Hyperbolic Systems and Transport Equations in Mathematical Biology

Summary The standard models for groups of interacting and moving individuals (from cell biology to vertebrate population dynamics) are reaction-diffusion models. They base on Brownian motion, which is characterized by one single parameter (diffusion coefficient). In particular for moving bacteria and (slime mold) amoebae, detailed information on individual movement behavior is available (speed, run times, turn angle distributions). If such information is entered into models for populations, then reaction-transport equations or hyperbolic equations (telegraph equations, damped wave equations) result. The goal of this review is to present some basic applications of transport equations and hyperbolic systems and to illustrate the connections between transport equations, hyperbolic models, and reaction-diffusion equations. Applied to chemosensitive movement (chemotaxis) functional estimates for the nonlinearities in the classical chemotaxis model (Patlak-Keller-Segel) can be derived, based on the individual behavior of cells and attractants. A detailed review is given on two methods of reduction for transport equations. First the construction of parabolic limits (diffusion limits) for linear and non-linear transport equations and then a moment closure method based on energy minimization principles. We illustrate the moment closure method on the lowest non-trivial case (two-moment closure), which leads to Cattaneo systems. Moreover we study coupled dynamical systems and models with quiescent states. These occur naturally if it is assumed that different processes, like movement and reproduction, do not occur simultaneously. We report on travelling front problems, stability, epidemic modeling, and transport equations with resting phases.