The Markov Structure of Population Growth

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.