# American Institute of Mathematical Sciences

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

 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 $x>0$.
Citation: Linlin Su, Thomas Nagylaki. Clines with directional selection and partial panmixia in an unbounded unidimensional habitat. Discrete and 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. [2] J. A. Endler, Geographic Variation, Speciation, and Clines, Princeton University Press, Princeton, 1977. [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. [4] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations Of Second Order, Springer, Berlin, 2001. [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. [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. [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. [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. [9] T. Nagylaki, Conditions for the existence of clines, Genetics, 80 (1975), 595-615. [10] T. Nagylaki, Clines with variable migration, Genetics, 83 (1976), 867-886. [11] T. Nagylaki, Clines with asymmetric migration, Genetics, 88 (1978), 813-827. [12] T. Nagylaki, The strong-migration limit in geographically structured populations, J. Math. Biol., 9 (1980), 101-114. doi: 10.1007/BF00275916. [13] T. Nagylaki, Introduction to Theoretical Population Genetics, Biomathematics, 21, Springer, Berlin, 1992. doi: 10.1007/978-3-642-76214-7. [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. [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. [16] T. Nagylaki, Clines with partial panmixia, Theor. Popul. Biol., 81 (2012), 45-68. doi: 10.1016/j.tpb.2011.09.006. [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. [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. [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. [20] D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1972), 979-1000.

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##### References:
 [1] B. Charlesworth and D. Charlesworth, Elements of Evolutionary Genetics, Roberts, Greenwood Village, 2010. [2] J. A. Endler, Geographic Variation, Speciation, and Clines, Princeton University Press, Princeton, 1977. [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. [4] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations Of Second Order, Springer, Berlin, 2001. [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. [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. [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. [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. [9] T. Nagylaki, Conditions for the existence of clines, Genetics, 80 (1975), 595-615. [10] T. Nagylaki, Clines with variable migration, Genetics, 83 (1976), 867-886. [11] T. Nagylaki, Clines with asymmetric migration, Genetics, 88 (1978), 813-827. [12] T. Nagylaki, The strong-migration limit in geographically structured populations, J. Math. Biol., 9 (1980), 101-114. doi: 10.1007/BF00275916. [13] T. Nagylaki, Introduction to Theoretical Population Genetics, Biomathematics, 21, Springer, Berlin, 1992. doi: 10.1007/978-3-642-76214-7. [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. [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. [16] T. Nagylaki, Clines with partial panmixia, Theor. Popul. Biol., 81 (2012), 45-68. doi: 10.1016/j.tpb.2011.09.006. [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. [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. [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. [20] D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1972), 979-1000.
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