Editing Galton–Watson process (section)

== Bisexual Galton&ndash;Watson process ==

In the classical family surname Galton&ndash;Watson process described above, only men need to be considered, since only males transmit their family name to descendants.  This effectively means that reproduction can be modeled as asexual.  (Likewise, if mitochondrial transmission is analyzed, only women need to be considered, since only females transmit their mitochondria to descendants.)

A model more closely following actual sexual reproduction is the so-called "bisexual Galton&ndash;Watson process", where only couples reproduce.{{citation needed|date=November 2020}}  (''Bisexual'' in this context refers to the number of sexes involved, not [[sexual orientation]].)  In this process, each child is supposed as male or female, independently of each other, with a specified probability, and a so-called "mating function" determines how many couples will form in a given generation.  As before, reproduction of different couples is considered to be independent of each other.  Now the analogue of the trivial case corresponds to the case of each male and female reproducing in exactly one couple, having one male and one female descendant, and that the mating function takes the value of the minimum of the number of males and females (which are then the same from the next generation onwards).

Since the total reproduction within a generation depends now strongly on the mating function, there exists in general no simple necessary and sufficient condition for final extinction as is the case in the classical Galton&ndash;Watson process.{{citation needed|date=November 2020}}  However, excluding the non-trivial case, the concept of the [[Net reproduction rate|averaged reproduction mean]] (Bruss (1984)) allows for a general sufficient condition for final extinction, treated in the next section.

===Extinction criterion===

If in the non-trivial case the [[Net reproduction rate|averaged reproduction mean]] per couple stays bounded over all generations and will not exceed 1 for a sufficiently large population size, then the probability of final extinction is always&nbsp;1.