All populations live in  environments where resources are bounded, either depleting them or living sustainably. In the latter case they don't erode the habitat carrying capacity, which thus remains the same. We study populations bounded by a finite constant carrying capacity. The mathematical interpretation of this would be that reproduction becomes subcritical when the population size exceeds the carrying capacity, whereas it is supercritical otherwise, i. e. while there is enough space and food. We shall discuss the life career of such populations, their establishment, growth, persistence, decay – and ultimate extinction.