Discrete-Time Branching Processes 43 Figure 2.9 Alternative representations of a process with sibling dependence. 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 G. Alsmeyer 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- Collections: Mathematics