You could say:
The resident and transient populations both remain constant over time.
The resident and transient populations each remain constant over time.
The word each can be used with several items but both with only two:
Monday, Tuesday, and Friday are each fully booked but my calendar is open both Wednesday and Thursday.
P.S. In English, "the population remains constant" and "the population remains stable", unless modified in some way, refer to the size of the population. But one could always add "in size", though it isn't necessary:
The resident and transient populations each remain constant in size over time.
The resident and transient populations each remain stable over time.
The demographic characteristics of a population are the basic determinants of how the population changes over time. If birth and death rates are equal, the population remains stable. The population will increase if birth rates exceed death rates, but will decrease if birth rates are lower than death rates. Life expectancy, another important factor, is the length of time individuals remain in the population. It is impacted by local resources, reproduction, and the overall health of the population. These demographic characteristics are often displayed in the form of a life table.
The Study of Population Dynamics