Stochastic differential equations for evolutionary dynamics with demographic noise and mutations

Area of Science:

• Evolutionary Biology
• Mathematical Biology
• Population Genetics

Background:

• Modeling evolutionary dynamics in finite populations is crucial for understanding biological diversity.
• Stochastic effects and mutations significantly influence evolutionary trajectories.
• Existing models often struggle to incorporate demographic noise and mutations simultaneously in finite populations.

Purpose of the Study:

• To present a general framework for describing evolutionary dynamics of multiple types in finite populations.
• To incorporate demographic noise and mutations within a unified mathematical approach.
• To validate the framework's accuracy against simulation-based results.

Main Methods:

• Development of a general framework based on stochastic differential equations (SDEs).
• Inclusion of demographic noise where population size rescales noise amplitude.
• Incorporation of mutations between types under specific conditions (μ not too small compared to 1/N).

Main Results:

• The framework accurately models evolutionary dynamics in large, finite populations with demographic noise.
• Excellent agreement between the SDE framework and simulation results was demonstrated for a rock-scissors-paper game with mutations.
• The framework's accuracy extends to small population sizes in the absence of mutations.

Conclusions:

• The proposed SDE framework offers a powerful tool for analyzing evolutionary dynamics in finite populations.
• The model effectively handles demographic noise and mutations, providing a more realistic representation of evolutionary processes.
• This approach bridges the gap between theoretical modeling and empirical observations in population genetics.