Existence and approximation of probability measure solutions to models of collective behaviors

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September 2011, 6(3): 561-596. doi: 10.3934/nhm.2011.6.561

Existence and approximation of probability measure solutions to models of collective behaviors

Andrea Tosin ^1, and Paolo Frasca ^1,

1.	Department of Mathematics, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

Received December 2010 Revised February 2011 Published August 2011

Abstract
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In this paper we consider first order differential models of collective behaviors of groups of agents, based on the mass conservation equation. Models are formulated taking the spatial distribution of the agents as the main unknown, expressed in terms of a probability measure evolving in time. We develop an existence and approximation theory of the solutions to such models and we show that some recently proposed models of crowd and swarm dynamics fit our theoretic paradigm.

Keywords: nonlocal flux., Systems of interacting agents, probability distribution, continuity equation.

Mathematics Subject Classification: Primary: 35L65, 35Q70; Secondary: 35Q9.

Citation: Andrea Tosin, Paolo Frasca. Existence and approximation of probability measure solutions to models of collective behaviors. Networks and Heterogeneous Media, 2011, 6 (3) : 561-596. doi: 10.3934/nhm.2011.6.561

References:

[1]	L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures," 2^nd edition, Birkhäuser Verlag, Basel, 2008.
[2]	N. Bellomo, "Modeling Complex Living Systems. A Kinetic Theory and Stochastic Game Approach," Birkhäuser, Boston, 2008.
[3]	J. A. Cañizo, J. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Models Methods Appl. Sci., 21 (2011), 515-539. doi: 10.1142/S0218202511005131.
[4]	C. Canuto, F. Fagnani and P. Tilli, A Eulerian approach to the analysis of rendez-vous algorithms, in "Proceedings of the 17th IFAC World Congress", (2008), 9039-9044.
[5]	E. Cristiani, P. Frasca and B. Piccoli, Effects of anisotropic interactions on the structure of animal groups, J. Math. Biol., 62 (2011), 569-588. doi: 10.1007/s00285-010-0347-7.
[6]	E. Cristiani, B. Piccoli and A. Tosin, Modeling self-organization in pedestrian and animal groups from macroscopic and microscopic viewpoints, in "Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences" (eds. G. Naldi, L. Pareschi and G. Toscani), Birkhäuser, Boston, (2010), 337-364.
[7]	R. J. LeVeque, "Numerical Methods for Conservation Laws," 2^nd edition, Birkhäuser Verlag, Basel, 1992.
[8]	B. Piccoli and A. Tosin, Pedestrian flows in bounded domains with obstacles, Contin. Mech. Thermodyn., 21 (2009), 85-107. doi: 10.1007/s00161-009-0100-x.
[9]	B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow, Arch. Ration. Mech. Anal., 199 (2011), 707-738. doi: 10.1007/s00205-010-0366-y.

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References:

[1]	L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures," 2^nd edition, Birkhäuser Verlag, Basel, 2008.
[2]	N. Bellomo, "Modeling Complex Living Systems. A Kinetic Theory and Stochastic Game Approach," Birkhäuser, Boston, 2008.
[3]	J. A. Cañizo, J. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Models Methods Appl. Sci., 21 (2011), 515-539. doi: 10.1142/S0218202511005131.
[4]	C. Canuto, F. Fagnani and P. Tilli, A Eulerian approach to the analysis of rendez-vous algorithms, in "Proceedings of the 17th IFAC World Congress", (2008), 9039-9044.
[5]	E. Cristiani, P. Frasca and B. Piccoli, Effects of anisotropic interactions on the structure of animal groups, J. Math. Biol., 62 (2011), 569-588. doi: 10.1007/s00285-010-0347-7.
[6]	E. Cristiani, B. Piccoli and A. Tosin, Modeling self-organization in pedestrian and animal groups from macroscopic and microscopic viewpoints, in "Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences" (eds. G. Naldi, L. Pareschi and G. Toscani), Birkhäuser, Boston, (2010), 337-364.
[7]	R. J. LeVeque, "Numerical Methods for Conservation Laws," 2^nd edition, Birkhäuser Verlag, Basel, 1992.
[8]	B. Piccoli and A. Tosin, Pedestrian flows in bounded domains with obstacles, Contin. Mech. Thermodyn., 21 (2009), 85-107. doi: 10.1007/s00161-009-0100-x.
[9]	B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow, Arch. Ration. Mech. Anal., 199 (2011), 707-738. doi: 10.1007/s00205-010-0366-y.

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