The kinetics of the signal v in model II is represented as follows, We here adopt Hill-type kinetics for the induction of the signal v by u$_{i}$ , where β is the Hill coefficient, representing the cooperativity in the induction, and K$_{v}$ denotes the threshold value for the signal induction. The parameter c$_{2}$ gives the release rate of v from each cell. Dependence of the stationary states on the total cell number is shown in Fig. 4. For a small N , all the cells always fall on a single-cluster state of u$_{(1)}$ . As N gets larger, the bifurcation to a two-cluster state occurs, where the cells take either u$_{(1)}$ or u$_{(2)}$ . Here, the single-cluster state of u$_{(1)}$ ( u$_{(2)}$ ) is realized only at a small (large) number of cells, respec- tively, so that there is a gap in the total number of cells between the two single-cluster states. The two-cluster state exists within this gap. To understand the observed dependence of the clustering behavior on the cell number, we first consider the stability of a single-cluster state. From du$_{i}$/dt = 0, dv/dt = 0, and u$_{i}$ = u$_{(}$$_{k}$$_{)}$ ( k = 1 , or 2) for i = 1 , . . . , N , we get By solving the above equations self-consistently, the solution curve of u is obtained as a function of the total cell number N (Fig. 5). For N < ˜ N ∗ $_{1}$, a single-cluster state of u$_{(1)}$ is always stable. When the cell number increases beyond ˜ N ∗ $_{1}$, this single-cluster state becomes unstable, while for much larger N such that N > ˜ N ∗ $_{2}$, the single-cluster state becomes stable again, where the cell state is u$_{(2)}$ (Fig. 5) . The threshold ˜ N ∗ 1 and ˜ N ∗ 2 are given by ˜ N ∗ 1 = v ∗ $_{2}$( K β v + u β $_{(1)}$( v ∗ $_{2}$)) / ( c$_{2}$u β $_{(1)}$( v ∗ $_{2}$)) and ˜ N ∗ 2 = v ∗ $_{1}$( K β v + u β $_{(2)}$( v ∗ $_{1}$)) / ( c$_{2}$u β $_{(2)}$( v ∗ $_{1}$)), 8 f ( u$_{(}$$_{k}$$_{)}$, v ) = 0 , v = c$_{2}$N u β ( k ) K β v + u β ( k ) . (8) dv ( t ) dt = g$_{2}$( u$_{1}$, . . . , u$_{N}$, v ) = c$_{2}$ N ∑ i =1 u β i ( t ) K β v + u β i ( t ) − v ( t ) . (7) model II