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June  2020, 40(6): 3837-3855. doi: 10.3934/dcds.2020169

Indefinite nonlinear diffusion problem in population genetics

Kimie Nakashima

Tokyo University of Marine Science and Technology, 4-5-7 Kounan, Minato-ku, Tokyo, 108-8477, Japan

Received  April 2019 Revised  January 2020 Published  March 2020

We study the following Neumann problem in one dimension,
$ \left\{ {\begin{array}{*{20}{l}}\begin{array}{l}{u_t} = du'' + g(x){u^2}(1 - u)\quad {\rm{in}}\quad (0,1) \times (0,\infty ),\;\\0 \le u \le 1\quad {\rm{in}}\quad (0,1) \times (0,\infty ),\;\\u'(0,t) = u'(1,t) = 0\quad {\rm{in}}\quad (0,\infty ),\end{array}\end{array}} \right.$
where
$ g $
 changes sign in
$ (0, 1) $
. This equation models the "complete dominance" case in population genetics of two alleles. It is known that this equation has a nontrivial stable steady state
$ U_d $
 for
$ d $
 sufficiently small. We show that
$ U_d $
 is a unique nontrivial steady state under a condition
$ \int_{0}^1\, g(x)\, dx\geq 0 $
 and some other additional condition.
Keywords: Reaction diffusion equation, singular perturbation, layers.
Mathematics Subject Classification: 35J25, 35B25.
Citation: Kimie Nakashima. Indefinite nonlinear diffusion problem in population genetics. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3837-3855. doi: 10.3934/dcds.2020169
References:
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

K. Nakashima, Multiple existence of indefinite nonlinear diffusion problem in population genetics, J. Differential Equations, work in progress. doi: 10.1016/j.jde.2019.11.082.  Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

K. Nakashima, Multiple existence of indefinite nonlinear diffusion problem in population genetics, J. Differential Equations, work in progress. doi: 10.1016/j.jde.2019.11.082.  Google Scholar

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