Skip to main content
SpringerLink
Log in

A Kinetic Theory Approach to the Modeling of Complex Living Systems

  • Chapter
  • First Online:
Active Particles, Volume 1

Abstract

In this chapter, a mathematical structure is derived to provide a general framework toward the modeling of space-homogeneous living systems, according to the kinetic theory for active particles. This structure can be adapted to study a variety of processes such as collective learning and social dynamics. Simple models regarding learning in a classroom and the dynamics of the criminality are presented to illustrate how the general modeling strategy operates in well-defined applications. Future research directions using the proposed approach are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Ajmone Marsan, G.: On the modelling and simulation of the competition for a secession under media influence by active particles methods and functional subsystems representation 57(5), 710–728 (2009)

    Google Scholar 

  2. Ajmone Marsan, G., Bellomo, N., Egidi, N.: Towards a methematical theory of complex socio-economical systems by functional subsystems representation. Kinetic Rel. Models 1(2), 249–278 (2008)

    Google Scholar 

  3. Ajmone Marsan, G., Bellomo, N., Gibelli, L.: Towards a systems approach to behavioral social dynamics. Math. Mod. Meth. Appl. Sci. 26(6), 1051–1093 (2016)

    MathSciNet  Google Scholar 

  4. Allen, B., Nowak, M.A.: Games on graphs. EMS Surv. Math. Sci. 1(1), 113–151 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Anderson, P.W.: More is different. Science 177(4047), 393–396 (1972)

    Article  Google Scholar 

  6. Axelrod, R.: The complexity of cooperation: Agent-based models of competition and collaboration. Princeton University Press, Princeton (1997)

    Google Scholar 

  7. Ballerini, M., Cabibbo, N., Candelier, R., Cisbani, E., Giardina, I., Lecomte, V., Orlandi, A., Parisi, G., Procaccini, A., Viale, M., Zdravkovic, V.: Interaction ruling animal collective behavior depends on topological rather than metric distance: evidence from a field study. Proc. Natl. Acad. Sci. USA 105(4), 1232–1237 (2008)

    Article  Google Scholar 

  8. Barabasi, A.L.: Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science and Everyday Life. Plume Editors (2002)

    Google Scholar 

  9. Barbante, P., Frezzotti, A., Gibelli, L.: A kinetic theory description of liquid menisci at the microscale. Kinetic Rel. Models 8(2), 235–254 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bellomo, N.: Modeling Complex Living Systems - A Kinetic Theory and Stochastic Game Approach. Birkhäuser, Boston, New York (2008)

    MATH  Google Scholar 

  11. Bellomo, N., Soler, J.: On the mathematical theory of the dynamics of swarms viewed as complex systems. Math. Mod. Meth. Appl. Sci. 22, Paper n.1140,006 (2012)

    Google Scholar 

  12. Bellomo, N., Herrero, M.A., Tosin, A.: On the dynamics of social conflicts: looking for the black swan. Kinetic Rel. Models 6, 459–479 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bellomo, N., Knopoff, D., Soler, J.: On the difficult interplay between life, “complexity”, and mathematical sciences. Math. Mod. Meth. Appl. Sci. 23, 1861–1913 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bellomo, N., Colasuonno, F., Knopoff, D., Soler, J.: From a systems theory of sociology to modeling the onset and evolution of criminality. Netw. Heterog. Media 10(3), 421–441 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bellouquid, A., De Angelis, E., Fermo, L.: Towards the modeling of vehicular traffic as a complex system: A kinetic theory approach. Math. Mod. Meth. Appl. Sci. 22(5), Paper n.1140,003 (2012)

    Google Scholar 

  16. Bellouquid, A., De Angelis, E., Knopoff, D.: From the modeling of the immune hallmarks of cancer to a black swan in biology. Math. Mod. Meth. Appl. Sci. 23, 949–978 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Berenji, B., Chou, T., D’Orsogna, M.R.: Recidivism and rehabilitation of criminal offenders: A carrot and stick evolutionary game. PLOS ONE 9, Paper n.885,531 (2014)

