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April  2015, 35(4): 1697-1741. doi: 10.3934/dcds.2015.35.1697

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, Austria
2. 

Department of Ecology and Evolution, University of Chicago, 1101 East 57th Street, Chicago, IL 60637

Received  August 2013 Revised  March 2014 Published  November 2014

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 $x>0$.
Keywords: population structure, long-distance migration., subdivided populations, Geographical structure, spatial structure, migration.
Mathematics Subject Classification: Primary: 35K57, 35R09, 92D10, 92D2.
Citation: Linlin Su, Thomas Nagylaki. Clines with directional selection and partial panmixia in an unbounded unidimensional habitat. Discrete & Continuous Dynamical Systems, 2015, 35 (4) : 1697-1741. doi: 10.3934/dcds.2015.35.1697
References:
[1]

B. Charlesworth and D. Charlesworth, Elements of Evolutionary Genetics, Roberts, Greenwood Village, 2010. Google Scholar

[2]

J. A. Endler, Geographic Variation, Speciation, and Clines, Princeton University Press, Princeton, 1977. Google Scholar

[3]

P. C. Fife and L. A. Peletier, Nonlinear diffusion in population genetics, Arch. Rat. Mech. Anal., 64 (1977), 93-109. doi: 10.1007/BF00280092.  Google Scholar

[4]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations Of Second Order, Springer, Berlin, 2001.  Google Scholar

[5]

Y. Lou and T. Nagylaki, A semilinear parabolic system for migration and selection in population genetics, J. Diff. Eqs., 181 (2002), 388-418. doi: 10.1006/jdeq.2001.4086.  Google Scholar

[6]

Y. Lou, T. Nagylaki and W.-M. Ni, An introduction to migration-selection PDE models, Disc. Cont. Dyn. Syst. A, 33 (2013), 4349-4373. doi: 10.3934/dcds.2013.33.4349.  Google Scholar

[7]

Y. Lou, W.-M. Ni and L. Su, An indefinite nonlinear diffusion problem in population genetics, II: Stability and multiplicity, Disc. Cont. Dyn. Syst. A, 27 (2010), 643-655. doi: 10.3934/dcds.2010.27.643.  Google Scholar

[8]

Y. Lou, T. Nagylaki and L. Su, An integro-PDE model from population genetics, J. Diff. Eqs., 254 (2013), 2367-2392. doi: 10.1016/j.jde.2012.12.006.  Google Scholar

[9]

T. Nagylaki, Conditions for the existence of clines, Genetics, 80 (1975), 595-615. Google Scholar

[10]

T. Nagylaki, Clines with variable migration, Genetics, 83 (1976), 867-886. Google Scholar

[11]

T. Nagylaki, Clines with asymmetric migration, Genetics, 88 (1978), 813-827. Google Scholar

[12]

T. Nagylaki, The strong-migration limit in geographically structured populations, J. Math. Biol., 9 (1980), 101-114. doi: 10.1007/BF00275916.  Google Scholar

[13]

T. Nagylaki, Introduction to Theoretical Population Genetics, Biomathematics, 21, Springer, Berlin, 1992. doi: 10.1007/978-3-642-76214-7.  Google Scholar

[14]

T. Nagylaki, Polymorphism in multiallelic migration-selection models with dominance, Theor. Popul. Biol., 75 (2009), 239-259. doi: 10.1016/j.tpb.2009.01.004.  Google Scholar

[15]

T. Nagylaki, The influence of partial panmixia on neutral models of spatial variation, Theor. Popul. Biol., 79 (2011), 19-38. doi: 10.1016/j.tpb.2010.08.006.  Google Scholar

[16]

T. Nagylaki, Clines with partial panmixia, Theor. Popul. Biol., 81 (2012), 45-68. doi: 10.1016/j.tpb.2011.09.006.  Google Scholar

[17]

T. Nagylaki, Clines with partial panmixia in an unbounded unidimensional habitat, Theor. Popul. Biol., 82 (2012), 22-28. doi: 10.1016/j.tpb.2012.02.008.  Google Scholar

[18]

T. Nagylaki and Y. Lou, The dynamics of migration-selection models, in Tutorials in Mathematical Biosciences IV (ed. A. Friedman), Lecture Notes in Math., 1922, Springer, Berlin, (2008), 117-170. doi: 10.1007/978-3-540-74331-6_4.  Google Scholar

