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Updated: Feb 23, 2026

Modeling the Size Spectrum for Macroinvertebrates and Fishes in Stream Ecosystems
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SCALING LIMITS OF A MODEL FOR SELECTION AT TWO SCALES.

Shishi Luo1, Jonathan C Mattingly2

  • 1Computer Science Division and Department of Statistics, University of California-Berkeley, Berkeley, CA 94720, United States of America.

Nonlinearity
|September 5, 2017
PubMed
Summary
This summary is machine-generated.

Population dynamics reveal opposing selection pressures within and between hosts. Mathematical models show how these conflicting forces, like viral replication rates, shape evolutionary trajectories and population genetics.

Keywords:
Fleming–Viot processMarkov chainsevolutionary dynamicslimiting behaviorscaling limits

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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.