From its early beginnings onwards mankind has put to test many different society forms, and this fact raises a complex of interesting questions. The objective of this paper is to present a general population model which takes essential features of any society into account and which gives interesting answers on the basis of only two natural hypotheses. One is that societies want to survive, the second one that individuals in a society would in general like to increase their standard of living. We start by presenting a mathematical model which may be seen as a particular type of a controlled branching process. All conditions of the model are justified and interpreted.After several preliminary results about societies in general we can show that two society forms should attract particular attention, both from a qualitative and a quantitative point of view. These are the so-called weakest-first society and the strongest-first society. In particular we prove then that these two societies stand out since they form an envelope of all possible societies in a sense we will make precise. This result (the Envelopment Theorem) is seen as significant because it is paralleled with precise survival criteria for the enveloping societies. Moreover, given that one of the "limiting" societies can be seen as an extreme form of communism, and the other one as being close to an extreme version of capitalism, we conclude that, remarkably, humanity is close to having already tested the limits.
Biology takes a special place among the other natural sciences because biological units, be they pieces of DNA, cells, or organisms, reproduce more or less faithfully. Like any other biological process, reproduction has a large random component. The theory of branching processes was developed especially as a mathematical counterpart to this most fundamental of biological processes. This active and rich research area allows us to determine extinction risks and predict the development of population composition, and also uncover aspects of a population's history from its current genetic composition. Branching processes play an increasingly important role in models of genetics, molecular biology, microbiology, ecology, and evolutionary theory. This book presents this body of mathematical ideas for a biological audience, but should also be enjoyable to mathematicians, if only for its rich stock of rich biological examples. It can be read by anyone with a basic command of calculus, matrix algebra, and probability theory. More advanced results from basic probability theory are treated in a special appendix. Let Zn be a branching process whose offspring distributions vary with n. It is shown that the sequence maxi, 0 P(Zn = i) has a limit. Denote this limit by M. It turns out that M is positive only if the offspring variables rapidly approach constants. Let cn be a sequence of constants and Wn = Zn cn. It will be proven that M = 0 is necessary and sufficient for the limit distribution functions of all convergent Wn to be continuous on (0, o). If M > 0 there is, up to an equivalence, only one sequence cn such that Wn has a limit distribution with jump points in (0, co). Necessary and sufficient conditions for -continuity of limit distributions are derived in terms of the offspring distributions of Zn We focus on supercritical decomposable (reducible) multitype branching processes. Types are partitioned into irreducible equivalence classes. In this context, extinction of some classes is possible without the whole process becoming extinct. We derive criteria for the almost-sure extinction of the whole process, as well as of a specific class, conditionally given the class of the initial particle. We give sufficient conditions under which the extinction of a class implies the extinction of another class or of the whole process. Finally, we show that the extinction probability of a specific class is the minimal nonnegative solution of the usual extinction equation but with added constraints. © Applied Probability Trust 2012. 