University of Tübingen
Lehrstuhl fuer Biomathematik
Auf der Morgenstelle 10
The various ways by which one can describe particles moving in continuous space differ by the state ascribed to the particle itself and then by the dynamical laws. In brownian motion or diffusion the state of the particle is its position in space, the particle has no individual velocity. A particle following a correlated random walk (in 1D) or a Pearson walk (in 2D and higher dimensions) has constant speed, it stops at times governed by a Poisson process and selects a new direction according to the uniform distribution on the sphere. In general transport processes, also the speed changes, and the distributions are non-uniform. Reaction transport equations have been introduced into biological modelling by H.Othmer, W.Alt and have been studied by several authors (e.g. S.Dunbar, T.Hillen). There is a strong formal connection to Boltzmann equations but in these change of velocity is controlled by collisons with other particles.
For some of these processes it can be shown that particle densities are solutions to partial differential equations, e.g. diffusion equations, damped wave equations, transport equations.
If we apply these processes in biological modelling, we have to incorporate nonlinearities and boundary conditions, usually there are several interacting species such as moving amoebae and chemical attractants in aggregation models, and the rates and speeds will depend on the densities of the species. Alignment (like schooling in fishes) can be incorporated by assuming that rates depend on direction.
For many such processes or equations one can show (either formally at the level of equations or by convergence results for solutions) that there is a diffusion limit. This is to say that under certain conditions, in the limit of large speeds and frequent change of direction, the standard reaction diffusion equations appear as limiting cases.
In some modelling problems the diffusion approximation satisfies the purpose, i.e., observed phenomena can be explained in terms of the model. In other cases it is necessary to choose a more refined model, away from the diffusion limit.
It is an interesting question to what extent the behavior of the refined models is qualitatively different from that of the reaction diffusion equation. A good example is the Pearson walk coupled to a standard nonlinearity which has been investigated by Hartmut Schwetlick. In the reaction diffusion case (Fisher's equation) the speed of travelling population fronts is the same in all space dimensions. In the case of the Pearson walk the speed decays markedly with increasing space dimension. Also the stationary solutions of reaction transport equations and of reaction diffusion equations may look rather different. On the other hand global attractors of the correponding dynamical systems have a rather similar structure, (in the scalar case).