[19]

K. Nakashima, W.-M. Ni and L. Su, An indefinite nonlinear diffusion problem in population genetics, I: Existence and limiting profiles, Disc. Cont. Dyn. Syst. A, 27 (2010), 617-641. doi: 10.3934/dcds.2010.27.617.  Google Scholar

[20]

D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1972), 979-1000.  Google Scholar

show all references

References:
[1]

B. Charlesworth and D. Charlesworth, Elements of Evolutionary Genetics, Roberts, Greenwood Village, 2010. Google Scholar

[2]

J. A. Endler, Geographic Variation, Speciation, and Clines, Princeton University Press, Princeton, 1977. Google Scholar

[3]

P. C. Fife and L. A. Peletier, Nonlinear diffusion in population genetics, Arch. Rat. Mech. Anal., 64 (1977), 93-109. doi: 10.1007/BF00280092.  Google Scholar

[4]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations Of Second Order, Springer, Berlin, 2001.  Google Scholar

[5]

Y. Lou and T. Nagylaki, A semilinear parabolic system for migration and selection in population genetics, J. Diff. Eqs., 181 (2002), 388-418. doi: 10.1006/jdeq.2001.4086.  Google Scholar

[6]

Y. Lou, T. Nagylaki and W.-M. Ni, An introduction to migration-selection PDE models, Disc. Cont. Dyn. Syst. A, 33 (2013), 4349-4373. doi: 10.3934/dcds.2013.33.4349.  Google Scholar

[7]

Y. Lou, W.-M. Ni and L. Su, An indefinite nonlinear diffusion problem in population genetics, II: Stability and multiplicity, Disc. Cont. Dyn. Syst. A, 27 (2010), 643-655. doi: 10.3934/dcds.2010.27.643.  Google Scholar

[8]

Y. Lou, T. Nagylaki and L. Su, An integro-PDE model from population genetics, J. Diff. Eqs., 254 (2013), 2367-2392. doi: 10.1016/j.jde.2012.12.006.  Google Scholar

[9]

T. Nagylaki, Conditions for the existence of clines, Genetics, 80 (1975), 595-615. Google Scholar

[10]

T. Nagylaki, Clines with variable migration, Genetics, 83 (1976), 867-886. Google Scholar

[11]

T. Nagylaki, Clines with asymmetric migration, Genetics, 88 (1978), 813-827. Google Scholar

[12]

T. Nagylaki, The strong-migration limit in geographically structured populations, J. Math. Biol., 9 (1980), 101-114. doi: 10.1007/BF00275916.  Google Scholar

[13]

T. Nagylaki, Introduction to Theoretical Population Genetics, Biomathematics, 21, Springer, Berlin, 1992. doi: 10.1007/978-3-642-76214-7.  Google Scholar

[14]

T. Nagylaki, Polymorphism in multiallelic migration-selection models with dominance, Theor. Popul. Biol., 75 (2009), 239-259. doi: 10.1016/j.tpb.2009.01.004.  Google Scholar

[15]

T. Nagylaki, The influence of partial panmixia on neutral models of spatial variation, Theor. Popul. Biol., 79 (2011), 19-38. doi: 10.1016/j.tpb.2010.08.006.  Google Scholar

[16]

T. Nagylaki, Clines with partial panmixia, Theor. Popul. Biol., 81 (2012), 45-68. doi: 10.1016/j.tpb.2011.09.006.  Google Scholar

[17]

T. Nagylaki, Clines with partial panmixia in an unbounded unidimensional habitat, Theor. Popul. Biol., 82 (2012), 22-28. doi: 10.1016/j.tpb.2012.02.008.  Google Scholar

[18]

T. Nagylaki and Y. Lou, The dynamics of migration-selection models, in Tutorials in Mathematical Biosciences IV (ed. A. Friedman), Lecture Notes in Math., 1922, Springer, Berlin, (2008), 117-170. doi: 10.1007/978-3-540-74331-6_4.  Google Scholar

[19]

K. Nakashima, W.-M. Ni and L. Su, An indefinite nonlinear diffusion problem in population genetics, I: Existence and limiting profiles, Disc. Cont. Dyn. Syst. A, 27 (2010), 617-641. doi: 10.3934/dcds.2010.27.617.  Google Scholar

[20]

D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1972), 979-1000.  Google Scholar

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