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

Abstract

Abstract. We study the Fokker Planck equation derived in the large system limit of the Markovian process describing the dynamics of quantitative traits. The Fokker-Planck 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 no-flux 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 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 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.

Dynamic maximum entropy provides accurate approximation of structured population dynamics

“…Here we present a dynamic maximum entropy (DME) method to approximate stochastic dynamics by a FPE [ 14 – 21 ]. The method is based on a combination of ME in statistical physics [ 1 ], which solves the stationary problem exactly, with a quasi-stationary assumption, as typically used in chemical kinetics [ 22 ] to reduce the number of equations.…”

“…The method is based on a combination of ME in statistical physics [ 1 ], which solves the stationary problem exactly, with a quasi-stationary assumption, as typically used in chemical kinetics [ 22 ] to reduce the number of equations. The method applies to stochastic dynamics with an explicit stationary distribution, even though its application is not limited to such problems (as shown in [ 21 ] a solution ansatz, which is not based on the stationary form can sometimes lead to more accurate approximation).…”

“…The method was further applied to a polygenic trait under stabilizing selection, mutation and drift in [ 27 ] and its limitations in the small-mutation regime were resolved in [ 16 ] by distinguishing the bulk and the boundary layers of the microscopic distributions of allele frequencies. Although accuracy of the method is still not rigorously understood, the authors of [ 21 ] showed an exponential convergence to equilibrium in the FPE by finding a positive spectral gap and investigated an alternative formulation of the DME, which is not exact in the stationary case, but leads to a lower error in the dynamical situation and avoids the problems with small mutation rate.…”

“…Although in most of the previous work the derivation of the DME method was presented phenomenologically, a rigorous derivation of the approximation can be found in [ 21 ]. Here we provide a comprehensive summary of the method.…”

“…The method, although derived from the assumption of quasi-stationarity, remains extremely accurate even far-from-equilibrium. Although numerical evidence of the method’s accuracy has been demonstrated in [ 14 – 16 , 21 ], explicit analysis is a challenging mathematical problem. This is analogous to problems in reaction kinetics, where reduction of dynamics based on the quasi-stationary assumption is often used but its validity has been shown rigorously only for a few basic systems using singular perturbation [ 22 – 24 ].…”