×

On the interaction between soft and hard sciences: the role of mathematical sciences. Looking ahead to research perspectives. (English) Zbl 1464.35354

Summary: This paper deals with a study of the conceivable contributions of mathematical sciences to the modeling of complex systems constituted by several interacting living entities, where interactions are nonlocal and nonlinearly additive. The first part of the presentation provides some reasonings and speculations on the specific features of living systems which have to be taken into account in the modeling approach. The second part is proposed as a review of known results focused on a modeling strategy. Some open problems are treated in the third part where hints to tackle them are brought to the readers attention in view of research perspectives. Some free speculations on the complex interplay between soft and hard sciences follows.

MSC:

35Q92 PDEs in connection with biology, chemistry and other natural sciences
35Q84 Fokker-Planck equations
92-10 Mathematical modeling or simulation for problems pertaining to biology
92B05 General biology and biomathematics
92D25 Population dynamics (general)
82C40 Kinetic theory of gases in time-dependent statistical mechanics
82C22 Interacting particle systems in time-dependent statistical mechanics
35K55 Nonlinear parabolic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Acemoglu, D.; Robinson, JA, Economic backwardness in political perspective, Am. Politi. Sci. Rev., 100, 115-131 (2006)
[2] Ajmone Marsan, G.; Bellomo, N.; Gibelli, L., Stochastic evolutionary differential games toward a systems theory of behavioral social dynamics, Math. Models Methods Appl. Sci., 26, 1051-1093 (2016) · Zbl 1414.91053
[3] Albi, G.; Bellomo, N.; Fermo, L.; Ha, S-Y; Kim, J.; Pareschi, L.; Poyato, D.; Soler, J., Vehicular traffic, crowds, and swarms: from kinetic theory and multiscale methods to applications and research perspectives, Math. Models Methods Appl. Sci., 29, 1901-2005 (2019) · Zbl 1431.35211
[4] Allen, B.; Nowak, MA, Games on graphs, EMS Surv. Math. Sci., 1, 113-151 (2014) · Zbl 1303.91040
[5] Arcuri, A.; Lanchier, N., Stochastic spatial model for the division of labor in social insects, Math. Models Methods Appl. Sci., 27, 45-73 (2017) · Zbl 1358.92103
[6] Arias, M.; Campos, J.; Soler, J., Cross-diffusion and traveling waves in porous-media flux-saturated Keller-Segel models, Math. Models Methods Appl. Sci., 28, 2103-2129 (2018) · Zbl 1411.35157
[7] Ball, P., Why Society is a Complex Matter (2012), Berlin: Springer, Berlin
[8] Ballerini, M.; Cabibbo, N.; Candelier, R.; Cavagna, A.; 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. Nat. Acad. Sci., 105, 1232-1237 (2008)
[9] Banasiak, J.; Lachowicz, M., Methods of Small Parameter in Mathematical Biology. Modeling and Simulation in Science, Engineering and Technology (2014), Boston: Birkhäuser, Boston · Zbl 1309.92012
[10] Beinhocker, E.: The Origin of Wealth: Evolution, Complexity and the Radical Remaking of Economics. Random House (2006)
[11] Bellomo, N.: Modeling Complex Living Systems: a Kinetic Theory and Stochastic Game Approach. Birkhäuser, Boston (2008) · Zbl 1140.91007
[12] Bellomo, N.; Bellouquid, A., On multiscale models of pedestrian crowds from mesoscopic to macroscopic, Commun. Math. Sci., 13, 1649-1664 (2015) · Zbl 1329.90029
