Population stabilizes when:
Net reproductive rate R0 measures generational replacement. An R0 of 1 means that, on average, each female produces exactly one daughter that survives through the relevant life-table schedule, replacing herself in the next generation. Under unchanging conditions, successive generations therefore maintain the same expected size. Values above 1 cause generational increase, and values below 1 cause decline. This replacement criterion is analogous to λ = 1 and r = 0, though R0, λ, and r refer to different temporal formulations and should not be substituted numerically without generation-time information. Population stabilization does not imply that births and deaths cease; it means gains and losses balance in expectation. Age structure can also cause transient changes even when R0 equals 1, because a population not initially at its stable age distribution may fluctuate before settling. Environmental and demographic stochasticity introduce further variation. Thus R0 = 1 identifies the deterministic threshold between increase and decrease, not a guarantee that every census will contain exactly the same number of organisms.
Ref: Ecology: Concepts and Applications, Molles, 9th Ed., Ch. 11