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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. [ABSTRACT FROM AUTHOR]

We consider the general scalar balance law ∂t u + Div f(t, x, u) = F(t, x, u) in several space dimensions and improve the results of Colombo, Mercier and Rosini who gave an estimate of the dependence of the solutions in terms of the flux f and the source F. The improvement is two-fold: first, the expression of the coefficients in these estimates are sharper and, second, we eliminate some regularity hypotheses, and thus extend significantly the applicability of these estimates. [ABSTRACT FROM AUTHOR]