[13] Bellomo, N.; Bellouquid, A.; Chouhad, N., From a multiscale derivation of nonlinear cross-diffusion models to Keller-Segel models in a Navier-Stokes fluid, Math. Models Methods Appl. Sci., 26, 2041-2069 (2016) · Zbl 1353.35038
[14] Bellomo, N., Bellouquid, A., Gibelli, L., Outada, N.: A Quest Towards a Mathematical Theory of Living Systems. Birkhäuser, New York (2017) · Zbl 1381.92001
[15] Bellomo, N.; Bellouquid, A.; Nieto, J.; Soler, J., On the asymptotic theory from microscopic to macroscopic growing tissue models: an overview with perspectives, Math. Models Methods Appl. Sci., 22, 1130001 (2012) · Zbl 1328.92023
[16] Bellomo, N.; Bellouquid, A.; Nieto, J.; Soler, J., On the multiscale modeling of vehicular traffic: from kinetic to hydrodynamics, Discrete Contin. Dyn. Syst. Ser. B, 19, 1869-1888 (2014) · Zbl 1302.35372
[17] Bellomo, N.; Bellouquid, A.; Tao, Y.; Winkler, M., Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25, 1663-1763 (2015) · Zbl 1326.35397
[18] 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, 421-441 (2015) · Zbl 1403.91278
[19] Bellomo, N.; Degond, P.; Tadmor, E., Active Particles, Volume 1: Advances in Theory, Models, and Applications (2017), Basel: Birkhäuser, Basel · Zbl 1368.00045
[20] Bellomo, N.; Gibelli, L.; Outada, N., On the interplay between behavioral dynamics and social interactions in human crowds, Kinet. Relat. Models, 12, 397-409 (2019) · Zbl 1420.91384
[21] Bellomo, N.; Herrero, MA; Tosin, A., On the dynamics of social conflicts: looking for the black swan, Kinet. Relat. Models, 6, 459-479 (2013) · Zbl 1276.82028
[22] Bellomo, N.; Knopoff, D.; Soler, J., On the difficult interplay between life complexity and mathematical sciences, Math. Models Methods Appl. Sci., 23, 1861-1913 (2013) · Zbl 1315.35137
[23] Bellomo, N.; Soler, J., On the mathematical theory of the dynamics of swarms viewed as a complex system, Math. Models Methods Appl. Sci., 22, 1140006 (2012) · Zbl 1242.92065
[24] Bellomo, N.; Winkler, M., A degenerate chemotaxis system with flux limitation: maximally extended solutions and absence of gradient blow-up, Commun. Partial Differ. Equ., 42, 436-473 (2017) · Zbl 1430.35166
[25] Bellouquid, A.; Chouhad, N., Kinetic models of chemotaxis towards the diffusive limit: asymptotic analysis, Math. Methods Appl. Sci., 39, 3136-3151 (2016) · Zbl 1342.35403
[26] Bellouquid, A.; De Angelis, E., From kinetic models of multicellular growing systems to macroscopic biological tissue models, Nonlinear Anal. Real World Appl., 12, 1111-1122 (2011) · Zbl 1203.92020
[27] Bellouquid, A.; De Angelis, E.; Knopoff, D., From the modeling of the immune hallmarks of cancer to a black swan in biology, Math. Models Methods Appl. Sci., 23, 949-978 (2013) · Zbl 1303.92040
[28] Bellouquid, A., Delitala, M.: Mathematical Modeling of Complex Biological Systems: a Kinetic Theory Approach. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Basel (2006) · Zbl 1178.92002
[29] Bertotti, ML; Modanese, G., From microscopic taxation and redistribution models to macroscopic income distributions, Phys. A, 390, 3782-3793 (2011)
[30] Bertozzi, AL; Rosado, J.; Short, MB; Wang, L., Contagion shocks in one dimension, J. Stat. Phys., 158, 647-664 (2015) · Zbl 1318.35135
[31] Bonacich, P.; Lu, P., Introduction to Mathematical Sociology (2012), Princeton: Princeton University Press, Princeton · Zbl 1244.91075
