Author(s):

Bodova, Katarina; Haskovec, Jan; Markowich, Peter

Article Title: 
Well posedness and maximum entropy approximation for the dynamics of quantitative traits

Affiliation 
IST Austria 
Abstract: 
We study the FokkerPlanck equation derived in the large system limit of the Markovian process describing the dynamics of quantitative traits. The FokkerPlanck equation is posed on a bounded domain and its transport and diffusion coefficients vanish on the domain's boundary. We first argue that, despite this degeneracy, the standard noflux boundary condition is valid. We derive the weak formulation of the problem and prove the existence and uniqueness of its solutions by constructing the corresponding contraction semigroup on a suitable function space. Then, we prove that for the parameter regime with high enough mutation rate the problem exhibits a positive spectral gap, which implies exponential convergence to equilibrium.Next, we provide a simple derivation of the socalled Dynamic Maximum Entropy (DynMaxEnt) method for approximation of observables (moments) of the FokkerPlanck solution, which can be interpreted as a nonlinear Galerkin approximation. The limited applicability of the DynMaxEnt method inspires us to introduce its modified version that is valid for the whole range of admissible parameters. Finally, we present several numerical experiments to demonstrate the performance of both the original and modified DynMaxEnt methods. We observe that in the parameter regimes where both methods are valid, the modified one exhibits slightly better approximation properties compared to the original one.

Keywords: 
Boundary condition; Dynamic maximum entropy; FokkerPlanck equation; Quantitative traits; Quasistationary approximation

Journal Title:

Physica D: Nonlinear Phenomena

ISSN:

01672789

Publisher:

Elsevier

Date Published:

20180101

Start Page: 
Epub ahead of print

Sponsor: 
JH and PM are funded by KAUST baseline funds and grant no. 1000000193 .

URL: 

DOI: 
10.1016/j.physd.2017.10.015

Notes: 
We thank Nicholas Barton (IST Austria) for his useful comments and suggestions.

Open access: 
yes (repository) 