# American Institute of Mathematical Sciences

September  2012, 32(9): 3081-3097. doi: 10.3934/dcds.2012.32.3081

## Conservation laws in mathematical biology

 1 The Ohio State University, Department of Mathematics, Columbus, OH 43210

Received  September 2011 Revised  March 2012 Published  April 2012

Many mathematical models in biology can be described by conservation laws of the form $$\tag{0.1} \frac{\partial{\bf{u}}}{\partial t} + \rm{div}(V{\bf{u}})=F(t,{\bf{x}}, {\bf{u}})\quad ({\bf{x}}=(x_1,\dots, x_n))$$ where ${\bf{u}}={\bf{u}}(t,{\bf{x}})$ is a vector $(u_1,\dots,u_k)$, ${\bf{F}}$ is a vector $(F_1,\dots,F_k)$, $V$ is a matrix with elements $V_{ij}(t,{\bf{x}},{\bf{u}})$, and $F_i(t,{\bf{x}}, {\bf{u}})$, $V_{ij}(t,{\bf{x}}, {\bf{u}})$ are nonlinear and/or non-local functions of ${\bf{u}}$. From a mathematical point of view one would like to establish, first of all, the existence and uniqueness of solutions under some prescribed initial (and possibly also boundary) conditions. However, the more interesting questions relate to establishing properties of the solutions that are of biological interest.
In this article we give examples of biological processes whose mathematical models are represented in the form (0.1). We describe results and present open problems.
Citation: Avner Friedman. Conservation laws in mathematical biology. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3081-3097. doi: 10.3934/dcds.2012.32.3081
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