Summary: Discrete-Time Branching Processes 43
Figure 2.9 Alternative representations of a process with sibling dependence.
of its members. The process of these macro-individuals has independently repro-
ducing elements, as different sibships are independent of each other. The numbers
of "macro children" need, however, not be distributed identically (e.g., we might
expect a large sibship to give birth to more children than a small one). Thus the
macro-process is a multi-type branching process, and in the GaltonWatson case
simply has sibship size as its type.
Example 2.10 Imagine a cell population in which if one cell divides, her sister cell cannot,
but dies without trace. Then each individual (= cell) can produce zero or two children, but
not independently. The two possibilities are that one sister cell divides, say with probability
p, or that none does. The population of sibling pairs is again a single-type process, which
produces one child (= pair) with probability p, and none with probability 1 - p. If there
are no dependences beyond that between sisters, the process of sister pairs is a classic
independently reproducing branching process. This is illustrated in Figure 2.9.
2.8 Sexual Reproduction
Up to now we have ignored that in species with sexual reproduction changes in
population size depend on the formation of couples. Sometimes this may be jus-