The extinction or explosion dichotomy, and the subsequent analysis of the role and character of extinction, left many questions unanswered. What is the form of the uninhibited growth of theoretical populations that do not die out? What determines its speed? Does the composition of the population stabilize, as numbers grow, by some Law-of-Large-Numbers effect? And what happens, in reality, as a result of the inevitable bounds imposed upon any population?

In the long run, we know there is no other way than extinction, but what happens before that event? What temporary stabilities, so-called quasi-stationarities would populations show in the contest between an intrinsic tendency toward expansion and exterior limitations? And what can the theoretical uninhibited growth tell us about the form of populations that develop for a long time in benign stable circumstances, which practically allow unlimited growth?

In more concrete terms, what can the mathematics of supercritical branching processes that do not die out, and of similar stochastic processes, tell us about doubling rates and distributions over ages, phases, cell mass, or DNA content in chemostats or other in vitro populations, or in young, quickly growing in vivo tumors?

Also, what can we conclude about population growth from observed age or body mass distributions in, say, caught fish or hunted moose or deer?

In this chapter we seek to answer such questions. First, we establish the Malthusian, exponential growth that is the mathematical alternative to extinction of freely reproducing populations in stable conditions. Then we proceed to the stabilization of composition (Section 6.2) and the meaning of reproductive value (Section 6.3).