Probability of fixation under weak selection: a branching process unifying approach

Area of Science:

• Population genetics
• Mathematical biology
• Evolutionary dynamics

Background:

• Classical population genetics models like Wright-Fisher, Haldane, and Fisher provide frameworks for understanding allele frequency changes.
• Continuous-state branching processes offer a flexible mathematical tool for modeling population dynamics.
• Density-dependence, a key ecological factor, influences population growth and stability.

Purpose of the Study:

• To unify and extend existing population genetics models using continuous-state branching processes.
• To analyze the fixation probability of mutant alleles under varying demographic traits (growth rate, reproduction variance, competition sensitivity).
• To investigate the impact of density-dependence on fixation probabilities in ecological contexts.

Main Methods:

• Utilizing continuous-state branching processes (independent and constant sum variants).
• Incorporating logistic density-dependence into population models.
• Deriving a master formula for fixation probability under weak selection.
• Analyzing invasibility coefficients related to demographic traits.

Main Results:

• Established a unified framework linking Haldane-Fisher models with Kimura's diffusion approximations via continuous-state branching processes.
• Demonstrated that increased reproduction variance (sigma) is always deleterious for mutant fixation.
• Derived specific formulas for invasibility coefficients (g(r), g(sigma), g(c)) across different models and population sizes.
• Showed that density-dependence in Model III alters fixation probabilities, leading to underestimation in growing populations and overestimation in declining ones compared to classical models.

Conclusions:

• The study provides a generalized framework for analyzing allele fixation in diverse population models.
• Reproduction variance is a critical factor that negatively impacts fixation probability.
• Density-dependence significantly modifies fixation dynamics, highlighting the limitations of simpler models in ecological settings.