SCALING LIMITS OF A MODEL FOR SELECTION AT TWO SCALES

Area of Science:

• Evolutionary Biology
• Mathematical Biology
• Population Genetics

Background:

• Population dynamics are central to evolutionary biology, especially when selection acts oppositely at different scales.
• Viral evolution exemplifies this: fast replication within hosts can be countered by reduced transmission due to host morbidity between hosts.

Purpose of the Study:

• To model and analyze population dynamics under opposing selective pressures at within-host and between-host scales.
• To investigate the mathematical consequences of these dual selective forces using stochastic processes.

Main Methods:

• A stochastic ball-and-urn process was employed to model the population dynamics.
• Weak convergence of the process was proven under two distinct scaling regimes.
• Analysis involved a deterministic nonlinear integro-partial differential equation and a measure-valued Fleming-Viot process.

Main Results:

• The first scaling yielded a nonlinear integro-partial differential equation where fixed points are Beta distributions.
• The stability of these Beta distributions depends on a parameter (λ) and initial data characteristics.
• The second scaling resulted in a Fleming-Viot process, a stochastic process relevant to population genetics.

Conclusions:

• The study provides a mathematical framework for understanding evolutionary dynamics with multi-scale selection.
• The derived mathematical models (integro-partial differential equation and Fleming-Viot process) offer insights into population genetics.
• The stability analysis of Beta distributions highlights key factors influencing population evolution under these conditions.