[32] Brugna, C.; Toscani, G., Kinetic models of opinion formation in the presence of personal conviction, Phys. Rev. E, 92, 052818 (2015)
[33] Brugna, C.; Toscani, G., Kinetic models for goods exchange in a multi-agent market, Phys. A Stat. Mech. Appl., 499, 362-375 (2018) · Zbl 1514.91208
[34] Burger, M.; Caffarelli, L.; Markowich, P., Partial differential equation models in the socio-economic sciences, Philos. Trans. R. Soc. A, 372, 20130406 (2014) · Zbl 1353.00004
[35] Burini, D.; Chouhad, N., Hilbert method toward a multiscale analysis from kinetic to macroscopic models for active particles, Math. Models Methods Appl. Sci., 27, 1327-1353 (2017) · Zbl 1372.35302
[36] Burini, D.; Chouhad, N., A multiscale view of nonlinear diffusion in biology: from cells to tissues, Math. Models Methods Appl. Sci., 29, 791-823 (2019) · Zbl 1427.35291
[37] Burini, D.; De Lillo, S., On the complex interaction between collective learning and social dynamics, Symmetry, 11, 967 (2019) · doi:10.3390/sym11080967
[38] Burini, D.; De Lillo, S.; Fioriti, G., Influence of drivers ability in a discrete vehicular traffic model, Int. J. Modern Phys. C, 28, 1750030 (2017)
[39] Burini, D.; De Lillo, S.; Gibelli, L., Collective learning modeling based on the kinetic theory of active particles, Phys. Life Rev., 16, 123-139 (2016)
[40] Burini, D., Gibelli, L., Outada, N.: 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 Simulations in Science, Engineering and Technology, pp 229-258. Birkhäuser, Basel (2017) · Zbl 1381.92001
[41] Camerer, CF, Behavioral Game Theory: Experiments in Strategic Interaction (2003), Princeton, NJ: Princeton University Press, Princeton, NJ · Zbl 1019.91001
[42] Caponigro, M.; Lai, AC; Piccoli, B., A nonlinear model of opinion formation on the sphere, Discrete Contin. Dyn. Syst. Ser. A, 35, 4241-4268 (2015) · Zbl 1333.91047
[43] Comte, A.: Cours De Philosophie Positive. Hermann, Paris (2012) · JFM 38.0092.05
[44] Corbin, G.; Hunt, A.; Klar, A.; Schneider, F.; Surulescu, C., Higher-order models for glioma invasion: from a two-scale description to effective equations for mass density and momentum, Math. Models Methods Appl. Sci., 28, 1771-1800 (2018) · Zbl 1411.92034
[45] Dabnoun, NMO; Mongiovì, MS, A contribution to the mathematical modeling of immune-cancer competition, Commun. Appl. Ind. Math., 9, 76-90 (2018) · Zbl 1423.92031
[46] De Angelis, E., On the mathematical theory of post-Darwinian mutations, selection, and evolution, Math. Models Methods Appl. Sci., 24, 2723-2742 (2014) · Zbl 1328.92050
[47] De Lillo, S.; Delitala, M.; Salvatori, M., Modelling epidemics and virus mutations by methods of the mathematical kinetic theory for active particles, Math. Models Methods Appl. Sci., 19, 1405-1425 (2009) · Zbl 1175.92035
[48] Diamond, J., Soft sciences are often harder than hard sciences, Discover, 8, 34-39 (1987)
[49] Dolfin, M.; Knopoff, D.; Leonida, L.; Patti, DMA, Escaping the trap of blocking: a kinetic model linking economic development and political competition, Kinet. Relat. Models, 10, 423-443 (2017) · Zbl 1414.91134
[50] Dolfin, M.; Lachowicz, M., Modeling altruism and selfishness in welfare dynamics: the role of nonlinear interactions, Math. Models Methods Appl. Sci., 24, 2361-2381 (2014) · Zbl 1303.91074
[51] Dolfin, M.; Lachowicz, M., Modeling opinion dynamics: how the network enhances consensus, Netw. Heterog. Media, 10, 877-896 (2015) · Zbl 1377.91142
