Clines with directional selection and partial panmixia in an unbounded unidimensional habitat

Linlin Su; Thomas Nagylaki

doi:10.3934/dcds.2015.35.1697

Article Contents

2015, Volume 35, Issue 4: 1697-1741. Doi: 10.3934/dcds.2015.35.1697

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Clines with directional selection and partial panmixia in an unbounded unidimensional habitat

Linlin Su^1, and
Thomas Nagylaki^2,

1.
Department of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna
2.
Department of Ecology and Evolution, University of Chicago, 1101 East 57th Street, Chicago, IL 60637

Received: July 31, 2013

Revised: February 28, 2014

Published: May 2017

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Abstract

In geographically structured populations, global panmixia can be regarded as the limiting case of long-distance migration. The effect of incorporating partial panmixia into diallelic single-locus clines maintained by migration and arbitrary directional selection in an unbounded unidimensional habitat is investigated. The population density is uniform. Migration and selection are both weak; the former is homogeneous and symmetric. Suppose that the spatial factor $g(x)$ in the scaled selection term satisfies $g'(x)\ge 0$ for every $x$ and the limiting values $p_{\pm}=p(\pm\infty)$ of the equilibrium gene frequency $p(x)$ exist and satisfy $0 < p_- < p_+ < 1$. Then (i) $p_- < p(x) < p_+$ for every $x\in\mathbb{R}$; (ii) $p'(x)>0$ for every $x\in\mathbb{R}$; (iii) for each given pair $p_-$ and $p_+$, there exists at most one equilibrium $p(x)$; (iv) the existence and multiplicity of $p(x)$ are determined under various conditions; (v) given two pairs $p_{1\pm}$ and $p_{2\pm}$ such that $p_{1\pm}>p_{2\pm}$, the ordering $p_1(x)>p_2(x)$ holds for every $x\in\mathbb{R}$; (vi) if the factor $f(p)$ $(\ge 0)$ in the scaled selection term is unimodal, as is the case when the selection coefficients do not depend on $p$, and in some other situations, then $p_{1\pm}>p_{2\pm}$; and (vii) in a step-environment that changes sign at $x=0$, under some assumptions on $f(p)$, the equilibria satisfy $p''(x)>0$ if $ x<0$ and $p''(x)<0$ if $x>0$.

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Mathematics Subject Classification: Primary: 35K57, 35R09, 92D10, 92D25.

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