Indefinite nonlinear diffusion problem in population genetics

Kimie Nakashima

doi:10.3934/dcds.2020169

Article Contents

2020, Volume 40, Issue 6: 3837-3855. Doi: 10.3934/dcds.2020169 open access

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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: March 31, 2019

Revised: December 31, 2019

Published: March 2020

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Abstract

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.

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Mathematics Subject Classification: 35J25, 35B25.

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References

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