Article

# Gene surfing in expanding populations

Lyman Laboratory of Physics, Harvard University, 17 Oxford Street, Cambridge, MA 02138, USA.
(Impact Factor: 1.53). 03/2008; 73(1):158-70. DOI: 10.1016/j.tpb.2007.08.008
Source: PubMed

ABSTRACT Large scale genomic surveys are partly motivated by the idea that the neutral genetic variation of a population may be used to reconstruct its migration history. However, our ability to trace back the colonization pathways of a species from their genetic footprints is limited by our understanding of the genetic consequences of a range expansion. Here, we study, by means of simulations and analytical methods, the neutral dynamics of gene frequencies in an asexual population undergoing a continual range expansion in one dimension. During such a colonization period, lineages can fix at the wave front by means of a "surfing" mechanism [Edmonds, C.A., Lillie, A.S., Cavalli-Sforza, L.L., 2004. Mutations arising in the wave front of an expanding population. Proc. Natl. Acad. Sci. 101, 975-979]. We quantify this phenomenon in terms of (i) the spatial distribution of lineages that reach fixation and, closely related, (ii) the continual loss of genetic diversity (heterozygosity) at the wave front, characterizing the approach to fixation. Our stochastic simulations show that an effective population size can be assigned to the wave that controls the (observable) gradient in heterozygosity left behind the colonization process. This effective population size is markedly higher in the presence of cooperation between individuals ("pushed waves") than when individuals proliferate independently ("pulled waves"), and increases only sub-linearly with deme size. To explain these and other findings, we develop a versatile analytical approach, based on the physics of reaction-diffusion systems, that yields simple predictions for any deterministic population dynamics. Our analytical theory compares well with the simulation results for pushed waves, but is less accurate in the case of pulled waves when stochastic fluctuations in the tip of the wave are important.

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##### Article: INSIDE DYNAMICS OF SOLUTIONS OF INTEGRO-DIFFERENTIAL EQUATIONS
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ABSTRACT: In this paper, we investigate the inside dynamics of the positive solutions of integro-differential equations \begin{equation*} \partial_t u(t,x)= (J\star u)(t,x) -u(t,x) + f(u(t,x)), \ t>0 \hbox{ and } x\in\R, \end{equation*} with both \emph{thin-tailed} and \emph{fat-tailed} dispersal kernels $J$ and a monostable reaction term $f.$ The notion of inside dynamics has been introduced to characterize traveling waves of some reaction-diffusion equations~\cite{GarGilHamRoq12}. Assuming that the solution is made of several fractions $\upsilon^i\ge 0$ ($i\in I \subset \N$), its inside dynamics is given by the spatio-temporal evolution of $\upsilon^i$. According to this dynamics, the traveling waves can be classified in two categories: pushed and pulled waves. For thin-tailed kernels, we observe no qualitative differences between the traveling waves of the above integro-differential equations and the traveling waves of the classical reaction-diffusion equations. In particular, in the KPP case ($f(u)\leq f'(0)u$ for all $u\in(0,1)$) we prove that all the traveling waves are pulled. On the other hand for fat-tailed kernels, the integro-differential equations do not admit any traveling waves. Therefore, to analyse the inside dynamics of a solution in this case, we introduce new notions of pulled and pushed solutions. Within this new framework, we provide analytical and numerical results showing that the solutions of integro-differential equations involving a fat-tailed dispersal kernel are pushed. Our results have applications in population genetics. They show that the existence of long distance dispersal events during a colonization tend to preserve the genetic diversity.
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