Stationary gene frequency distribution in the environment fluctuating between two distinct states

Area of Science:

• Population Genetics
• Theoretical Biology
• Mathematical Ecology

Background:

• Stochastic dynamical systems are crucial for modeling biological processes.
• Environmental fluctuations significantly impact population genetics models.
• Existing models often lack exact solutions for complex environmental dynamics.

Purpose of the Study:

• To develop a general method for finding stationary distributions in stochastic systems.
• To derive the exact stationary gene frequency distribution for a fluctuating environment model.
• To compare derived distributions with existing diffusion equation models.

Main Methods:

• Developed a general method for stationary distribution calculation.
• Applied the method to a genic selection model with a two-state Markovian environment.
• Derived limiting stationary distributions and compared them to diffusion equations.

Main Results:

• An exact stationary gene frequency distribution was obtained for the fluctuating environment model.
• Several limiting stationary distributions were derived.
• One derived distribution matched a heuristically derived diffusion equation solution.

Conclusions:

• The presented method offers a robust approach for analyzing stochastic genetic models.
• The findings provide a theoretical basis for understanding gene frequency dynamics in fluctuating environments.
• This work clarifies relationships between discrete models and continuous diffusion approximations.