English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We study a class of processes that are akin to the Wright-Fisher model, withtransition probabilities weighted in terms of the frequency-dependent fitnessof the population types. By considering an approximate weak formulation of thediscrete problem, we are able to derive a corresponding continuous weakformulation for the probability density. Therefore, we obtain a family ofpartial differential equations (PDE) for the evolution of the probabilitydensity, and which will be an approximation of the discrete process in thejoint large population, small time-steps and weak selection limit. If thefitness functions are sufficiently regular, we can recast the weak formulationin a more standard formulation, without any boundary conditions, butsupplemented by a number of conservation laws. The equations in this family canbe purely diffusive, purely hyperbolic or of convection-diffusion type, withfrequency dependent convection. The particular outcome will depend on theassumed scalings. The diffusive equations are of the degenerate type; using aduality approach, we also obtain a frequency dependent version of the Kimuraequation without any further assumptions. We also show that the convectiveapproximation is related to the replicator dynamics and provide some estimateof how accurate is the convective approximation, with respect to theconvective-diffusion approximation. In particular, we show that the mode, butnot the expected value, of the probability distribution is modelled by thereplicator dynamics. Some numerical simulations that illustrate the results arealso presented.

The frequency-dependent Wright-Fisher model: diffusive and non-diffusive approximations