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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
| 1. | Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China |
| 2. | Department of Mathematics, Southern University of Science and Technology, Shenzhen 518055, China |
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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 $.
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 and Continuous Dynamical Systems,
2020, 40
(6)
: 4019-4037.
doi: 10.3934/dcds.2020056
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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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [14] |
T. Nagylaki,
Clines with partial panmixia, Theor. Popul. Biol., 81 (2012), 45-68.
doi: 10.1016/j.tpb.2011.09.006. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [24] |
K. Nakashima, Multiple existence of indefinite nonlinear diffusion problem in population genetics, submitted. |
| [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.
|
| [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. |
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [14] |
T. Nagylaki,
Clines with partial panmixia, Theor. Popul. Biol., 81 (2012), 45-68.
doi: 10.1016/j.tpb.2011.09.006. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [24] |
K. Nakashima, Multiple existence of indefinite nonlinear diffusion problem in population genetics, submitted. |
| [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.
|
| [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. |
| [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. |
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)
Figure 8.
Superimposition of the phase portrait $ C^- $ of (18) and its image under $ \Phi_{\theta_{-}, \theta_{+ }} $ with the phase portraits of (15)
Figure 9.
Nonmonotone cline with $ f(u) = 2u^2(1-u) $
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