ABSTRACT
Prom observations and experiment it is known that the dynamics of de velopment of their community - that is, of a sufficiently large population of amoebas in small distance from each other - can occasionally be quite complex. For example, depending on external conditions amoebas gather in huge (up to hundreds of thousands) clusters, which start to move as a single unit, though the individuality of each amoeba is conserved. It is noticed that this macroscopic “organized” motion occurs towards a higher concentration
of some chemical substance, developed by the amoebas. The mathematical model of dynamics of a cluster is based on the following assumptions:
1) the distance between amoebas is small as compared with the sizes of the clusters (hundreds of microns), the latter can be considered as a “continuous medium” and one can introduce a concentration N(x, y , 2, t) - the number of amoebas in unit volume;
2) the process is one-dimensional, i.e. the concentration of amoebas and other quantities are functions only of coordinate x and time t ;
3) amoebas are not born and do not die in a process of macroscopic motion, i.e. the characteristic time of motion (several hours) is small relative to the characteristic times of multiplication and life duration of amoebas;
4) the individual motion of amoebas in the absence of stimulating exter nal influences (food, heat, etc.) is random, chaotic; there are no preferred directions and each amoeba with equal probability can move both to the right and to the left;
5) if there is an “attracting” chemical substance in the medium, then to the own disordered motion of amoebas a directed movement towards the area of higher density of this substance is added.
