The Feller diffusion conditioned on a single ancestral founder.

Area of Science:

• Population genetics
• Stochastic processes
• Mathematical biology

Background:

• Feller diffusion models describe population dynamics.
• Understanding ancestral origins is key in population genetics.
• Previous models lacked detailed distributional properties.

Purpose of the Study:

• To analyze distributional properties of Feller diffusion conditioned on a single ancestor.
• To determine ancestor distributions and coalescent times.
• To calculate joint distributions of time since the most recent common ancestor and population size.

Main Methods:

• Novel interpretation of Feller's solution using Poisson-distributed families.
• Analysis of ancestral distributions at intermediate times.
• Calculation of joint densities for coalescent times and population size.

Main Results:

• Exact solutions derived for supercritical, critical, and subcritical diffusions.
• Asymptotic forms provided for supercritical diffusions under exponential growth.
• Determined distributions for ancestor counts and coalescent times.

Conclusions:

• The novel approach offers exact solutions for Feller diffusion models.
• Provides insights into population structure and ancestral history.
• Applicable to various population dynamics scenarios, including unbounded growth.