Aggregation Induced by Diffusing and Nondiffusing Media

Abstract

Gathering of individuals is a widespread phenomenon in biology. Reasons are, for instance, to give each other shelter, to reproduce, to explore new regions, to feed or to endure starvation conditions. In the case of myxobacteria (Dworkin and Kaiser 1993) it is known that they glide cooperatively and aggregate under starvation conditions. During gliding they prefer to use paths which were laid down by themselves. When the nal aggregation takes place they glide in streams towards developing mounds which later grow to form so-called fruiting bodies. Other examples of collective aggregation are known from larvae, e.g. of the bark beetle Dendroctonus micans (Deneubourg et. al 1990) which clumps to feeding groups. Group feeding often improves individual survival, or allows better exploitation of food resources (Tsubaki 1981, Tsubaki and Shiotsu 1982). In many cases, the aggregation process occurs via exchange of chemical signals between the entities. Chemotaxis is one of the major communication mechanisms, and has been found, for instance, in the aggregation of cells, like human leucocytes (granulocytes) (Gruler and Bois eury-Chevance 1994; Tranquillo and Alt 1994) and for the slime mold amoebae (Keller and Segel 1970). In order to discuss the dynamic process of aggregation, we rst introduce a discrete model suitable for lattice-based computer simulations, where particles interact by changing the surface they move on, locally. This model can be applied either to the formation of trails or to the formation of aggregates, as found for myxobacteria and insect larvae. The discrete model is approximated by a PDE-system, a so-called chemotaxis system. We discuss critical parameters of the aggregation process, such as initial population density, production rate of the chemotactic substance, and the presence or lack of di usion of this substance, in dependence on the chemotactic sensitivity.