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/

Zendy Plus

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

Zendy Tools

Zendy Open

We study the Fokker-Planck equation derived in the large system limit of theMarkovian process describing the dynamics of quantitative traits. TheFokker-Planck equation is posed on a bounded domain and its transport anddiffusion coefficients vanish on the domain's boundary. We first argue that,despite this degeneracy, the standard no-flux boundary condition is valid. Wederive the weak formulation of the problem and prove the existence anduniqueness of its solutions by constructing the corresponding contractionsemigroup on a suitable function space. Then, we prove that for the parameterregime with high enough mutation rate the problem exhibits a positive spectralgap, which implies exponential convergence to equilibrium.  Next, we provide a simple derivation of the so-called Dynamic Maximum Entropy(DynMaxEnt) method for approximation of moments of the Fokker-Planck solution,which can be interpreted as a nonlinear Galerkin approximation. The limitedapplicability of the DynMaxEnt method inspires us to introduce its modifiedversion that is valid for the whole range of admissible parameters. Finally, wepresent several numerical experiments to demonstrate the performance of boththe original and modified DynMaxEnt methods. We observe that in the parameterregimes where both methods are valid, the modified one exhibits slightly betterapproximation properties compared to the original one.

Well posedness and Maximum Entropy Approximation for the Dynamics of Quantitative Traits