American Institute of Mathematical Sciences

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June  2020, 40(6): 4019-4037. doi: 10.3934/dcds.2020056

Monotone and nonmonotone clines with partial panmixia across a geographical barrier

Yantao Wang 1,2, and  Linlin Su 2,,

1. 

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
2. 

Department of Mathematics, Southern University of Science and Technology, Shenzhen 518055, China

* Corresponding author: sull@sustech.edu.cn

Received  July 2019 Revised  August 2019 Published  October 2019

The number of clines (i.e., nonconstant equilibria) maintained by viability selection, migration, and partial global panmixia in a step-environment with a geographical barrier is investigated. Our results extend the results of T. Nagylaki (2016, Clines with partial panmixia across a geographical barrier, Theor. Popul. Biol. 109) from the no dominance case to arbitrary dominance and to various other selection functions. Unexpectedly, besides the usual monotone clines, we discover nonmonotone clines with both equal and unequal limits at $ \pm\infty $.

Keywords: Migration, selection, global panmixia, geographical barrier, cline.
Mathematics Subject Classification: 35K57, 92D10, 35A01, 35A02, 35A24.
Citation: Yantao Wang, Linlin Su. Monotone and nonmonotone clines with partial panmixia across a geographical barrier. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 4019-4037. doi: 10.3934/dcds.2020056
References:
[1]

R. Bürger, A survey of migration-selection models in population genetics, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 883-959.  doi: 10.3934/dcdsb.2014.19.883.  Google Scholar

[2]

G. Feltrin and E. Sovrano, Three positive solutions to an indefinite Neumann problem: A shooting method, Nonlinear Anal., 166 (2018), 87-101.  doi: 10.1016/j.na.2017.10.006.  Google Scholar

[3]

G. Feltrin and E. Sovrano, An indefinite nonlinear problem in population dynamics: High multiplicity of positive solutions, Nonlinearity, 31 (2018), 4137-4161.  doi: 10.1088/1361-6544/aac8bb.  Google Scholar

[4]

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

[5]

J. Hofbauer and L. L. Su, Global stability in diallelic migration-selection models, J. Math. Anal. Appl., 428 (2018), 677-695.  doi: 10.1016/j.jmaa.2015.03.034.  Google Scholar

[6]

J. Hofbauer and L. L. Su, Global stability of spatially homogeneous equilibria in migration-selection models, SIAM J. Appl. Math., 76 (2016), 578-597.  doi: 10.1137/15M1027504.  Google Scholar

[7]

F. Li, K. Nakashima and W.-M. Ni, Non-local effects in an integro-PDE model from population genetics, Eur. J. Appl. Math., 28 (2017), 1-41.  doi: 10.1017/S0956792515000601.  Google Scholar

[8]

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

[9]

Y. Lou and T. Nagylaki, Evolution of a semilinear parabolic system for migration and selection in population genetics, J. Differential Equations, 204 (2004), 292-322.  doi: 10.1016/j.jde.2004.01.009.  Google Scholar

[10]

Y. Lou and T. Nagylaki, Evolution of a semilinear parabolic system for migration and selection without dominance, J. Differential Equations, 225 (2006), 624-665.  doi: 10.1016/j.jde.2006.01.012.  Google Scholar

[11]

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

[12]

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

[13]

T. Nagylaki, The diffusion model for migration and selection, Some Mathematical Questions in Biology—Models in Population Biology, Lectures Math. Life Sci., Amer. Math. Soc., Providence, RI, 20 (1989), 55-75.   Google Scholar

[14]

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

[15]

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

[16]

T. Nagylaki, Clines with partial panmixia across a geographical barrier, Theor. Popul. Biol., 109 (2016), 28-43.  doi: 10.1016/j.tpb.2016.01.002.  Google Scholar

[17]

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

[18]

T. Nagylaki, L. L. Su, I. Alevy and T. F. Dupont, Clines with partial panmixia in an environmental pocket, Theor. Popul. Biol., 95 (2014), 24-32.  doi: 10.1016/j.tpb.2014.05.003.  Google Scholar

[19]

