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Lehrstuhl für Biomathematik
Auf der Morgenstelle 10
72076 Tübingen
Tel.: 07071/29 78827
e-mail: thomas.hillen@uni-tuebingen.de(thomas.hillen@uni-tuebingen.de)

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Hyperbolic Models for Spatial Spread and Reactions

The classical models to describe spatial motion and interaction of particles are reaction diffusion systems They are often used to describe biological, chemical or physical experiments, but they suffer from the deficiencies of the heat equation, i.e. infinite propagation speeds are possible. Even Einstein (1906) criticises the fact that in Brownian motion successive steps are uncorrelated. Many applications, especially in biological population dynamics, require an adaptation for finite propagation speeds. Moreover the interplay of reaction and motion cannot be neglected in such situations. For this reason I consider two approaches which respect finite speeds and reactions. The first is based on individual motion of particles ( velocity jump process ), whereas the second is used in thermodynamics to describe heat propagation with finite speeds ( Cattaneo systems ). Both models meet in a very spacial case of the one dimensional correlated random walk .

The velocity jump process is described by a system of (infinitely many) coupled integro-differential equations (Othmer, Dunbar, Alt (1988)). It is closely adapted to the biological reality but they are mathematicaly hard to handle. Nevertheless the theory of the Boltzmann equation can be used as a tool book (e.g. Cercignani (1975)).

The second approach generalizes the equations of Brownian motion in such a way that finite particle speed results. This is done by Cattaneo (1948) in the context of heat propagation with finite speed. It is closely related to a telegraph equation , hence a damped wave equation.

In a limiting process ( parabolic limit ) the diffusion equation is an approximation to the Cattaneo system and to the velocity jump process, where the kernel is symmetric with respect to the velocities and the absolut value of the velocities is constant.

The one dimensional correlated random walk (Taylor (1920), Goldstein (1951), Kac (1956)) is a special case of both the velocity jump process and the Cattaneo system. Particles move along the real axis with a constant velocity and they change their direction with respect to a Poisson process. A simple hyperbolic system follows. It becomes more interesting if we introduce (i) boundary conditions, (ii) reactions of the particles and (iii) more than one particle types. Here the interplay of motion and reaction is not neglected. The resulting hyperbolic model is called reaction random walk system or reaction telegraph equation and is studied in detail by Hadeler and the author (1994-1996) (See the publications below).

Results

Results on local and global existence are given in the framework of operator semigroups. Following Brayton and Miranker (1964) a Lyapunov function is constructed via a variational problem. The existence of a global attractor follows. To find invariant regions a vectorfield is identified which is supposed to point inward. Then a positivity result follows which is important for applications. For linear analysis we gave a condition such that the non compact part of the solution semigroup decays. Then stability of stationary solutions is determined by the eigenvalues of the infinitessimal generator. For an activator-inhibitor Turing model this leads to pattern formation. Hadeler considered travelling wave fronts for reaction random walk systems and for epidemic models with correlated random walk. Moreover he describes the connections of the correlated random walk to the velocity jump process and to the Cattaneo system.

Since the Cattaneo system is closely related to a damped wave equation the introduction of (i) boundary conditions, (ii) reactions and (iii) more then one particle type is straight forward. But here no interplay between motion and reaction is assumed. The construction of a Lyapunov function to find the global attractor is done recently.

To model interactions of particles in combination with the velocity jump process is not as obviuos as in the previous case. One has to distinguish between birth-, death-, and interaction terms. Nonlinear velocity jump models arise, which are hard to analyse mathematically. The qualitative behavior of the solutions is influenced by variuos parameters. First of all the symmetries of the space of velocities and the symmetrie of the kernel are crucial. Boundary conditions have to be introduced, such that the problem is well posed. And the connetction between reaction and motion is a major challenge. Nevertheless some results can be taken from the studies of the Boltzman equation in neutron transport theory (see Schwetlick

Applications

There are several applications of the veocity jump process, e.g. to the motion of myxobacteria where the parameter of changing direction is density dependent (Pfistner (1990)) or to the motion of Dictyostelium discoideum by Othmer e.a. (1996) where a chemotaxis term is added.

The Cattaneo system is used to describe the effect of second sound in heat transport theory (Joseph, Preziosi (1988)).

The whole project is related to my studies in the SFB 382 "Verfahren und Algorithmen zur Simulation Physikalischer Prozesse auf Höchstleistungsrechnern" project C3 "Hyperbolische Systeme als Modelle für Ausbreitung und Reaktionen.

Publications