LOOKING FOR NEW PARADIGMS TOWARDS A BIOLOGICAL-MATHEMATICAL THEORY OF COMPLEX MULTICELLULAR SYSTEMS

2006 ◽  
Vol 16 (07) ◽  
pp. 1001-1029 ◽  
Author(s):  
NICOLA BELLOMO ◽  
GUIDO FORNI
Keyword(s):  
Kinetic Theory ◽  
Complex Systems ◽  
Critical Analysis ◽  
Mathematical Theory ◽  
Classical Mechanics ◽  
Biological Functions ◽  
Living Matter ◽  
Research Perspectives ◽  
Active Particles ◽  
Large Systems

This paper deals with the development of new paradigms based on the methods of the mathematical kinetic theory for active particles to model the dynamics of large systems of interacting cells. Interactions are ruled, not only by laws of classical mechanics, but also by a few biological functions which are able to modify the above laws. The paper technically shows, also by reasoning on specific examples, how the theory can be applied to model large complex systems in biology. The last part of the paper deals with a critical analysis and with the indication of research perspectives concerning the challenging target of developing a biological-mathematical theory for the living matter.

MATHEMATICAL METHODS AND TOOLS OF KINETIC THEORY TOWARDS MODELLING COMPLEX BIOLOGICAL SYSTEMS

2005 ◽  
Vol 15 (11) ◽  
pp. 1639-1666 ◽  
Author(s):  
ABDELGHANI BELLOUQUID ◽  
MARCELLO DELITALA
Keyword(s):  
Kinetic Theory ◽  
Evolution Equations ◽  
Classical Mechanics ◽  
Mathematical Framework ◽  
Mathematical Approach ◽  
Biological Functions ◽  
Mathematical Methods ◽  
Research Perspectives ◽  
Large Systems ◽  
The One

This paper develops a variety of mathematical tools to model the dynamics of large systems of interacting cells. Interactions are ruled not only by laws of classical mechanics, but also by some biological functions. The mathematical approach is the one of kinetic theory and non-equilibrium statistical mechanics. The paper deals both with the derivation of evolution equations and with the design of specific models consistent with the above-mentioned mathematical framework. Various hints for research perspectives are proposed in the last part of the paper.

Active particles methods and challenges in behavioral systems

2020 ◽  
Vol 30 (04) ◽  
pp. 653-658 ◽  
Author(s):  
N. Bellomo ◽  
F. Brezzi ◽  
J. Soler
Keyword(s):  
Qualitative Analysis ◽  
Critical Analysis ◽  
Mathematical Approach ◽  
Special Issue ◽  
Behavioral Systems ◽  
Research Perspectives ◽  
Active Particles ◽  
Challenging Research ◽  
Large Systems

This paper first provides an introduction to the mathematical approach to the modeling, qualitative analysis, and simulation of large systems of living entities, specifically self-propelled particles. Subsequently, a presentation of the papers published in this special issue follows. Finally, a critical analysis of the overall contents of the issue is proposed, thus leading to define some challenging research perspectives.

ON THE ASYMPTOTIC THEORY FROM MICROSCOPIC TO MACROSCOPIC GROWING TISSUE MODELS: AN OVERVIEW WITH PERSPECTIVES

2012 ◽  
Vol 22 (01) ◽  
pp. 1130001 ◽  
Author(s):  
N. BELLOMO ◽  
A. BELLOUQUID ◽  
J. NIETO ◽  
J. SOLER
Keyword(s):  
Kinetic Theory ◽  
Asymptotic Analysis ◽  
Critical Analysis ◽  
Asymptotic Theory ◽  
Blow Up ◽  
Cellular Interactions ◽  
Biological Functions ◽  
Active Particles ◽  
Tissue Models ◽  
Macroscopic Equations

This paper proposes a review and critical analysis of the asymptotic limit methods focused on the derivation of macroscopic equations for a class of equations modeling complex multicellular systems by methods of the kinetic theory for active particles. Cellular interactions generate both modification of biological functions and proliferative/destructive events. The asymptotic analysis deals with suitable parabolic, hyperbolic, and mixed limits. The review includes the derivation of the classical Keller–Segel model and flux limited models that prevent non-physical blow up of solutions.

On the mathematical theory of post-Darwinian mutations, selection, and evolution

2014 ◽  
Vol 24 (13) ◽  
pp. 2723-2742 ◽  
Author(s):  
E. De Angelis
Keyword(s):  
Kinetic Theory ◽  
Qualitative Analysis ◽  
Mathematical Theory ◽  
Darwinian Selection ◽  
Research Perspectives ◽  
Active Particles ◽  
Selection Processes

This paper is devoted to the modeling, qualitative analysis and simulation of Darwinian selection phenomena and their evolution. The approach takes advantage of the mathematical tools of the kinetic theory of active particles which are applied to describe the selective dynamics of evolution processes. The first part of the paper focuses on a mathematical theory that has been developed to describe mutations and selection processes. The second part deals with different modeling strategies and looks ahead to research perspectives.

ON A MATHEMATICAL THEORY OF COMPLEX SYSTEMS ON NETWORKS WITH APPLICATION TO OPINION FORMATION

2013 ◽  
Vol 24 (02) ◽  
pp. 405-426 ◽  
Author(s):  
D. KNOPOFF
Keyword(s):  
Kinetic Theory ◽  
Complex Systems ◽  
Mathematical Theory ◽  
Stochastic Games ◽  
Mathematical Structure ◽  
Opinion Formation ◽  
Active Particles ◽  
Mathematical Equations

This paper presents a development of the so-called kinetic theory for active particles to the modeling of living, hence complex, systems localized in networks. The overall system is viewed as a network of interacting nodes, mathematical equations are required to describe the dynamics in each node and in the whole network. These interactions, which are nonlinearly additive, are modeled by evolutive stochastic games. The first conceptual part derives a general mathematical structure, to be regarded as a candidate towards the derivation of models, suitable to capture the main features of the said systems. An application on opinion formation follows to show how the theory can generate specific models.

