Geometrical models of territory. I. Models for synchronous and asynchronous settlement of territories.

Abstract There is often the case in which it can be assumed that the condition of a habitat of territorial animals is uniform and that the habitat is divided into territories without any gap. If this were the case, the density and the spatial pattern of territories will be mainly affected by the interaction between individuals and by the settling process of territories. This paper presents geometrical models of the establishment of territories, where it is supposed that the repulsive forces act among individuals and that all individuals are identical in strength. In the case of synchronous settlement of territories, individuals will adjust the position of their territorial centres until a stable set of boundaries is obtained. A model of this process is constructed under the assumption that each territory is regarded as a Voronoi polygon. In the case of asynchronous settlement, on the contrary, the adjustment of centres will not occur. This settling process is studied as a problem of random sequential packing of circles with a fixed radius. The models for respective settling processes explain well the existing data of observations.