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Mathematics, complexity and multiscale features of large systems of self-propelled particles. (English) Zbl 1398.92024

Summary: This issue is devoted to complex systems in life sciences. Some perspective ideas on possible objectives of future research are extracted from the contents of this issue and brought to the reader’s attention. The final ambitious aim is the development of a mathematical theory for complex living systems.

MSC:

92C10 Biomechanics
35Q92 PDEs in connection with biology, chemistry and other natural sciences
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[1] 1. S. Ahn, H.-O. Bae, S.-Y. Ha, Y. Kim and H. Lim, Application of flocking mechanism to the modeling of stochastic volatility, Math. Models Methods Appl. Sci.23 (2013) 1603-1628. [Abstract] genRefLink(128, ’S0218202516020012BIB1’, ’000318877500002’); · Zbl 1266.91097
[2] 2. B. Allen and M. A. Nowak, Games on networks, EMS Survey Math. Sci.1 (2013) 113-151. genRefLink(16, ’S0218202516020012BIB2’, ’10.4171 · Zbl 1303.91040
[3] 3. V. V. Aristov, Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows (Springer-Verlag, 2001). · Zbl 0983.82011
[4] 4. P. Ball (ed.), Why Society is a Complex Matter (Springer-Verlag, 2012). genRefLink(16, ’S0218202516020012BIB4’, ’10.1007
[5] 5. J. Banasiak, A. Falkiewicz and P. Namayanja, Asymptotic state lumping in transport and diffusion problems on networks with applications to population problems, Math. Models Methods Appl. Sci.26 (2016) 215-247. [Abstract] · Zbl 1355.92084
[6] 6. P. Barbante, A. Frezzotti and L. Gibelli, A kinetic theory description of liquid menisci at the microscale Kinet. Relat. Mod.8 (2015) 235-254. genRefLink(16, ’S0218202516020012BIB6’, ’10.3934 · Zbl 1362.82044
[7] 7. M. Batty, The New Science of Cities (MIT Press, 2013).
[8] 8. E. Beinhocker, Complex new world: Translating new economic thinking into public policy, The Institute for New Economic Thinking at the Oxford Martin School (2012).
[9] 9. N. Bellomo and A. Bellouquid, On multiscale models of pedestrian crowds from mesoscopic to macroscopic, Commun. Math. Sci.13 (2015) 1649-1664. genRefLink(16, ’S0218202516020012BIB9’, ’10.4310
[10] 10. N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, On the multiscale modeling of vehicular traffic: From kinetic to hydrodynamics, Discrete Contin. Dynam. Syst. Ser. B19 (2014) 1869-1888. genRefLink(16, ’S0218202516020012BIB10’, ’10.3934
[11] 11. N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci.25 (2015) 1663-1763. [Abstract] genRefLink(128, ’S0218202516020012BIB11’, ’000355690000002’); · Zbl 1326.35397
[12] 12. N. Bellomo and F. Brezzi, Complex systems: New challenges with modeling headaches, Math. Models Methods Appl. Sci.24 (2014) 213-219. [Abstract] genRefLink(128, ’S0218202516020012BIB12’, ’000328363800001’); · Zbl 1280.00033
[13] 13. N. Bellomo and F. Brezzi, Traffic, crowds, and dynamics of self-organized particles: New trends and challenges, Math. Models Methods Appl. Sci.25 (2015) 395-400. [Abstract] genRefLink(128, ’S0218202516020012BIB13’, ’000351124000001’); · Zbl 1308.92110
[14] 14. N. Bellomo and L. Gibelli, Toward a mathematical theory of behavioral-social dynamics for pedestrian crowds, Math. Models Methods Appl. Sci.25 (2015) 2417-2437. [Abstract] genRefLink(128, ’S0218202516020012BIB14’, ’000361572300002’); · Zbl 1325.91042
[15] 15. N. Bellomo, M. A. Herrero and A. Tosin, On the dynamics of social conflicts looking for the black swan, Kinet. Relat. Mod.6 (2013) 459-479. genRefLink(16, ’S0218202516020012BIB15’, ’10.3934
[16] 16. N. Bellomo, D. Knopoff and J. Soler, On the difficult interplay between life, ”complexity”, and mathematical sciences, Math. Models Methods Appl. Sci.23 (2013) 1861-1913. [Abstract] genRefLink(128, ’S0218202516020012BIB16’, ’000321774700003’); · Zbl 1315.35137
[17] 17. A. Bellouquid, J. Nieto and L. Urrutia, About the kinetic description of fractional diffusion equations modeling chemotaxis, Math. Models Methods Appl. Sci.26 (2016) 249-268. [Abstract] · Zbl 1333.35298
[18] 18. A. Borzi and S. Wongkaew, Modeling and control through leadership of a refined flocking system, Math. Models Methods Appl. Sci.25 (2015) 565-585. [Abstract] genRefLink(128, ’S0218202516020012BIB18’, ’000356064500003’);
[19] 19. M. Caponigro, M. Fornasier, B. Piccoli and E. Trélat, Sparse stabilization and control of alignment models, Math. Models Methods Appl. Sci.25 (2015) 561-564. [Abstract] genRefLink(128, ’S0218202516020012BIB19’, ’000351124000006’); · Zbl 1331.49003
