An integro-PDE model from population genetics

Abstract We investigate an integro-partial differential equation that models the evolution of the frequencies for two alleles at a single locus under the joint action of migration, selection, and partial panmixia (i.e., global random mating). We extend previous analyses [T. Nagylaki, Clines with partial panmixia, Theor. Popul. Biol. 81 (2012) 45–68] on the maintenance of both alleles from conservative to arbitrary migration and prove the uniqueness and global asymptotic stability of the nontrivial equilibrium. For conservative migration, we show that increasing the rate of panmixia makes it harder to maintain the allele with the smaller average fitness in the population. In terms of the selection function, we estimate the dependence on the panmictic rate of the minimal value of the selection intensity for the persistence of the allele with the smaller average fitness. We also show that, at least in an average sense, increasing panmixia flattens the cline.

[1]  Wenxian Shen,et al.  Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats , 2012 .

[2]  C. Cosner,et al.  Spatial Ecology via Reaction-Diffusion Equations , 2003 .

[3]  Thomas Nagylaki,et al.  Clines with partial panmixia. , 2012, Theoretical population biology.

[4]  Thomas Nagylaki,et al.  The Dynamics of Migration–Selection Models , 2008 .

[5]  Yuan Lou,et al.  An introduction to migration-selection pde models , 2013 .

[6]  Wei-Ming Ni,et al.  An indefinite nonlinear diffusion problem in population gentics,I: Existence and limiting profiles , 2010 .

[7]  Yuan Lou,et al.  An indefinite nonlinear diffusion problem in population genetics, II: Stability and multiplicity , 2010 .

[8]  Peter Hess,et al.  On positive solutions of a linear elliptic eigenvalue problem with neumann boundary conditions , 1982 .

[9]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[10]  T. Nagylaki.  The influence of partial panmixia on neutral models of spatial variation. , 2011, Theoretical population biology.

[11]  H. Weinberger,et al.  Maximum principles in differential equations , 1967 .

[12]  Stefan Senn,et al.  On a nonlinear elliptic eigenvalue problem with Neumann boundary conditions, with an application to population genetics , 1983 .

[13]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[14]  Yuan Lou,et al.  Random dispersal vs. non-local dispersal , 2009 .

[15]  Thomas Nagylaki,et al.  Clines with partial panmixia in an unbounded unidimensional habitat. , 2012, Theoretical population biology.

[16]  J. Coville.  On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators , 2010, 1106.5137.

[17]  Georg Hetzer,et al.  Effects of spatial variations and dispersal strategies on principal eigenvalues of dispersal operators and spreading speeds of monostable equations , 2013 .

[18]  Song-Sun Lin,et al.  On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function , 1980 .

[19]  Wendell H. Fleming,et al.  A selection-migration model in population genetics , 1975 .

[20]  F. M. Arscott,et al.  PERIODIC‐PARABOLIC BOUNDARY VALUE PROBLEMS AND POSITIVITY , 1992 .

[21]  V. Hutson,et al.  The evolution of dispersal , 2003, Journal of mathematical biology.

[22]  T. Nagylaki,et al.  The strong-migration limit in geographically structured populations , 1980, Journal of mathematical biology.

[23]  Yuan Lou,et al.  A Semilinear Parabolic System for Migration and Selection in Population Genetics , 2002 .

[24]  J. Saut.  Remarks on a non linear equation arising in population genetics , 1978 .

[25]  Wei-Ming Ni,et al.  The Mathematics of Diffusion , 2011 .

[26]  Edward Norman Dancer,et al.  On domain perturbation for super-linear Neumann problems and a question of Y. Lou,W-M Ni and L. Su , 2012 .