that A ( v ) > 0 and B ( v ) > 0. For the existence of a two-cluster state, v has to satisfy v ∗ 1 < v < v ∗ $_{2}$, that is, N$_{(1)}$ ( N,v ∗ $_{1}$) < N$_{(1)}$ ( N,v ) < N$_{(1)}$ ( N,v ∗ $_{2}$) for each N . By inserting Eq. (9) into this expression, it is shown that N$_{(1)}$ ( N,v ∗ $_{1}$) and N$_{(1)}$ ( N,v ∗ $_{2}$), i.e., the lower and upper bounds of N$_{(1)}$ , decay linearly with N , with the slope of A ( v ∗ $_{1}$) and A ( v ∗ $_{2}$). In fact, a linear decrease in N$_{(1)}$ with the increase in N is clearly discernible in Fig. 6. 

Next, we evaluate the value of the slope A ( v ). Eq. (9) is written as A ( v ) = { ( u$_{(2)}$/K$_{v}$ ) $^{β}$+ ( u$_{(2)}$/u$_{(1)}$ ) $^{β}$} / { 1 − ( u$_{(2)}$/u$_{(1)}$ ) $^{β}$} . If u$_{(2)}$ ≪ u$_{(1)}$ and u$_{(2)}$ ≪ K$_{v}$ are satisfied, that is the case for the parameters used in Fig. 6, A ( v ) is much smaller than unity. As a result, the decrease in N$_{(1)}$ with N is slow, and N$_{(1)}$ is sustained at a same level over a wide range of N , satisfying N$_{(1)}$ ( v ∗ $_{1}$) < N$_{(1)}$ ( v ) < N$_{(1)}$ ( v ∗ $_{2}$) (Fig. 6). 

By increasing the Hill-coefficient β , A ( v ) becomes much smaller than unity which asymptotically go to zero, even if the value of u$_{(2)}$ is the same level as u$_{(1)}$ or K$_{v}$ as is shown in Fig. 7. Note that the conditions u$_{(2)}$ < u$_{(1)}$ and u$_{(2)}$ < K$_{v}$ have to be satisfied. The value of the slope A ( v ) shows an exponential decrease with β . Hence, N$_{(1)}$ is sustained at an almost constant level and the population size regulation of cell type 1 is realized with a sufficiently large β . 

## 5 Model III: Proportion Preservation of Two Cell Types 

For precise body plan or for tissue homeostasis, proportion regulation of the number of each cell type is required. The fraction of each cell type has to be sustained at a certain range, against the change in the total number of cells. Here we modify the kinetics of v in the previous model II to seek for the possibility of the proportion regulation. With this modification, we will show that the population fraction of the two types of cells is kept at a certain level against the change of N . 

Here, the kinetics of v is modified as follows, 