[20] 20. P. Degond, F. Delebecque and D. Peurichard, Continuum model for linked fibers with alignment interactions, Math. Models Methods Appl. Sci.26 (2016) 269-318. [Abstract] · Zbl 1341.82057
[21] 21. M. Di Francesco and S. Fagioli, A nonlocal swarm model for predators-prey interactions, Math. Models Methods Appl. Sci.26 (2016) 319-355. [Abstract] · Zbl 1334.35357
[22] 22. G. Dimarco and L. Pareschi, Numerical methods for kinetic equations, Acta Numer.23 (2014) 369-520. genRefLink(16, ’S0218202516020012BIB22’, ’10.1017
[23] 23. G. P. Ghiroldi and L. Gibelli, A direct method for the Boltzmann equation based on a pseudo-spectral velocity space discretization, J. Comput. Phys.258 (2014) 568-584. genRefLink(16, ’S0218202516020012BIB23’, ’10.1016 · Zbl 1349.76207
[24] 24. H. Gintis, Game Theory Evolving, 2nd edn. (Princeton Uni v. Press, 2009). · Zbl 1161.91005
[25] 25. M. Gromov, In a search for a structure, Part 1: On entropy, http://www.ihes.fr/ gromov/PDF/structre-serch-entropy-july5-2012. · Zbl 1364.82009
[26] 26. H. L. Hartwell, J. J. Hopfield, S. Leibler and A. W. Murray, From molecular to modular cell biology, Nature Rev.402 (1999) c47-c52. genRefLink(16, ’S0218202516020012BIB26’, ’10.1038
[27] 27. D. Helbing, Quantitative Sociodynamics. Stochastic Methods and Models of Social Interaction Processes, 2nd edn. (Springer, 2010).
[28] 28. M. A. Herrero and J. Soler, Cooperation, competition, organization: The dynamics of interacting living populations, Math. Models Methods Appl. Sci.25 (2015) 2407-2415. [Abstract] genRefLink(128, ’S0218202516020012BIB28’, ’000361572300001’); · Zbl 1325.92007
[29] 29. J. Hofbauer and K. Sigmund, Evolutionary game dynamics, Bull. Amer. Math. Soc.40 (2003) 479-519. genRefLink(16, ’S0218202516020012BIB29’, ’10.1090 · Zbl 1049.91025
[30] 30. G. Jona-Lasinio, La Matematica Come Linguaggio delle Scienze della Natura, Public talk, Centro E. De Giorgi, SNS, Pisa (2004).
[31] 31. I. Kant, Critique of the Power of Judgement (Cambridge Univ. Press, 2002).
[32] 32. A. Kirman, Complex Economics: Individual and Collective Rationality (Routledge, 2011).
[33] 33. J. M. Lasry and P. L. Lions, Mean field games, Japan J. Math.2 (2007) 229-260. genRefLink(16, ’S0218202516020012BIB33’, ’10.1007
[34] 34. Z. Li and S.-Y. Ha, Uniqueness and well-ordering of emergent phase-locked states for Kuramoto model with frustration and inertia, Math. Models Methods Appl. Sci.26 (2016) 357-382. [Abstract] · Zbl 1355.92146
[35] 35. E. Mayr, Populations, Species, and Evolution (Harvard Univ. Press, 1970).
[36] 36. E. Mayr, What Evolution Is (Basic Books, 2001).
[37] 37. J. Nash, Noncooperative games, Ann. Math.54 (1951) 287-295. genRefLink(16, ’S0218202516020012BIB37’, ’10.2307
[38] 38. J. Nash, Essentials of Game Theory (Elgar, 1996).
[39] 39. M. A. Nowak, Evolutionary Dynamics: Exploring the Equations of Life (Harvard Univ. Press, 2006). · Zbl 1115.92047
[40] 40. L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods (Oxford Univ. Press, 2013). · Zbl 1330.93004
[41] 41. B. Piccoli, N. Pouradier Duteil and B. Scharf, Optimal control of a collective migration model, Math. Models Methods Appl. Sci.26 (2016) 383-417. [Abstract] · Zbl 1342.92308
[42] 42. P. Romanczuk, M. Bar, W. Ebeling, B. Lindner and L. Schimansky-Geier, Active Brownian particles, from individual to collective stochastic dynamics, Eur. Phys. J.202 (2012) 1-162. genRefLink(128, ’S0218202516020012BIB42’, ’000301985100001’);
[43] 43. F. C. Santos, J. M. Pacheco and T. Lenaerts, Evolutionary dynamics of social dilemmas in structured heterogeneous populations, Proc. Natl. Acad. Sci. USA103 (2006) 3490-3494. genRefLink(16, ’S0218202516020012BIB43’, ’10.1073
[44] 44. M. Scheffer, J. Bascompte, W. A. Brock, V. Brovkin, S. R. Carpenter, V. Dakos, H. Held, E. H. van Nes, M. Rietkerk and G. Sugihara, Early-warning signals for critical transitions, Nature461 (2009) 53-59. genRefLink(16, ’S0218202516020012BIB44’, ’10.1038
[45] 45. E. Schrödinger, What is Life? The Physical Aspect of the Living Cell (Cambridge Univ. Press, 1933).
[46] 46. K. Sigmund, The Calculus of Selfishness, Princeton University Series in Theoretical and Computational Biology (Princeton Univ. Press, 2011). · Zbl 1338.91008
[47] 47. N. N. Taleb, The Black Swan: The Impact of the Highly Improbable (Random House, 2007).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.