Most, but certainly not all, population mathematics is deterministic. And on the population level it is usually also in order to disregard all those variations and fluctuations that can never be accounted for -and therefore might be called random. At least this is true for big populations. But one of the purposes of mathematical population dynamics is to serve as a bridge between individual properties and properties of the population as a whole. We may, for example, want to find out what population increase or age distribution should result from certain reproductive behaviours or life spans, or conversely we might like to know what can be inferred about individual, say, cells from more easily made observations on whole populations. This presupposes a theory of population evolution built upon a description of individual life. But individuals vary, and it is preposterous, indeed, to believe that we can formulate a deterministic theory about individual life. So we need a formulation allowing variation between individuals, seemingly in the same conditions, i. e. we need a stochastic description. The most general such models (though still not general enough!) are those furnished by the theory of branching processes. For deterministic theories explicitly built upon the same philosophy cf. Metz and Diekmann, 1986. To say that a realistic stochastic description is necessary, does not say that it is easy. Most stochastic descriptions are as unrealistic as are (the underlying assumptions of) most deterministic models. This is particularly true for the usual Markovian models of population growth, corresponding on the deterministic side to traditional differential equations approaches.
Indeed, consider individuals living and reproducing independently, according to the same probability law. If population size in itself is taken to be Markovian in real time, so that future size is influenced only by the present population size, then the age distribution must be irrelevant. This, in its turn, implies that life spans must be exponentially distributed and child bearing occur as an age-homogeneous Poisson process during life, possibly augmented by splitting at death. Hence, individuals do not age, a property that does not seem very This work has been supported by the Swedish Natural Science Research Council.