Fixation in large populations: a continuous view of a discrete problem

Area of Science:

• Evolutionary biology
• Population genetics
• Mathematical modeling

Background:

• Birth-death processes govern evolutionary dynamics in finite populations.
• Existing models often rely on weak-selection approximations, limiting applicability.
• Understanding fixation probability is crucial for evolutionary outcomes.

Purpose of the Study:

• To derive a continuous approximation for fixation probability beyond the weak-selection limit.
• To analyze evolutionary dynamics in large, finite populations with two types.
• To re-examine classical concepts like evolutionary stable strategies (ESS) and risk dominance in finite populations.

Main Methods:

• Derivation of a continuous approximation for fixation probability using birth-death processes.
• Analysis of three emergent regimes: selection-driven, balanced, and quasi-neutral.
• Asymptotic approximations for evolutionary dynamics with one or two equilibria.
• Continuous restatements of ESSN and risk-dominant strategies for large populations.

Main Results:

• A fixation probability approximation valid beyond the weak-selection limit.
• Identification of selection-driven dynamics that can occur with or without weak selection.
• Demonstration that long-term dynamics in large finite populations can differ significantly from infinite population models, even under weak selection.
• The Hawk and Dove game exhibits a 'one-half law' for fixation patterns.
• Restatements of ESSN and risk-dominant strategies yield new insights, including generalized one-third laws and the concept of critical frequency.

Conclusions:

• The derived continuous approximation provides a more comprehensive understanding of fixation dynamics in large, finite populations.
• Finite population effects can lead to substantial deviations from predictions based on infinite population models.
• The redefinition of classical evolutionary concepts in the context of finite populations offers new analytical tools and insights.