[52] Dolfin, M.; Leonida, L.; Outada, N., Modeling human behavior in economics and social science, Phys. Life Rev., 22-23, 1-21 (2017)
[53] Elaiw, A.; Al-Turki, Y.; Alghamdi, M., A critical analysis of behavioural crowd dynamics—from a modelling strategy to kinetic theory methods, Symmetry, 11, 851 (2019)
[54] Engwer, C.; Stinner, C.; Surulescu, C., On a structured multiscale model for acid-mediated tumor invasion: the effects of adhesion and proliferation, Math. Models Methods Appl. Sci., 27, 1355-1390 (2017) · Zbl 1371.35149
[55] Furioli, G.; Pulvirenti, A.; Terraneo, E.; Toscani, G., Fokker-Planck equations in the modeling of socio-economic phenomena, Math. Models Methods Appl. Sci., 27, 115-158 (2017) · Zbl 1362.35305
[56] Gächter, S.; Schulz, JF, Intrinsic honesty and the prevalence of rule violations across societies, Nature, 531, 496-499 (2016)
[57] Galam, S., Sociophysics (2012), New York: Springer, New York · Zbl 1141.91668
[58] Gino, F.; Pierce, L., The abundance effect: unethical behavior in the presence of wealth, Organ. Behav. Hum. Decis. Process., 109, 142-155 (2009)
[59] Gintis, H.: Game Theory Evolving, 2nd edn. Princeton University Press, Princeton (2009) · Zbl 1161.91005
[60] Guéant, O., Lasry, J.-M., Lions, P.-L.: Mean field games and applications. In: Cousin, A., et al. (eds.) Paris-Princeton Lectures on Mathematical Finance 2010. Lecture Notes in Mathematics, vol. 2003, pp 205-266. Springer, Berlin (2011) · Zbl 1205.91027
[61] Gualandi, S.; Toscani, G., Call center service times are lognormal: a Fokker-Planck description, Math. Models Methods Appl. Sci., 28, 1513-1527 (2018) · Zbl 1398.35247
[62] Hartwell, LH; Hopfield, JJ; Leibler, S.; Murray, AW, From molecular to modular cell biology, Nature, 402, C47-C52 (1999)
[63] Hegselmann, R.; Krause, U., Opinion dynamics and bounded confidence: models, analysis, and simulations, J. Artif. Soc. Soc. Simul., 5, 2, 2 (2002)
[64] Hegselmann, R.; Krause, U., Opinion dynamics under the influence of radical groups, charismatic and leaders, and other constant signals: a simple unifying model, Netw. Heterog. Media, 10, 477-509 (2015) · Zbl 1332.90049
[65] Helbing, D.: Quantitative Sociodynamics: Stochastic Methods and Models of Social Interaction Processes, 2nd edn. Springer, Berlin (2010) · Zbl 1213.91002
[66] Herrero, M., On the role of mathematics in biology, J. Math. Biol., 54, 887-889 (2007) · Zbl 1180.92003
[67] Hilbert, D., Mathematical problems, Bull. Am. Math. Soc., 8, 437-479 (1902) · JFM 33.0976.07
[68] Hofbauer, J.; Sigmund, K., Evolutionary game dynamics, Bull. Am. Math. Soc., 40, 479-519 (2003) · Zbl 1049.91025
[69] Kant, I., Critique of the Power of Judgment (2002), Cambridge: Cambridge University Press, Cambridge
[70] Knopoff, D., On the modeling of migration phenomena on small networks, Math. Models Methods Appl. Sci., 23, 541-563 (2013) · Zbl 1357.91035
[71] Knopoff, D., On a mathematical theory of complex systems on networks with application to opinion formation, Math. Models Methods Appl. Sci., 24, 405-426 (2014) · Zbl 1281.91134
[72] Knopoff, D.; Nieto, J.; Urrutia, L., Numerical simulation of a multiscale cell motility model based on the kinetic theory of active particles, Symmetry, 11, 1003 (2019) · doi:10.3390/sym11081003
