THE DYNAMICS ON THE CELL SPACES WITH $ p $ STATES UNDER THE LOCAL TRANSFORMATIONS SATISFYING THE PRINCIPLE OF LOCAL MAJORITY

Abstract

In order to understand biological phenomena, a new mathematical field was proposed by T. Kitagawa in 1970 and it was named bio mathematics. T. Kitagawa and M. Yamaguchi discussed cell spaces which are one of approaches to establish biomathematics. More than two states are introduced in the cell space. By algebraic methods, in a cell space with four or two states it is proved that a set of all configurations is a vector space and a set of all possible stable configurations is a vector subspace. Thus, in the cell space with four or two states any stable configuration is represented by a superposition of elementary stable configurations. Furthermore, it is proved that in the cell space with two states, there exists one to one correspondence between the set of stable configurations and the set of garden of Eden configurations. The invariant boundary is introduced in the celI space with two states, which generate some variant cell subspaces whose configuration oscillates among definite configurations. The possible positions of this cell subspace are investigated.