active particles. This term
encompasses the idea that these particles have the ability to express special
strategies generally addressing to their well-being and hence do not follow laws of
classical mechanics as they can think, namely possess both an individual and a
collective intelligence [84]. Due to this specific feature, interactions between par-
ticles are nonlinearly additive. In fact, the strategy developed by each particle
depends on that expressed by the other particles, and in some cases develops a
collective intelligence of the whole viewed as a swarm. Moreover, it often happens
that all these events occur in a nonlinear manner.
An important conceptual contribution to describing interactions within an
evolutive mathematical framework is offered by the theory of evolutive games
[186, 189]. Once suitable models of the dynamics at the scale of individuals have
been derived, methods of the kinetic theory suggest to describe the overall system
vii
viii Preface
by a probability distribution over the microstate of the particles, while a balance
of the number of particles within the elementary volume of the space of the
microstates provides the time and space dynamics of the said distribution, viewed as
a dependent variable. Quantities at the macroscale are useful in several applications,
and these can be obtained from averaged moments of the dependent variable.
The hallmarks that have been presented above are somehow analogous to those
proposed in the book [26], where the approach, however, was limited to linear
interactions. Therefore, this present monograph provides, in the authors’ belief,
a far more advanced approach, definitely closer to physical reality. Moreover, an
additional feature is the search of a link between mutations and selections from
post-Darwinist theories [173, 174] to game theory and evolution. Indeed, appli-
cations, based on very recent papers proposed by several researchers, have been
selected for physical systems, where nonlinearities appear to play an important role
in the dynamics. Special attention is paid to the onset of a rare, not predictable
event, called black swan according to the definition offered by Taleb [230].
The contents of the book are presented at the end of the first chapter after some
general speculations on the complexity of living systems and on conceivable paths
that mathematics can look for an effective interplay with their interpretation. Some
statements can possibly contribute to understanding the conceptual approach and
the personal style of presentation:
• The study of models, corresponding to a number of case studies developed in
the research activity of the author and coworkers, motivated the derivation of
mathematical structures, which have the ability to capture the most important
complexity features. This formal framework can play the role of paradigms in
the derivation of specific models, where the lack of a background field theory
creates a huge conceptual difficulty very hard to tackle.
• Each chapter is concluded by a critical analysis, proposed with two goals:
focusing on the developments needed for improving the efficacy of the proposed
methods and envisaging further applications, possibly in fields different from
those treated in this book. Applications cover a broad range of fields, including
biology, social sciences, and applied sciences in general. The common feature of
all these applications is a mathematical approach, where all of them are viewed
as living, hence complex, systems.
• The authors of this book do not naively claim that the final objective of pro-
viding a mathematical theory of living systems has been fully achieved. It is
simply claimed that a contribution to this challenging and fascinating research
field is proposed and brought to the attention of future generations of applied
mathematicians.
Finally, I wish to mention that this book represents what has been achieved until
now. Hopefully, new results can be obtained in future activities. However, I decided
to write a book, in collaboration with Abdelghani Bellouquid, Livio Gibelli, and
Nisrine Outada, according to the feeling that defining the state of the art at this
stage is a necessary step to look forward. Abdelghani, Livio, and Nisrine were kind
Preface ix
enough to allow me to write this Preface, as my experience in the field was
developed in a longer (not deeper) lapse of time. Therefore, I have stories to tell, but
mainly persons to thank.
I mentioned that many results have been achieved by various authors. Among
them the coworkers are very many and I will not mention them explicitly, as they
appear in the bibliography. However, I would like to acknowledge the contribution
of some scientists who have motivated the activity developed in this book.
The first hint is from Helmut Neunzert, who is arguably the first to understand that
a natural development of the mathematical kinetic theory needed to be addressed
to systems far from that of molecular fluids. Namely, the pioneer ideas on vehicular
traffic by Prigogine should have been applied, according to his hint, also to biology
and applied sciences. He organized a fruitful, small workshop in Kaiserslautern,
where discussions, critical analysis, and hints left a deep trace in my mind.
Subsequently I met Wolfgang Alt, who also had the feeling that methods of
kinetic theory and statistical mechanics in general could find an interesting area of
application in biology. He invited me to an Oberwolfach workshop devoted to
mathematical biology, although I had never made, as a mathematician, a contri-
bution to the specific field of the meeting. I was a sort of a guest scientist, who was
lucky to have met, on that occasion, Lee Segel. His pragmatic way of developing
research activity opened my eyes and convinced me to initiate a twenty-year
activity, which is still going on and looks forward.
However, I still wish to mention three more lucky events. The first one is the
collaboration with Guido Forni, an outstanding immunologist who helped me to
understand the complex and multiscale essence of biology and of the immune
competition in particular. Indeed, my first contributions are on the applications of
mathematics to the immune competition. Recently, I met Constantine Dafermos in
Rome, who strongly encouraged me to write this book to leave a trace on the
interplay between mathematics and life. Finally, I had the pleasure to listen the
opening lecture of Giovanni Jona Lasinio at the 2012 meeting of the Italian
Mathematical Union. I am proud to state that I do share with him the idea that
evolution is a key feature of all living systems and that mathematics should take
into account this specific feature.
My special thanks go to Abdelghani Bellouquid. He has been a precious
coworker for me and several colleagues. For my family, he has been one of us.
I have to use the past tense, as he passed away when the book was reaching the end
of the authors’ efforts. My family and I, the two other authors of this book, and
many others will never forget him.
The approach presented in this book was certainly challenging and certainly on
the border of my knowledge and ability. In many cases, I have been alone with my
thoughts and speculations. However, my scientific friends know that I have never
been really alone, as my wife Fiorella was always close to me. Without her, this
book would have simply been a wish.
Turin, Italy Nicola Bellomo
January 2017
Contents
1 On the “Complex” Interplay Between Mathematics
and Living Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 A Quest Through Three Scientific Contributions . . . . . . . . . . . . . . 3
1.3 Five Key Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Complexity Features of Living Systems . . . . . . . . . . . . . . . . . . . . . 5
1.5 Rationale Toward Modeling and Plan of the Book. . . . . . . . . . . . . 11
2 A Brief Introduction to the Mathematical Kinetic Theory
of Classical Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1 Plan of the Chapter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Phenomenological Derivation of the Boltzmann Equation . . . . . . . 16
2.2.1 Interaction dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.2 The Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.3 Properties of the Boltzmann equation . . . . . . . . . . . . . . . . . 23
2.3 Some Generalized Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.1 The BGK model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.2 The discrete Boltzmann equation . . . . . . . . . . . . . . . . . . . . 26
2.3.3 Vlasov and Enskog equations . . . . . . . . . . . . . . . . . . . . . . . 28
2.4 Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5 Critical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3 On the Search for a Structure: Toward a Mathematical
Theory to Model Living Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1 Plan of the Chapter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 A Representation of Large Living Systems . . . . . . . . . . . . . . . . . . 35
3.3 Mathematical Structures for Systems with Space Homogeneity . . . 39
3.3.1 A phenomenological description of games . . . . . . . . . . . . . 39
3.3.2 Modeling interactions by tools of game theory . . . . . . . . . . 42
3.3.3 Mathematical structures for closed systems . . . . . . . . . . . . . 44
xi
