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Non-local effects in an integro-PDE model from population genetics

Published online by Cambridge University Press:  20 November 2015

F. LI ,
K. NAKASHIMA  and
W.-M. NI
F. LI
Affiliation:
Center for Partial Differential Equations, East China Normal University, Minhang, Shanghai 200241, People's Republic of China email: fli@cpde.ecnu.edu.cn
K. NAKASHIMA
Affiliation:
Tokyo University of Marine Science and Technology, 4-5-7 Konan, Minato-ku, Tokyo 108-8477, Japan email: nkimie@kaiyodai.ac.jp
W.-M. NI
Affiliation:
Center for Partial Differential Equations, East China Normal University, Minhang, Shanghai 200241, People's Republic of China email: fli@cpde.ecnu.edu.cn School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA email: weiming.ni@gmail.com
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Abstract

In this paper, we study the following non-local problem:

\begin{equation*}\begin{cases}\displaystyle u_t=d{1\over\rho}\nabla\cdot(\rho V\nabla u)+b(\bar{u}-u)+ g(x) u^2(1-u) &\displaystyle \quad \textrm{in} \; \Omega\times (0,\infty),\\[3pt]\displaystyle 0\leq u\leq 1 & \quad\displaystyle \textrm{in}\ \Omega\times (0,\infty),\\[3pt]\displaystyle \nu \cdot V\nabla u=0 &\displaystyle \quad \textrm{on} \; \partial\Omega\times (0,\infty).\vspace*{-2pt}\end{cases}\end{equation*}
This model, proposed by T. Nagylaki, describes the evolution of two alleles under the joint action of selection, migration, and partial panmixia – a non-local term, for the complete dominance case, where g(x) is assumed to change sign at least once to reflect the diversity of the environment. First, properties for general non-local problems are studied. Then, existence of non-trivial steady states, in terms of the diffusion coefficient d and the partial panmixia rate b, is obtained under different signs of the integral ∫Ωg(x)dx. Furthermore, stability and instability properties for non-trivial steady states, as well as the trivial steady states u ≡ 0 and u ≡ 1 are investigated. Our results illustrate how the non-local term – namely, the partial panmixia – helps the migration in this model.

Keywords

partial panmixianon-local effectsnon-trivial steady statesstability

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Type
Papers
Information
European Journal of Applied Mathematics , Volume 28 , Issue 1 , February 2017 , pp. 1 - 41
Copyright
Copyright © Cambridge University Press 2015 

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