T. Nagylaki, L. L. Su and T. F. Dupont, Uniqueness and multiplicity of clines in an environmental pocket, Theor. Popul. Biol., (2019). doi: 10.1016/j.tpb.2019.07.006.  Google Scholar

[20]

T. Nagylaki and K. Zeng, Clines with complete dominance and partial panmixia in an unbounded unidimensional habitat, Theor. Popul. Biol., 93 (2014), 63-74.   Google Scholar

[21]

T. Nagylaki and K. Zeng, Clines with partial panmixia across a geographical barrier in an environmental pocket, Theor. Popul. Biol., 110 (2016), 1-11.  doi: 10.1016/j.tpb.2016.03.003.  Google Scholar

[22]

K. Nakashima, The uniqueness of indefinite nonlinear diffusion problem in population genetics, part Ⅰ, J. Differential Equations, 261 (2016), 6233-6282.  doi: 10.1016/j.jde.2016.08.041.  Google Scholar

[23]

K. Nakashima, The uniqueness of an indefinite nonlinear diffusion problem in population genetics, part Ⅱ, J. Differential Equations, 264 (2018), 1946-1983.  doi: 10.1016/j.jde.2017.10.014.  Google Scholar

[24]

K. Nakashima, Multiple existence of indefinite nonlinear diffusion problem in population genetics, submitted. Google Scholar

[25]

J. Piálek and N. H. Barton, The spread of an advantageous allele across a barrier: The effects of random drift and selection against heterozygotes, Genetics, 145 (1997), 493-504.   Google Scholar

[26]

E. Sovrano, A negative answer to a conjecture arising in the study of selection-migration models in population genetics, J. Math. Biol., 76 (2018), 1655-1672.  doi: 10.1007/s00285-017-1185-7.  Google Scholar

[27]

L. L. Su and T. Nagylaki, Clines with directional selection and partial panmixia in an unbounded unidimensional habitat, Discrete Contin. Dyn. Syst., 35 (2015), 1697-1741.  doi: 10.3934/dcds.2015.35.1697.  Google Scholar

show all references

References:
[1]

R. Bürger, A survey of migration-selection models in population genetics, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 883-959.  doi: 10.3934/dcdsb.2014.19.883.  Google Scholar

[2]

G. Feltrin and E. Sovrano, Three positive solutions to an indefinite Neumann problem: A shooting method, Nonlinear Anal., 166 (2018), 87-101.  doi: 10.1016/j.na.2017.10.006.  Google Scholar

[3]

G. Feltrin and E. Sovrano, An indefinite nonlinear problem in population dynamics: High multiplicity of positive solutions, Nonlinearity, 31 (2018), 4137-4161.  doi: 10.1088/1361-6544/aac8bb.  Google Scholar

[4]

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

[5]

J. Hofbauer and L. L. Su, Global stability in diallelic migration-selection models, J. Math. Anal. Appl., 428 (2018), 677-695.  doi: 10.1016/j.jmaa.2015.03.034.  Google Scholar

[6]

J. Hofbauer and L. L. Su, Global stability of spatially homogeneous equilibria in migration-selection models, SIAM J. Appl. Math., 76 (2016), 578-597.  doi: 10.1137/15M1027504.  Google Scholar

[7]

F. Li, K. Nakashima and W.-M. Ni, Non-local effects in an integro-PDE model from population genetics, Eur. J. Appl. Math., 28 (2017), 1-41.  doi: 10.1017/S0956792515000601.  Google Scholar

[8]

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

[9]

Y. Lou and T. Nagylaki, Evolution of a semilinear parabolic system for migration and selection in population genetics, J. Differential Equations, 204 (2004), 292-322.  doi: 10.1016/j.jde.2004.01.009.  Google Scholar

[10]

Y. Lou and T. Nagylaki, Evolution of a semilinear parabolic system for migration and selection without dominance, J. Differential Equations, 225 (2006), 624-665.  doi: 10.1016/j.jde.2006.01.012.  Google Scholar

[11]

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

[12]

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

[13]

