An integro-PDE model from population genetics
type: Item
instance of: scholarly article
title: An integro-PDE model from population genetics
zbMATH Open document ID: 1264.35047
author: Yuan Lou
author: Thomas Nagylaki
author: Linlin Su
published in: Journal of Differential Equations
publication date: 2013-03-04T00:00:00Z
review text: This paper is concerned with an integro-partial differential equation that models the evolution of the frequencies for two alleles at a single locus under the joint action of migration, selection, and partial panmixia. Let \(\Omega\) be a bounded region in \(\mathbb{R}^N\), let \(P(x,t)\) and \(1-P(x,t)\) be the frequencies of the two alleles at location \(x\) and generation \(t\), let \(V(x)\) be one half of the \(N\times N\) symmetric, positive definite, single generation covariance matrix of local migration, and let the vector \(M(x)\) represent the single generation mean displacement by local migration. Define a vector \(b\) by \(b(x)=\rho^{-1}\nabla\cdot (\rho V)-M\). Normalize the population \(\rho(x)\) by \(\int_{\Omega}\rho(x)dx=1\) and define the average of a function \(u(x,t)\) by \(\bar u(t)=\int_{\Omega}\rho(x)u(x,t)dx\). Then the equation can be written as \[ \begin{cases} P_t=\sum_{i,j=1}^N V_{ij}P_{x_ix_j}+b\cdot \nabla P+B(\bar P-P)+\lambda g(x)f(P) & \text{in } \Omega\times (0,\infty) \\ \nu \cdot V\nabla P=0 & \text{on } \partial \Omega\times (0,\infty) \end{cases} \] under the initial condition \(0\leq P(x,0)\leq 0\) in \(\Omega\), \(0<\bar P(0)<1\). Here \(B\) is a positive constant that represents the scaled rate of global panmixia, \(\lambda\) denotes the selection intensity, the functions \(g(x)\) and \(f(P)\) denote the spatial and frequency dependence of selection respectively, and \(\nu\) is the out-normal vector along the boundary \(\partial \Omega\). One of the main results obtained by the authors is the existence, uniqueness and global asymptotic stability of the nontrivial equilibrium, which implies the persistence of both alleles. Parameter dependence for the maintenance of the alleles is also studied. This work extends results in \textit{T. Nagylaki} [``Clines with partial panmixia'', Theor. Popul. Biol. 81, 45--68 (2012)] on conservative migration to arbitrary migration.
reviewed by: Yaping Liu
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35R09
Mathematics Subject Classification ID: 92D10
Mathematics Subject Classification ID: 92D25
zbMATH DE Number: 6140822
zbMATH Keywords: nonlocal reaction
zbMATH Keywords: long-distance migration
zbMATH Keywords: stability of nontrivial equilibrium
zbMATH Keywords: persistence of both alleles
MaRDI profile type: Publication
full work available at URL: j.jde.2012.12.006
OpenAlex ID: W2067085894
DOI: 10.1016/J.JDE.2012.12.006
description: scientific article; zbMATH DE number 6140822