Most mathematical studies on expanding populations have focused on the rate of range expansion of a population. However, the genetic consequences of population expansion remain an understudied body of theory. Describing an expanding population as a traveling wave solution derived from a classical reaction-diffusion model, we analyze the spatio-temporal evolution of its genetic structure. We show that the presence of an Allee effect (i.e., a lower per capita growth rate at low densities) drastically modifies genetic diversity, both in the colonization front and behind it. With an Allee effect (i.e., pushed colonization waves), all of the genetic diversity of a population is conserved in the colonization front. In the absence of an Allee effect (i.e., pulled waves), only the furthest forward members of the initial population persist in the colonization front, indicating a strong erosion of the diversity in this population. These results counteract commonly held notions that the Allee effect generally has adverse consequences. Our study contributes new knowledge to the surfing phenomenon in continuous models without random genetic drift. It also provides insight into the dynamics of traveling wave solutions and leads to a new interpretation of the mathematical notions of pulled and pushed waves. R apid increases in the number of biological invasions by alien organisms (1) and the movement of species in response to their climatic niches shifting as a result of climate change have caused a growing number of empirical and observational studies to address the phenomenon of range expansion. Numerous mathematical approaches and simulations have been developed to analyze the processes of these expansions (2, 3). Most results focus on the rate of range expansion (4), and the genetic consequences of range expansion have received little attention from mathematicians and modelers (5). However, range expansions are known to have an important effect on genetic diversity (6, 7) and generally lead to a loss of genetic diversity along the expansion axis due to successive founder effects (8). Simulation studies have already investigated the role of the geometry of the invaded territory (9-11), the importance of long-distance dispersal and the shape of the dispersal kernel (12-14), the effects of local demography (15), or existence of a juvenile stage (13). Further research is needed to obtain mathematical results supporting these empirical and simulation studies, as such results could determine the causes of diversity loss and the factors capable of increasing or reducing it.In a simulation study using a stepping-stone model with a lattice structure, Edmonds et. al (16) analyzed the fate of a neutral mutation present in the leading edge of an expanding population. Although in most cases the mutation remains at a low frequency in its original position, in some cases the mutation increases in frequency and propagates among the leading edge. This phenomenon is described as "surfing" (15). Surfing is caused by the strong genetic drift ...

In this paper, we study the spreading properties of the solutions of an integro-differential equation of the form u t = J * u − u + f (u). We focus on equations with slowly decaying dispersal kernels J(x) which correspond to models of population dynamics with long-distance dispersal events. We prove that for kernels J which decrease to 0 slower than any exponentially decaying function, the level sets of the solution u propagate with an infinite asymptotic speed. Moreover, we obtain lower and upper bounds for the position of any level set of u. These bounds allow us to estimate how the solution accelerates, depending on the kernel J: the slower the kernel decays, the faster the level sets propagate. Our results are in sharp contrast with most results on this type of equation, where the dispersal kernels are generally assumed to decrease exponentially fast, leading to finite propagation speeds.

We investigate the inside structure of one-dimensional reaction-diffusion traveling fronts. The reaction terms are of the monostable, bistable or ignition types. Assuming that the fronts are made of several components with identical diffusion and growth rates, we analyze the spreading properties of each component. In the monostable case, the fronts are classified as pulled or pushed ones, depending on the propagation speed. We prove that any localized component of a pulled front converges locally to 0 at large times in the moving frame of the front, while any component of a pushed front converges to a well determined positive proportion of the front in the moving frame. These results give a new and more complete interpretation of the pulled/pushed terminology which extends the previous definitions to the case of general transition waves. In particular, in the bistable and ignition cases, the fronts are proved to be pushed as they share the same inside structure as the pushed monostable critical fronts. Uniform convergence results and precise estimates of the left and right spreading speeds of the components of pulled and pushed fronts are also established. RésuméOn s'intéresse à la structure interne des fronts progressifs de réaction-diffusion en dimension 1. Les termes de réaction sont du type monostable, bistable ou ignition. Les fronts étant décomposés en une somme de composantes ayant des taux de diffusion et de croissance identiques, nous analysons dans cet article les propriétés d'expansion de chaque composante. Dans le cas monostable, il est connu que les fronts peuvent être classés en fronts tirés et poussés, suivant leur vitesse de propagation. On montre que chaque composante localisée d'un front tiré converge vers 0 en temps grand localement dans le repère du front, alors que chaque composante d'un front poussé converge vers une proportion strictement positive du front dans le repère mobile. Ces résultats fournissent une interprétation nouvelle et plus complète de la terminologie « fronts tirés -fronts poussés », qui étend les définitions antérieures au cas de fronts généralisés de transition. Pour des non-linéarités du type bistable ou ignition, on démontre que les fronts sont poussés, au sens qu'ils vérifient les mêmes propriétés que les fronts monostables critiques poussés. On établit également des résultats de convergence uniforme et des estimations précisées des vitesses d'expansion à gauche et à droite des composantes des fronts tirés et poussés.