T. Nagylaki, The diffusion model for migration and selection, Some Mathematical Questions in Biology—Models in Population Biology, Lectures Math. Life Sci., Amer. Math. Soc., Providence, RI, 20 (1989), 55-75.   Google Scholar

[14]

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

[15]

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

[16]

T. Nagylaki, Clines with partial panmixia across a geographical barrier, Theor. Popul. Biol., 109 (2016), 28-43.  doi: 10.1016/j.tpb.2016.01.002.  Google Scholar

[17]

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

[18]

T. Nagylaki, L. L. Su, I. Alevy and T. F. Dupont, Clines with partial panmixia in an environmental pocket, Theor. Popul. Biol., 95 (2014), 24-32.  doi: 10.1016/j.tpb.2014.05.003.  Google Scholar

[19]

T. Nagylaki, L. L. Su and T. F. Dupont, Uniqueness and multiplicity of clines in an environmental pocket, Theor. Popul. Biol., (2019). doi: 10.1016/j.tpb.2019.07.006.  Google Scholar

[20]

T. Nagylaki and K. Zeng, Clines with complete dominance and partial panmixia in an unbounded unidimensional habitat, Theor. Popul. Biol., 93 (2014), 63-74.   Google Scholar

[21]

T. Nagylaki and K. Zeng, Clines with partial panmixia across a geographical barrier in an environmental pocket, Theor. Popul. Biol., 110 (2016), 1-11.  doi: 10.1016/j.tpb.2016.03.003.  Google Scholar

[22]

K. Nakashima, The uniqueness of indefinite nonlinear diffusion problem in population genetics, part Ⅰ, J. Differential Equations, 261 (2016), 6233-6282.  doi: 10.1016/j.jde.2016.08.041.  Google Scholar

[23]

K. Nakashima, The uniqueness of an indefinite nonlinear diffusion problem in population genetics, part Ⅱ, J. Differential Equations, 264 (2018), 1946-1983.  doi: 10.1016/j.jde.2017.10.014.  Google Scholar

[24]

K. Nakashima, Multiple existence of indefinite nonlinear diffusion problem in population genetics, submitted. Google Scholar

[25]

J. Piálek and N. H. Barton, The spread of an advantageous allele across a barrier: The effects of random drift and selection against heterozygotes, Genetics, 145 (1997), 493-504.   Google Scholar

[26]

E. Sovrano, A negative answer to a conjecture arising in the study of selection-migration models in population genetics, J. Math. Biol., 76 (2018), 1655-1672.  doi: 10.1007/s00285-017-1185-7.  Google Scholar

[27]

L. L. Su and T. Nagylaki, Clines with directional selection and partial panmixia in an unbounded unidimensional habitat, Discrete Contin. Dyn. Syst., 35 (2015), 1697-1741.  doi: 10.3934/dcds.2015.35.1697.  Google Scholar

Figure 1.   
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Figure 2.   
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Figure 3.   
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Figure 4.  A sketch of the situation in Remark 4(ⅱ); where (A) shows that the straight line $ \beta(p-\bar p) $ is tangent to the graph of $ f(p) $ at $ p_1 $, and (B) shows the corresponding phase portraits of (15)
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Fig. 4 except that the straight line $ \beta(p-\bar p) $ is tangent to the graph of $ f(p) $ at $ p_2 $">Figure 5.  Same as Fig. 4 except that the straight line $ \beta(p-\bar p) $ is tangent to the graph of $ f(p) $ at $ p_2 $
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Fig. 6">Figure 7.  Phase portraits of (15) in Lemma 3.10 with $ I<II $, $ I = II $, and $ I>II $, respectively; where $ I $ and $ II $ stand for the areas enclosed by the graphs of $ L(p) $ and $ f(p) $ as shown in Fig. 6
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Figure 6.  Graph of $ f(p) $ and the straight line $ \beta(p-\bar{p}) $ if $ m = 3 $
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Figure 8.  Superimposition of the phase portrait $ C^- $ of (18) and its image under $ \Phi_{\theta_{-}, \theta_{+ }} $ with the phase portraits of (15)
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Figure 9.  Nonmonotone cline with $ f(u) = 2u^2(1-u) $
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