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 & Continuous Dynamical Systems - A, 2015, 35 (4) : 1697-1741. doi: 10.3934/dcds.2015.35.1697
References:
[1]

B. Charlesworth and D. Charlesworth, Elements of Evolutionary Genetics,, Roberts, (2010). Google Scholar

[2]

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

[3]

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

[4]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations Of Second Order,, Springer, (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. 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. 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. 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. doi: 10.1016/j.jde.2012.12.006. Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

T. Nagylaki, Introduction to Theoretical Population Genetics,, Biomathematics, 21 (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. 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. doi: 10.1016/j.tpb.2010.08.006. Google Scholar

[16]

T. Nagylaki, Clines with partial panmixia,, Theor. Popul. Biol., 81 (2012), 45. 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. 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), 1922 (2008), 117. 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. 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. Google Scholar

show all references

References:
[1]

B. Charlesworth and D. Charlesworth, Elements of Evolutionary Genetics,, Roberts, (2010). Google Scholar

[2]

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

[3]

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

[4]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations Of Second Order,, Springer, (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. 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. 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. 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. doi: 10.1016/j.jde.2012.12.006. Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

T. Nagylaki, Introduction to Theoretical Population Genetics,, Biomathematics, 21 (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. 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. doi: 10.1016/j.tpb.2010.08.006. Google Scholar

[16]

T. Nagylaki, Clines with partial panmixia,, Theor. Popul. Biol., 81 (2012), 45. 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. 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), 1922 (2008), 117. 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. 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. Google Scholar

[1]

Fabio Augusto Milner, Ruijun Zhao. A deterministic model of schistosomiasis with spatial structure. Mathematical Biosciences & Engineering, 2008, 5 (3) : 505-522. doi: 10.3934/mbe.2008.5.505

[2]

Alessandro Bertuzzi, Alberto d'Onofrio, Antonio Fasano, Alberto Gandolfi. Modelling cell populations with spatial structure: Steady state and treatment-induced evolution. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 161-186. doi: 10.3934/dcdsb.2004.4.161

[3]

Hélène Leman, Sylvie Méléard, Sepideh Mirrahimi. Influence of a spatial structure on the long time behavior of a competitive Lotka-Volterra type system. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 469-493. doi: 10.3934/dcdsb.2015.20.469

[4]

Cui-Ping Cheng, Wan-Tong Li, Zhi-Cheng Wang. Asymptotic stability of traveling wavefronts in a delayed population model with stage structure on a two-dimensional spatial lattice. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 559-575. doi: 10.3934/dcdsb.2010.13.559

[5]

Ihab Haidar, Alain Rapaport, Frédéric Gérard. Effects of spatial structure and diffusion on the performances of the chemostat. Mathematical Biosciences & Engineering, 2011, 8 (4) : 953-971. doi: 10.3934/mbe.2011.8.953

[6]

Sepideh Mirrahimi. Adaptation and migration of a population between patches. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 753-768. doi: 10.3934/dcdsb.2013.18.753

[7]

Natalia L. Komarova. Spatial stochastic models of cancer: Fitness, migration, invasion. Mathematical Biosciences & Engineering, 2013, 10 (3) : 761-775. doi: 10.3934/mbe.2013.10.761

[8]

Liang Zhang, Zhi-Cheng Wang. Spatial dynamics of a diffusive predator-prey model with stage structure. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1831-1853. doi: 10.3934/dcdsb.2015.20.1831

[9]

Reinhard Bürger. A survey of migration-selection models in population genetics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 883-959. doi: 10.3934/dcdsb.2014.19.883

[10]

Grégoire Allaire, Alessandro Ferriero. Homogenization and long time asymptotic of a fluid-structure interaction problem. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 199-220. doi: 10.3934/dcdsb.2008.9.199

[11]

Rong Liu, Feng-Qin Zhang, Yuming Chen. Optimal contraception control for a nonlinear population model with size structure and a separable mortality. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3603-3618. doi: 10.3934/dcdsb.2016112

[12]

Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051

[13]

Anthony Tongen, María Zubillaga, Jorge E. Rabinovich. A two-sex matrix population model to represent harem structure. Mathematical Biosciences & Engineering, 2016, 13 (5) : 1077-1092. doi: 10.3934/mbe.2016031

[14]

Yueding Yuan, Zhiming Guo, Moxun Tang. A nonlocal diffusion population model with age structure and Dirichlet boundary condition. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2095-2115. doi: 10.3934/cpaa.2015.14.2095

[15]

Hongbin Guo, Michael Yi Li. Impacts of migration and immigration on disease transmission dynamics in heterogeneous populations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2413-2430. doi: 10.3934/dcdsb.2012.17.2413

[16]

Jianjun Tian, Bai-Lian Li. Coalgebraic Structure of Genetic Inheritance. Mathematical Biosciences & Engineering, 2004, 1 (2) : 243-266. doi: 10.3934/mbe.2004.1.243

[17]

V. Balaji, I. Biswas and D. S. Nagaraj. Principal bundles with parabolic structure. Electronic Research Announcements, 2001, 7: 37-44.

[18]

Toshikazu Kuniya, Yoshiaki Muroya. Global stability of a multi-group SIS epidemic model for population migration. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1105-1118. doi: 10.3934/dcdsb.2014.19.1105

[19]

Shangbin Cui, Meng Bai. Mathematical analysis of population migration and its effects to spread of epidemics. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2819-2858. doi: 10.3934/dcdsb.2015.20.2819

[20]

Shuling Yan, Shangjiang Guo. Dynamics of a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1559-1579. doi: 10.3934/dcdsb.2018059

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (19)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]