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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: [email protected]
K. NAKASHIMA
Affiliation:
Tokyo University of Marine Science and Technology, 4-5-7 Konan, Minato-ku, Tokyo 108-8477, Japan email: [email protected]
W.-M. NI
Affiliation:
Center for Partial Differential Equations, East China Normal University, Minhang, Shanghai 200241, People's Republic of China email: [email protected] School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA email: [email protected]
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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
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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References

[1] Brown, K. J. & Hess, P. (1990) Stability and uniqueness of positive solutions for a semi-linear elliptic boundary value problem. Differ. Integral Equ. 3 (2), 201–207.Google Scholar
[2] Coville, J. (2006) On uniqueness and monotonicity of solutions of non-local reaction diffusion equation. Ann. Mat. Pura Appl. (4) 185 (3), 461–485.Google Scholar
[3] Fleming, W. H. (1975) A selection-migration model in population genetics. J. Math. Biol. 2 (3), 219–233.Google Scholar
[4] Friedman, A. (1964) Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J. Google Scholar
[5] Henry, D. (1981) Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Vol. 840, Springer-Verlag, Berlin-New York.CrossRefGoogle Scholar
[6] Hutson, V., Shen, W. & Vickers, G. T. (2008) Spectral theory for nonlocal dispersal with periodic or almost-periodic time dependence. Rocky Mt. J. Math. 38 (4), 1147–1175.Google Scholar
[7] Li, F., Nakashima, K. & Ni, W.-M. (2008) Stability from the point of view of diffusion, relaxation and spatial inhomogeneity. Discrete Contin. Dyn. Syst. 20 (2), 259–274.CrossRefGoogle Scholar
[8] Li, F. & Ni, W.-M. (2009) On the global existence and finite time blow-up of shadow systems. J. Differ. Equ. 247 (6), 1762–1776.Google Scholar
[9] Li, F. & Yip, N. (2014) Finite time blow-up of parabolic system with nonlocal terms. Indiana Univ. Math. J. 63 (3), 783–829.Google Scholar
[10] Lou, Y. & Nagylaki, T. (2002) A semilinear parabolic system for migration and selection in population gentics. J. Differ. Equ. 181 (2), 388–418.Google Scholar
[11] Lou, Y., Nagylaki, T. & Su, L. (2013) An integro-PDE model from population genetics. J. Differ. Equ. 254 (6), 2367–2392.Google Scholar
[12] Lou, Y., Ni, W.-M. & Su, L. (2010) An indefinite nonlinear diffusion problem in population genetics. II. Stability and multiplicity. Discrete Contin. Dyn. Syst. 27 (2), 643–655.Google Scholar
[13] Lunardi, A. (1995) Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser Verlag, Basel.Google Scholar
[14] Mora, X. (1983) Semilinear parabolic problems define semiflows on C k spaces. Trans. Amer. Math. Soc. 278 (1), 21–55.Google Scholar
[15] Nagylaki, T. (1975) Conditions for the existence of clines. Genetics 80 (3), 595–615.Google Scholar
[16] Nagylaki, T. (2011) The influence of partial panmixia on neutral models of spatial variation. Theor. Popul. Biol. 79 (1–2), 19–38.Google Scholar
[17] Nagylaki, T. (2012) Clines with partial panmixia. Theor. Popul. Biol. 81 (1), 45–68.Google Scholar
[18] Nagylaki, T. (2012) Clines with partial panmixia in an unbounded unidimensional habitat. Theor. Popul. Biol. 82 (1), 22–28.Google Scholar
[19] Nakashima, K., Ni, W.-M. & Su, L. (2010) An indefinite nonlinear diffusion problem in population genetics. I. Existence and limiting profiles. Discrete Contin. Dyn. Syst. 27 (2), 617–641.CrossRefGoogle Scholar
[20] Rawal, N. & Shen, W. (2012) Criteria for the existence and lower bounds of principal eigenvalues of time periodic nonlocal dispersal operators and applications. J. Dyn. Diff. Eqs. 24 (4), 927–954.CrossRefGoogle Scholar
[21] Senn, S. (1983) On a nonlinear elliptic eigenvalue problem with Neumann boundary conditions, with an application to population genetics. Commun. Partial Differ. Equ. 8 (11), 1199–1228.Google Scholar
[22] Shen, W. & Vickers, G. T. (2007) Spectral theory for general nonautonomous/random dispersal evolution operators. J. Differ. Equ. 235 (1), 262–297.Google Scholar
[23] Slatkin, M. (1973) Gene flow and selection in a cline. Genetics 75 (4), 73–756.Google Scholar
[24] Winkler, M. (2010) Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model. J. Differ. Equ. 248 (12), 2889–2905.Google Scholar