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The expected number of type j individuals in generation n satisfies d E[Z n j ] = E[E[Z n j |Z n−1 ]] = E[ d Z n−1,h m h j ] = h=1 E[Z n−1,h ]m h j . 26) Discrete-Time Branching Processes 25 If we further introduce the notational convention that the expectation of a random vector is the vector of component expectations, E [Z n ] = (E [Z n1 ] , . . 27) we can summarize these relations for h = 1, 2, . . 28) according to the established conventions of linear algebra. Iteration then leads on to E[Z n ]T = E[Z 0 ]T M n .

46) 12 ⎩ m n −1 , if m = 1 . 3) . 47) After 10 generations, we thus expect around 58 individuals to be infected and more than 113 people to have been hit since the outbreak of the disease. 3 Most bacteria contain plasmids, pieces of extra-chromosomal DNA that can cause resistance against antibiotics. Before cell division, plasmids are doubled, so that Branching Processes: Variation, Growth, and Extinction of Populations 30 there are two copies of the original set. We consider the case in which both daughter cells contain the same number of plasmids at division, but not necessarily the same plasmids.

In real life, however, individuals usually affect each other’s offspring numbers as well as types or states in various ways. Whereas classic branching processes with independent individuals have a long history, only recently has it become possible to treat various types of interaction mathematically. 1 and either assume that generations are non-overlapping or that there is no Discrete-Time Branching Processes 39 aging. First, consider situations in which offspring numbers depend upon population size.

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