[73] Knopoff, D.; Sánchez, JM, A kinetic model for horizontal transfer and bacterial antibiotic resistance, Int. J. Biomath., 10, 1750051 (2017) · Zbl 1373.92062
[74] Lachowicz, M.; Leszczyński, H.; Puźniakowska-Galuch, E., Diffusive and anti-diffusive behavior for kinetic models of opinion dynamics, Symmetry, 11, 1024 (2019) · doi:10.3390/sym11081024
[75] Lasry, J-M; Lions, P-L, Mean field games, Jpn. J. Math., 2, 229-260 (2007) · Zbl 1156.91321
[76] Liu, L.; Chen, X.; Szolnoki, A., Evolutionary dynamics of cooperation in a population with probabilistic corrupt enforcers and violators, Math. Models Methods Appl. Sci., 29, 2127-2149 (2019) · Zbl 1428.91005
[77] May, RM, Uses and abuses of mathematics in biology, Science, 303, 790-793 (2004)
[78] Mayr, E., What Evolution Is (2001), New York: Basic Books, New York
[79] Nash, J., Essentials of Game Theory (1996), Cheltenham: Elgar, Cheltenham
[80] Nowak, MA, Evolutionary Dynamics: Exploring the Equations of Life (2006), Cambridge: Harvard University Press, Cambridge · Zbl 1115.92047
[81] Pareschi, L.; Toscani, G., Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods (2013), Oxford: Oxford University Press, Oxford · Zbl 1330.93004
[82] Piff, PK; Stancato, DM; Côté, S.; Mendoza-Denton, R.; Keltner, D., Higher social class predicts increased unethical behavior, Proc. Nat. Acad. Sci., 109, 4086-4091 (2014)
[83] Prigogine, I.; Herman, R., Kinetic Theory of Vehicular Traffic (1971), New York: Elsevier, New York · Zbl 0226.90011
[84] Reed, MC, Why is mathematical biology so hard?, Not. Am. Math. Soc., 51, 338-342 (2004) · Zbl 1168.92303
[85] Roth, S., Mathematics and biology: a Kantian view on the history of pattern formation theory, Dev. Genes Evol., 221, 255-279 (2011)
[86] Salvi, S., Corruption corrupts: Society-level rule violations affect individuals’ intrinsic honesty, Nature, 531, 456-457 (2016)
[87] Schrödinger, E., What is Life? The Physical Aspect of the Living Cell (1944), Cambridge: Cambridge University Press, Cambridge · Zbl 1254.01052
[88] Sigmund, K., The Calculus of Selfishness. Princeton Series in Theoretical and Computational Biology (2011), Princeton: Princeton University Press, Princeton · Zbl 1189.91010
[89] Tao, Y.; Winkler, M., Large time behavior in a forager-exploiter model with different taxis strategies for two groups in search of food, Math. Models Methods Appl. Sci., 29, 2151-2182 (2019) · Zbl 1425.35200
[90] Thaler, RH, Behavioral economics: past, present, and future, Am. Econ. Rev., 106, 1577-1600 (2016)
[91] Thaler, RH; Sunstein, C., Nudge: Improving Decisions about Health, Wealth, and Happiness (2016), New York: Penguin, New York
[92] Toscani, G., Kinetic models of opinion formation, Commun. Math. Sci., 4, 481-496 (2006) · Zbl 1195.91128
[93] Wang, L.; Short, MB; Bertozzi, AL, Efficient numerical methods for multiscale crowd dynamics with emotional contagion, Math. Models Methods Appl. Sci., 27, 205-230 (2017) · Zbl 1359.35197
[94] Weinberg, RA, The Biology of Cancer (2007), New York: Garland Sciences - Taylor and Francis, New York
[95] Woese, CR, A new biology for a new century, Microbiol. Mol. Biol. Rev., 68, 173-186 (2004)
[96] Zhigun, A.; Surulescu, C.; Hunt, A., A strongly degenerate diffusion-haptotaxis model of tumour invasion under the go-or-grow dichotomy hypothesis, Math. Methods Appl. Sci., 41, 2403-2428 (2018) · Zbl 1390.35383
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.