Journal of Nonlinear Science

, Volume 22, Issue 1, pp 39–61 | Cite as

An Analytical Framework to Describe the Interactions Between Individuals and a Continuum

Article

Abstract

We consider a discrete set of individual agents interacting with a continuum. Examples might be a predator facing a huge group of preys, or a few shepherd dogs driving a herd of sheep. Analytically, these situations can be described through a system of ordinary differential equations coupled with a scalar conservation law in several space dimensions. This paper provides a complete well-posedness theory for the resulting Cauchy problem. A few applications are considered in detail and numerical integrations are provided.

Keywords

Mixed P.D.E.–O.D.E. problems Conservation laws Ordinary differential equations 

Mathematics Subject Classification (2000)

35L65 34A12 37N99 

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References

  1. Borsche, R., Colombo, R.M., Garavello, M.: On the coupling of systems of hyperbolic conservation laws with ordinary differential equations. Nonlinearity 23, 2749–2770 (2010) MathSciNetMATHCrossRefGoogle Scholar
  2. Bressan, A., De Lellis, C.: Existence of optimal strategies for a fire confinement problem. Commun. Pure Appl. Math. 62(6), 789–830 (2009) MATHCrossRefGoogle Scholar
  3. Capasso, V., Micheletti, A., Morale, D.: Stochastic geometric models, and related statistical issues in tumour-induced angiogenesis. Math. Biosci. 214(1–2), 20–31 (2008) MathSciNetMATHCrossRefGoogle Scholar
  4. Colombo, R.M., Mercier, M., Rosini, M.D.: Stability and total variation estimates on general scalar balance laws. Commun. Math. Sci. 7(1), 37–65 (2009) MathSciNetMATHGoogle Scholar
  5. Colombo, R.M., Herty, M., Mercier, M.: Control of the continuity equation with a non-local flow. ESAIM Control Optim. Calc. Var. 17, 353–379 (2011). doi:10.1051/cocv/2010007 MathSciNetMATHCrossRefGoogle Scholar
  6. Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides. Kluwer Academic, Dordrecht (1988). Translated from the Russian Google Scholar
  7. Grimm, J., Grimm, W.: Deutsche Sagen, 2nd edn. Nicolaische Verlagsbichhandlung, Berlin (1865) Google Scholar
  8. Hoff, W.D., van der Horst, M.A., Nudel, C.B., Hellingwerf, K.J.: In: Prokaryotic Phototaxis, vol. 571, pp. 25–49. Springer, Berlin (2009). Chapter 2 Google Scholar
  9. Kružkov, S.N.: First order quasilinear equations with several independent variables. Mat. Sb. (N.S.) 81(123), 228–255 (1970) MathSciNetGoogle Scholar
  10. Lattanzio, C., Maurizi, A., Piccoli, B.: Moving bottlenecks in car traffic flow: a pde-ode coupled model. Preprint (2010) Google Scholar
  11. Lécureux-Mercier, M.: Improved stability estimates on general scalar balance laws. J. Hyperbolic Differ. Equ. (2011, to appear) Google Scholar
  12. LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002) CrossRefGoogle Scholar
  13. Serre, D.: Chute libre d’un solide dans un fluide visqueux incompressible. Existence. Jpn. J. Appl. Math. 4(1), 99–110 (1987) MATHCrossRefGoogle Scholar
  14. Vázquez, J.L., Zuazua, E.: Large time behavior for a simplified 1D model of fluid-solid interaction. Commun. Partial Differ. Equ. 28(9–10), 1705–1738 (2003) MATHCrossRefGoogle Scholar
  15. Witman, G.B.: Chlamydomonas phototaxis. Trends Cell Biol. 3(11), 403–408 (1993) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsBrescia UniversityBresciaItaly
  2. 2.UFR SciencesUniversité d’OrléansOrléans cedex 2France

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