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Taming the diffusion approximation through a controlling-factor WKB method.

Jayant Pande1, Nadav M Shnerb1

  • 1Department of Physics, Bar-Ilan University, Ramat-Gan IL52900, Israel.

Physical Review. E
|January 20, 2021
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Summary
This summary is machine-generated.

The diffusion approximation (DA) for population dynamics fails when selection direction changes. A new WKB method offers a controlled, more accurate DA for wider applications, including fixation probability calculations.

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Area of Science:

  • Population dynamics
  • Evolutionary biology
  • Mathematical biology

Background:

  • The diffusion approximation (DA) is a standard tool for analyzing stochastic population dynamics in fields like genetics and ecology.
  • DA relies on smoothness assumptions that fail when selection direction reverses, limiting its applicability.
  • This limitation is particularly problematic in evolutionary scenarios with fluctuating selection pressures.

Purpose of the Study:

  • To develop a controlled and more accurate approximation for stochastic population dynamics.
  • To extend the applicability of diffusion approximations to regimes where standard methods fail.
  • To introduce a scalable numerical method for analyzing complex population dynamics.

Main Methods:

  • Employed the Wentzel-Kramers-Brillouin (WKB) large-deviations method, which requires smoothness of the logarithm of quantities.
  • Combined WKB with asymptotic matching techniques for controlled derivation of improved approximations.
  • Developed a scalable, population-size-independent WKB-based numerical technique.

Main Results:

  • Demonstrated how to derive the diffusion approximation in a controlled manner using WKB.
  • Showcased the ability to produce improved approximations valid over wider parameter ranges.
  • Successfully applied the WKB method to calculate fixation probabilities in two-type competition models.

Conclusions:

  • The WKB method provides a robust alternative to standard diffusion approximations for population dynamics.
  • This approach enhances accuracy and extends applicability, especially in scenarios with changing selection.
  • The developed numerical technique offers a scalable solution for complex evolutionary and ecological modeling.