    Google Scholar 

  18. Berliner, D.C., Calfee, R.C. (eds.): Handbook of Educational Psychology. MacMillan, New York (1996)

    Google Scholar 

  19. Bertotti, M.L., Delitala, M.: From discrete kinetic and stochastic game theory to modelling complex systems in applied sciences. Math. Mod. Meth. Appl. Sci. 14(7), 1061–1084 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Bertotti, M.L., Delitala, M.: Conservation laws and asymptotic behavior of a model of social dynamics. Nonlinear Anal. Real World Appl. 9(1), 183–196 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Binder, K. (ed.): Applications of the Monte Carlo Method in Statistical Physics. Springer-Verlag, Heidelberg (1987)

    Google Scholar 

  22. Bird, G.A.: Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Oxford University Press, Oxford (UK) (1994)

    Google Scholar 

  23. Bisin, A., Verdier, T.: The economics of cultural transmission and the dynamics of preferences. J. Econ. Theory 97, 298–319 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  24. Bonabeau, E., Dorigo, M., Theraulaz, G.: Swarm Intelligence: From Natural to Artificial Systems. Oxford University Press, Oxford (1999)

    MATH  Google Scholar 

  25. Bordogna, C.M., Albano, E.V.: Theoretical description of teaching-learning processes: A multidisciplinary approach. Phys. Rev. Lett. 87(11), 118,701 (2001)

    Google Scholar 

  26. Burini, D., De Lillo, S., Gibelli, L.: Collective learning dynamics modeling based on the kinetic theory of active particles. Phys. Life Rev. 16, 123–139 (2016)

    Article  Google Scholar 

  27. Bush, R.R., Mosteller, F.: Stochastic models for learning. John Wiley & Sons, Inc., Oxford (1955)

    Book  MATH  Google Scholar 

  28. Camerer, C.F.: Behavioral Game Theory: Experiments in Strategic Interaction. Princeton University Press, Princeton (2003)

    MATH  Google Scholar 

  29. Castellano, C., Fortunato, S., Loreto, V.: Statistical physics of social dynamics. Rev. Mod. Phys. 81, 591–646 (2009)

    Article  Google Scholar 

  30. Couzin, I.D.: Collective minds. Nature 445, 715 (2007)

    Article  Google Scholar 

  31. Cucker, F., Smale, S.: On the mathematical foundations of learning. Bull. Amer. Mathe. Soc. 39(1), 1–49 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  32. D’Acci, L.: Mathematize urbes by humanizing them. Cities as isobenefit landscapes: psycho-economical distances and personal isobenefit lines. Landscape Urban Plan. 139, 63–81 (2015)

    Google Scholar 

  33. De Angelis, E.: On the mathematical theory of post-Darwinian mutations, selection, and evolution. Math. Mod. Meth. Appl. Sci. 24(13), 2723–2742 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  34. Dimarco, G., Pareschi, L.: Numerical methods for kinetic equations. Acta Num. 23(12), 369–520 (2014)

    Article  MathSciNet  Google Scholar 

  35. Dolfin, M., Lachowicz, M.: Modeling altruism and selfishness in welfare dynamics: the role of nonlinear interactions. Math. Mod. Meth. Appl. Sci. 24, 2361–2381 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  36. Epstein, J.M., Axtell, R.: Growing artificial societies: social science from the bottom up. The MIT Press, Boston (1996)

    Google Scholar 

  37. Estes, W.K., Suppes, P.: Foundations of Statistical Learning Theory, II. The Stimulus Sampling Model. Tech. Rep. 26, Stanford University, Institute for Mathematical Studies in the Social Sciences, Stanford University (1959)

    Google Scholar 

  38. Fajnzylber, P., Lederman, D., Loayza, N.: Inequality and violent crime. J. Law. Econ. 45(1), 1–40 (2002)

    Article  Google Scholar 

  39. Felson, M.: What every mathematician should know about modelling crime. Eur. J. Appl. Math. 21, 275–281 (2010)

    Article  MathSciNet  Google Scholar 

  40. Frezzotti, A., Ghiroldi, G.P., Gibelli, L.: Solving model kinetic equations on GPUs. Comput. Fluids 50, 136–146 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  41. Frezzotti, A., Ghiroldi, G.P., Gibelli, L.: Solving the Boltzmann equation on GPUs. Comput. Phys. Commun. 182, 2445–2453 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  42. Frezzotti, A., Ghiroldi, G.P., Gibelli, L., Bonucci, A.: DSMC simulation of rarefied gas mixtures flows driven by arrays of absorbing plates. Vacuum 103, 57–67 (2014)

