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cell goes beyond some threshold. Accordingly, the total cell number N is also a time- dependent variable. As the cells share the same medium, they interact with other cells through uptake from the medium and exchange of chemicals with it. 

The state of cell l is expressed by P different metabolites, ⃗x ( l ) = ( x ( l ) 1 , . . . , x ( l ) P ) $^{T}$, and the nutrients are ⃗ X = ( X$_{1}$, . . . , X$_{Q}$ ) $^{T}$, ( Q ≤ P ). The dynamics of the i -th metabolite in cell l is given as follows, 

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A change in the concentration of the i -th nutrient in the medium with the volume V is given by, 

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S$_{i}$ is the external source of the nutrient, D$_{env}$ is the diffusion constant between the nutri- ent reservoir and the medium, and D is that across the cell membrane. Each cell grows through uptake of nutrients and changing them to other metabolites by Eq. (17). As the cells share the same medium, they interact with each other through competition for nutrients. Here we confine our consideration only to the behavior of nutrients { X$_{i}$ } in the stationary states for fixed N , and to obtain the behavior of the stationary states as a function of N . 

Because the stationary state satisfies the condition dx ( l ) i /dt = 0 and dX$_{i}$/dt = 0, 

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From Eq. (19), possible stationary states of each cell, i.e., stationary solutions of ⃗x ( l $^{)}$, are obtained as a function of ⃗ X . Next, we describe how ⃗ X varies with N . As in the previous 

