By Haccou P., Jagers P., Vatutin V.A.

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11) Hence, E[Z n ] = mE[Z n−1 ], and repetition yields E[Z n ] = mE[Z n−1 ] = · · · = m n E[Z 0 ] . 12) Discrete-Time Branching Processes 19 Of course, if the initial number Z 0 is fixed and known, we need not use its expectation. Populations with m < 1, m = 1, or m > 1 are called subcritical, critical, and supercritical, respectively. 12), we see that as n → ∞ the average size of a subcritical population goes to zero and that of a critical population is stable, whereas the average size of a supercritical population grows unboundedly.

If seeds are uniformly and independently distributed over the sites (think of the feathered seeds of dandelions), determination of the number of occupied sites is a classic probabilistic problem called the occupancy problem [see Feller (1957), Chistyakov et al. (1978), or, for a textbook exposition, Durrett (1995)]. Indeed, as shown in these references, the exact probability that exactly k of the sites remain empty if X n = x is pk (x, s) = s k s−k (−1) j j =0 s−k j 1− k+ j s x . 69) sx for which see any of the references above.

Thus, if in one season there are x seeds, the population has size z, where s − z is (approximately) Poisson distributed. In particular, the expected population size is s(1 − e−x/s ). When the number of young that can be produced depends on a limited population resource that cannot be mirrored through population size, we may have a more complicated situation. For example, past population sizes may affect current resources. 76) somewhat inadvertently referred to as the total population size in branching process literature.

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