    Article  Google Scholar 

  43. Galam, S.: Sociophysics: A Physicists Modeling of Psycho-political Phenomena. Springer, New York (2012)

    Book  Google Scholar 

  44. Galam, S., Moscovici, S.: Towards a theory of collective phenomena: Consensus and attitude changes in groups. Eur. J. Soc. Psy. 21, 49–74 (1991)

    Article  Google Scholar 

  45. Gintis, H.: The bounds of reason: Game Theory and the unification of the behavioral sciences. Princeton University Press, Princeton and Oxford (2009)

    MATH  Google Scholar 

  46. Gintis, H.: Game Theory Evolving: A Problem-Centered Introduction to Modeling Strategic Interaction. Princeton University Press, Princeton and Oxford (2009)

    MATH  Google Scholar 

  47. Harsanyi, J.C.: Games with incomplete information played by “bayesian” players, i-iii part i. the basic model. Manage. Sci. 14(3), 159–182 (1967)

    Article  MATH  Google Scholar 

  48. Helbing, D.: Stochastic and Boltzmann-like models for behavioral changes, and their relation to game theory. Physica A 193(2), 241–258 (1993)

    Article  MathSciNet  Google Scholar 

  49. Helbing, D.: Quantitative sociodynamics: Stochastic methods and models of social interaction processes. Springer Verlag (2010)

    Google Scholar 

  50. Helbing, D., Yu, W.: The outbreak of cooperation among success-driven individuals under noisy conditions. Proc. Nat. Acad. Sci. 106(10), 3680–3685 (2009)

    Article  Google Scholar 

  51. Hofbauer, J., Sigmund, K.: Evolutionary game dynamics. Bull. Amer. Mathe. Soc. 40(4), 479–519 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  52. Knopoff, D.: On the modeling of migration phenomena on small networks. Math. Mod. Meth. Appl. Sci. 23(3), 541–563 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  53. Knopoff, D.: On a mathematical theory of complex systems on networks with application to opinion formation. Math. Mod. Meth. Appl. Sci. 24(2), 405–426 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  54. Kraus, M.W., Piff, P.K., Keltner, D.: Social class, sense of control, and social explanation. J. Pers. Soc. Psychol. 97(6), 992–1004 (2009)

    Article  Google Scholar 

  55. Kraus, M.W., Piff, P.K., Mendoza-Denton, R., Rheinschmidt, M.L., Keltner, D.: Social class, solipsism, and contextualism: How the rich are different from the poor. Psychol. Rev. 119(3), 546–572 (2012)

    Article  Google Scholar 

  56. Lasry, J.M., Lions, P.L.: Jeux à champ moyen. i le cas stationnaire. CR. Acad. Sci. I-Math. 343(9), 619–625 (2006)

    Google Scholar 

  57. Lasry, J.M., Lions, P.L.: Jeux à champ moyen. ii horizon fini et contrôle optimal. CR. Acad. Sci. I-Math. 343(10), 679–684 (2006)

    Google Scholar 

  58. Latané, B.: The psychology of social impact. Am. Physchol. 36(4), 343–356 (1981)

    Google Scholar 

  59. Lave, J., Wenger, E.: Situated Learning: Legitimate Peripheral Partecipation. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  60. May, R.M.: Uses and abuses of mathematics in biology. Science 303, 790–793 (2004)

    Article  Google Scholar 

  61. Mayr, E.: What Evolution Is. Basic Books, New York (2001)

    Google Scholar 

  62. Morgenstern, O., Von Neumann, J.: Theory of Games and Economic Behavior. Princeton University Press (1953)

    Google Scholar 

  63. Motsch, S., Tadmor, E.: Heterophilious dynamics enhances consensus. Siam Rev. 56(4), 577–621 (2014)

    Google Scholar 

  64. Myatt, D.P., Wallace, C.: An evolutionary analysis of the volunteer’s dilemma. Game Econ. Behav. 62, 67–76 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  65. Nash, J.: Non-cooperative games. Ann. Math. 54(2), 286–295 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  66. Nowak, M.A.: Evolutionary Dynamics. Exploring the Equations of Life. Harvard University Press, Cambridge (MA) (2006)

