model III dv ( t ) dt = g$_{3}$( u$_{1}$,...,u$_{N}$,v ) = c$_{v}$$_{1}$ N ∑ i =1 u β i ( t ) ˜ K β v + u β i ( t ) − c$_{v}$$_{2}$v ( t ) N ∑ i ′ =1 ˜ K β v ˜ K β v + u β i ′ ( t ) − v ( t ) (12) N$_{(1)}$ ( N,v ) N = ˜ A ( v ) + ˜ B ( v ) N , (13) ˜ A ( v ) = [ 1+ c v 1 u β (1) ( v ) $_{−}$c v 2 ˜ K β v v c v 2 ˜ K β v v$_{−}$c v 1 u β (2) ( v ) !  ˜ K β v + u β (2) ( v ) ˜ K β v + u β (1) ( v ) ! ] − 1 , (14) ˜ B ( v ) = v c$_{v}$$_{1}$ n u β $_{(1)}$( v ) / ( ˜ K β v + u β $_{(1)}$( v )) − u β $_{(2)}$( v ) / ( ˜ K β v + u β $_{(2)}$( v )) o + c$_{v}$$_{2}$ ˜ K β $_{v}$v n 1 / ( ˜ K β $_{v}$+ u β $_{(2)}$( v )) − 1 / ( ˜ K β v + u β $_{(1)}$( v )) o . (15) The modification to model II is just an addition of the second term in Eq. (12). In other words, each cell in this model also contributes to the degradation of the signal v . As in the previous model, the cellular states fall on stationary states, and the bifur- cation of the stationary state from a single-cluster to two-cluster states are observed with the increase in N (Fig. 8). Here, we first note that the two-cluster state remains stable over a wide range of N . Indeed, non-zero N$_{(1)}$ exists so that v ∗ 1 < v ( N$_{(1)}$,N ) < v ∗ 2 is satisfied even for sufficiently large N . Next, we study the population distribution of two cell types. As shown in Fig. 9, the ratio N$_{(1)}$/N stays at a constant level against the change of N . In the same way as in the previous section, the dependency of N$_{(1)}$ on v and N for a two-cluster state is written as, Here, ˜ B ( v ) > 0 is always satisfied. Because v satisfies v ∗ 1 < v < v ∗ 2 for the existence of a two-cluster state, N$_{(1)}$/N is within the range ( ˜ A ( v ∗ $_{1}$)+ ˜ B ( v ∗ $_{1}$) /N ) < N$_{(1)}$ ( N,v ) N < ( ˜ A ( v ∗ $_{2}$)+ ˜ B ( v ∗ $_{2}$) /N ) for each N . As a result, when N is sufficiently large, the possible range of N$_{(1)}$/N is given 11