Collective dynamics in science and society

2021 ◽  
pp. 1-6
Author(s):  
N. Bellomo ◽  
F. Brezzi
Keyword(s):  
Critical Analysis ◽  
Large Scale ◽  
Particle Methods ◽  
Science And Society ◽  
Special Issue ◽  
Collective Dynamics ◽  
Multiscale Problems ◽  
Research Perspectives ◽  
Active Particles ◽  
Large Systems

This editorial paper presents the articles published in a special issue devoted to active particle methods applied to modeling, qualitative analysis, and simulation of the collective dynamics of large systems of interacting living entities in science and society. The modeling approach refers to the mathematical tools of behavioral swarms theory and to the kinetic theory of active particles. Applications focus on classical problems of swarms theory, on crowd dynamics related to virus contagion problems, and to multiscale problems related to the derivation of models at a large scale from the mathematical description at the microscopic scale. A critical analysis of the overall contents of the issue is proposed, with the aim to provide a forward look to research perspectives.

ON THE DIFFICULT INTERPLAY BETWEEN LIFE, "COMPLEXITY", AND MATHEMATICAL SCIENCES

2013 ◽  
Vol 23 (10) ◽  
pp. 1861-1913 ◽  
Author(s):  
N. BELLOMO ◽  
D. KNOPOFF ◽  
J. SOLER
Keyword(s):  
Kinetic Theory ◽  
Complex Systems ◽  
Case Studies ◽  
Mathematical Theory ◽  
Stochastic Games ◽  
Mathematical Structure ◽  
Living Systems ◽  
Active Particles ◽  
Suitable Generalization ◽  
Mathematical Sciences

This paper presents a revisiting, with developments, of the so-called kinetic theory for active particles, with the main focus on the modeling of nonlinearly additive interactions. The approach is based on a suitable generalization of methods of kinetic theory, where interactions are depicted by stochastic games. The basic idea consists in looking for a general mathematical structure suitable to capture the main features of living, hence complex, systems. Hopefully, this structure is a candidate towards the challenging objective of designing a mathematical theory of living systems. These topics are treated in the first part of the paper, while the second one applies it to specific case studies, namely to the modeling of crowd dynamics and of the immune competition.

HYBRID TWO SCALES MATHEMATICAL TOOLS FOR ACTIVE PARTICLES MODELLING COMPLEX SYSTEMS WITH LEARNING HIDING DYNAMICS

2007 ◽  
Vol 17 (02) ◽  
pp. 171-187 ◽  
Author(s):  
C. CATTANI ◽  
A. CIANCIO
Keyword(s):  
Applied Sciences ◽  
Kinetic Theory ◽  
Differential Equations ◽  
Complex Systems ◽  
Ordinary Differential Equations ◽  
Preliminary Analysis ◽  
Mathematical Structures ◽  
Active Particles ◽  
Model Complex ◽  
Large Systems

This paper deals with the derivation of hybrid mathematical structures to describe the behavior of large systems of active particles by ordinary differential equations with stochastic coefficients whose evolution is modelled by equations of the mathematical kinetic theory. A preliminary analysis shows how the above tools can be used to model complex systems of interest in applied sciences, with special attention to the immune competition.

FROM THE MATHEMATICAL KINETIC THEORY TO MODELLING COMPLEX SYSTEMS IN APPLIED SCIENCES

2004 ◽  
Vol 18 (04n05) ◽  
pp. 487-500 ◽  
Author(s):  
NICOLA BELLOMO
Keyword(s):  
Applied Sciences ◽  
Kinetic Theory ◽  
Statistical Mechanics ◽  
Complex Systems ◽  
Nonequilibrium Statistical Mechanics ◽  
Classical Mechanics ◽  
Large Complex ◽  
Large Systems

This paper deals with the design of a nonequilibrium statistical mechanics theory developed to model large systems of interacting individuals. Interactions being ruled, not only by laws of classical mechanics, but also by some intelligent or organized behaviour. The paper technically shows, also by reasoning on specific examples, how the theory can be applied to model large complex systems in natural and applied sciences.

Hilbert method toward a multiscale analysis from kinetic to macroscopic models for active particles

2017 ◽  
Vol 27 (07) ◽  
pp. 1327-1353 ◽  
Author(s):  
D. Burini ◽  
N. Chouhad
Keyword(s):  
Kinetic Theory ◽  
Multiscale Analysis ◽  
Second Step ◽  
Theory Approach ◽  
Type Method ◽  
Macroscopic Models ◽  
Active Particles ◽  
Time Space ◽  
Large Systems ◽  
Hilbert Type

This paper develops a Hilbert type method to derive models at the macroscopic scale for large systems of several interacting living entities whose statistical dynamics at the microscopic scale is delivered by kinetic theory methods. The presentation is in three steps, where the first one presents the structures of the kinetic theory approach used toward the aforementioned analysis; the second step presents the mathematical method; while the third step provides a number of specific applications. The approach is focused on a simple system and with a binary mixture, where different time-space scalings are used. Namely, parabolic, hyperbolic, and mixed in the case of a mixture.

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