    MATH  Google Scholar 

  67. Ormerod, P.: Crime: Economic incentives and social networks. IEA Hobart Paper 151(4), 1–54 (2005)

    Google Scholar 

  68. Pareschi, L., Toscani, G.: Interacting Multiagent Systems - Kinetic equations and Monte Carlo methods. Oxford University Press, Oxford (UK) (2013)

    MATH  Google Scholar 

  69. Piaget, J.: The Child’s Conception of the World. Harcourt, Brace, New York (1929)

    Google Scholar 

  70. Piff, P.K., Stancato, D.M., Côté, S., Mendoza-Denton, R., Keltner, D.: Higher social class predicts increased unethical behavior. Proc. Natl. Acad. Sci. USA 109(11), 4086–4091 (2014)

    Article  Google Scholar 

  71. Rufus, I.: Differential games: A mathematical theory with applications to warfare and pursuit, control and optimization. John Wiley and Sons, New York (1965)

    MATH  Google Scholar 

  72. Schelling, T.C.: Dynamic models of segregation. J. Math. Sociol. 1, 143–186 (1971)

    Article  MATH  Google Scholar 

  73. Smith, M.J.: The stability of a dynamic model of traffic assignment - an application of a method of Lyapunov. Transport. Sci. 18(3), 245–252 (1984)

    Article  MathSciNet  Google Scholar 

  74. Szabó, G., Fáth, G.: Evolutionary games on graphs. Phys. Rep. 446, 97–216 (2007)

    Article  MathSciNet  Google Scholar 

  75. Taleb, N.N.: The Black Swan: The Impact of the Highly Improbable. Random House, New York City (2007)

    Google Scholar 

  76. Vega-Redondo, F.: Evolution, Games, and Economic Behaviour. Oxford University Press, Oxford (1996)

    Book  MATH  Google Scholar 

  77. Vygotsky, L.S.: Mind in Society: Development of Higher Psychological Processes. Harvard University Press, Cambridge, Massachusettes London, England (1978)

    Google Scholar 

  78. Wagner, W.: A convergence proof of bird’s direct simulation Monte-Carlo method for the Boltzmann equation. J. Stat. Phys. 66, 1011–1044 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  79. Webb, G.F.: Theory of Nonlinear Age-dependent Population Dynamics. Marcel Dekker, New York (1985)

    MATH  Google Scholar 

  80. Weidlich, W., Haag, G.: Concepts and Models of a Quantitative Sociology The Dynamics of Interacting Populations. Springer-Verlag, Berlin (1983)

    Book  MATH  Google Scholar 

  81. Wolpert, D.H.: Information theory - the bridge connecting bounded rational game theory and statistical physics. In: D. Braha, A.A. Minai, Y. Bar-Yam (eds.) Complex Engineered Systems, Understanding complex systems, pp. 262–290. Springer, Princeton (2006)

    Chapter  Google Scholar 

  82. Yakovenko, V.M., Barkley Rosser, J.: Statistical physics of social dynamics. Rev. Mod. Phys. 4, 1703–1725 (2009)

    Google Scholar 

  83. Young, H.P.: An evolutionary model of bargaining. J. Econ. Theory 59, 145–168 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  84. Zhang, J.: A dynamic model of residential segregation. J. Math. Sociol. 28, 147–170 (2004)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Politecnico di Torino, Turin, Italy

    Diletta Burini & Livio Gibelli

  2. Faculté des Sciences Semlalia, Marrakesh, Morocco

    Nisrine Outada

  3. Laboratoire Jacques Louis-Lions, Université Pierre et Marie Curie, Paris, France

    Nisrine Outada

Authors
  1. Diletta Burini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Livio Gibelli
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nisrine Outada
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Livio Gibelli .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, King Abdulaziz University, Jeddah, Saudi Arabia

    Nicola Bellomo

  2. Department of Mathematics, Imperial College London, London, United Kingdom

    Pierre Degond

  3. CSCAMM, University of Maryland, College Park, USA

    Eitan Tadmor

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Burini, D., Gibelli, L., Outada, N. (2017). A Kinetic Theory Approach to the Modeling of Complex Living Systems. In: Bellomo, N., Degond, P., Tadmor, E. (eds) Active Particles, Volume 1 . Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-49996-3_6

Download citation

Publish with us

Policies and ethics

Search

Navigation