Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-24T01:35:20.307Z Has data issue: false hasContentIssue false
  • English
  • Français

A particle system approach to aggregation phenomena

Part of: Mathematical biology in general Equations of mathematical physics and other areas of application Parabolic equations and systems Physiological, cellular and medical topics Special processes

Published online by Cambridge University Press:  12 July 2019

Franco Flandoli  and
Marta Leocata
Franco Flandoli*
Affiliation:
Scuola Normale Superiore di Pisa
Marta Leocata*
Affiliation:
University of Pisa
*
*Postal address: Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, Pisa, Italy. Email address: franco.flandoli@sns.it
**Postal address: Dipartimento di Matematica, University of Pisa, Largo Pontecorvo 5, Pisa, Italy. Email address: leocata@mail.dm.unipi.it
Get access Rights & Permissions [Opens in a new window]

Abstract

Inspired by a PDE–ODE system of aggregation developed in the biomathematical literature, we investigate an interacting particle system representing aggregation at the level of individuals. We prove that the empirical density of the individual converges to the solution of the PDE–ODE system.

Keywords

Aggregation phenomenapopulation dynamicmacroscopic limitparticle systemsemigroup approach

MSC classification

Primary: 35K57: Reaction-diffusion equations 35Q92: PDEs in connection with biology and other natural sciences 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 92B05: General biology and biomathematics 92C17: Cell movement (chemotaxis, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 1 , March 2019 , pp. 282 - 306
Copyright
© Applied Probability Trust 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Armstrong, N. J., Painter, K. J. and Sherratt, J. A. (2006). A continuum approach to modelling cell-cell adhesion. J. Theoret. Biol. 243, 98–113.CrossRefGoogle ScholarPubMed
Bergh, J. and Löfström, J. (1976). Interpolation Spaces. An Introduction Springer, Berlin.CrossRefGoogle Scholar
Deroulers, C., Aubert, M., Badoual, M. and Grammaticos, B. (2009). Modeling tumor cell migration: From microscopic to macroscopic models. Phys. Rev. E 79, 14pp.CrossRefGoogle ScholarPubMed
Dyson, J., Gourley, S. A. and Webb, G. F. (2013). A non-local evolution equation model of cell-cell adhesion in higher dimensional space. J. Biol. Dyn. 7, 68–87.CrossRefGoogle ScholarPubMed
Flandoli, F., Leimbach, M. and Olivera, C. (2019). Uniform convergence of proliferating particles to the FKPP equation. J. Math. Anal. Appl. 473, 27–52.CrossRefGoogle Scholar
Karatzas, I. and Shreve, S. E. (1988). Brownian Motion and Stochastic Calculus. Springer, New York.CrossRefGoogle Scholar
Kipnis, C. and Landim, C. (2013). Scaling limits of interacting particle systems, Vol. 320. Springer, Berlin.Google Scholar
Neklydov, M. and Trevisan, D. (2016). A particle system approach to cell-cell adhesion models. Preprint. Available at https://arxiv.org/abs/1601.05241.Google Scholar
Oelschläger, K. (1985). A law of large numbers for moderately interacting diffusion processes. Z. Wahrscheinlichkeitsth. 69, 279–322.CrossRefGoogle Scholar
Painter, K. J., Armstrong, N. J., Sherratt, J. A. (2010). The impact of adhesion on cellular invasion processes in cancer and development. J. Theoret. Biol. 264, 1057–1067.CrossRefGoogle ScholarPubMed
Perumpanani, A. J., Sherratt, J. A., Norbury, J., and Byrne, H. M. (1996). Biological inferences from a mathematical model for malignant invasion. Invasion Metastasis, 16, 209-221.Google ScholarPubMed
Simon, J. (1987). Compact sets in the space Lp(0, T;B). Ann. Mat. Pura Appl. (4) 146, 65–96.CrossRefGoogle Scholar
Sznitman, A.S. (1991). Topics in propagation of chaos. In Ecole d’Ete de Probabilites de Saint-Flour XIX–1989 (Lecture Notes Math. 1464), Springer, Berlin, pp. 165–251.Google Scholar