A reconsideration of Galton's problem (using a two-sex population)

Area of Science:

• Population Dynamics
• Mathematical Biology
• Stochastic Processes

Background:

• Francis Galton's problem concerns the extinction of surnames among notable men.
• Traditional Galton-Watson branching processes model asexual reproduction.
• Extinction dynamics in two-sex populations require specialized models.

Purpose of the Study:

• To promote the bisexual Galton-Watson branching process as a model for surname extinction.
• To adapt the bisexual process for analyzing Francis Galton's problem.
• To derive and apply a condition for the extinction of male-induced properties in two-sex species.

Main Methods:

• Introduction of a scheme to adapt the bisexual branching process.
• Presentation of a necessary and sufficient condition for male-induced property extinction.
• Development of an approach to calculate extinction probability in bisexual populations.

Main Results:

• A novel condition for the certain extinction of male-induced properties is established.
• An extinction probability calculation method is proposed and applied to surname data.
• Comparison of bisexual model extinction probabilities with traditional asexual models.

Conclusions:

• The bisexual Galton-Watson branching process offers a relevant framework for surname extinction.
• The derived extinction condition and calculation method are applicable to two-sex populations.
• Bisexual reproduction dynamics significantly influence extinction probabilities compared to asexual models.