As far as I know, cells that are actively dividing (like immortalized cancer cell lines) will not reach a stable state when they occupy the whole space available for growth (similar to the rabbit population). This is unless some sort of genetic engineering techniques are used to stop or slow down the cell division rate.

For other cells with little or no cell division activities, they usually have a limited lifespan (not immortalized like cancer cell lines) and might die before they manage to fill the whole space. Whether they will reach a stable state or not, I'm not sure as I only worked with immortalized cells for my MSc.