An Analytical Framework to Describe the Interactions Between Individuals and a Continuum
- 131 Downloads
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.
KeywordsMixed P.D.E.–O.D.E. problems Conservation laws Ordinary differential equations
Mathematics Subject Classification (2000)35L65 34A12 37N99
Unable to display preview. Download preview PDF.
- Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides. Kluwer Academic, Dordrecht (1988). Translated from the Russian Google Scholar
- Grimm, J., Grimm, W.: Deutsche Sagen, 2nd edn. Nicolaische Verlagsbichhandlung, Berlin (1865) Google Scholar
- 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
- Lattanzio, C., Maurizi, A., Piccoli, B.: Moving bottlenecks in car traffic flow: a pde-ode coupled model. Preprint (2010) Google Scholar
- Lécureux-Mercier, M.: Improved stability estimates on general scalar balance laws. J. Hyperbolic Differ. Equ. (2011, to appear) Google Scholar