A simple mathematical model of gradual Darwinian evolution: Emergence of a Gaussian trait distribution in adaptation along a fitness gradient

We consider a simple mathematical model of gradual Darwinian evolution in continuous time and continuous trait space, due to intraspecific competition for common resource in an asexually reproducing population in constant environment, while far from evolutionary stable equilibrium. The model admits exact analytical solution. In particular, Gaussian distribution of the trait emerges from generic initial conditions.Comment: 21 pages, 2 figures, as accepted to J Math Biol 2013/03/1

Recolonisation by diffusion can generate increasing rates of spread

International audienceDiffusion is one of the most frequently used assumptions to explain dispersal. Diffusion models and in particular reaction–diffusion equations usually lead to solutions moving at constant speeds, too slow compared to observations. As early as 1899, Reid had found that the rate of spread of tree species migrating to northern environments at the beginning of the Holocene was too fast to be explained by diffusive dispersal. Rapid spreading is generally explained using long distance dispersal events, modelled through integro-differential equations (IDEs) with exponentially unbounded (EU) kernels, i.e. decaying slower than any exponential. We show here that classical reaction–diffusion models of the Fisher–Kolmogorov–Petrovsky–Piskunov type can produce patterns of colonisation very similar to those of IDEs, if the initial population is EU at the beginning of the considered colonisation event. Many similarities between reaction–diffusion models with EU initial data and IDEs with EU kernels are found; in particular comparable accelerating rates of spread and flattening of the solutions. There was previously no systematic mathematical theory for such reaction–diffusion models with EU initial data. Yet, EU initial data can easily be understood as consequences of colonisation–retraction events and lead to fast spreading and accelerating rates of spread without the long distance hypothesis

The spatio-temporal dynamics of neutral genetic diversity

International audienceThe notions of pulled and pushed solutions of reaction-dispersal equations introduced by Garnier et al. (2012) and Roques et al. (2012) are based on a decomposition of the solutions into several components. In the framework of population dynamics, this decomposition is related to the spatio-temporal evolution of the genetic structure of a population. The pulled solutions describe a rapid erosion of neutral genetic diversity, while the pushed solutions are associated with a maintenance of diversity. This paper is a survey of the most recent applications of these notions to several standard models of population dynamics, including reaction-diffusion equations and systems and integro-differential equations. We describe several counterintuitive results, where unfavorable factors for the persistence and spreading of a population tend to promote diversity in this population. In particular, we show that the Allee effect, the existence of a competitor species, as well as the presence of climate constraints are factors which can promote diversity during a colonization. We also show that long distance dispersal events lead to a higher diversity, whereas the existence of a nonreproductive juvenile stage does not affect the neutral diversity in a range-expanding population

Two classes of nonlocal Evolution Equations related by a shared Traveling Wave Problem

We consider reaction-diffusion equations and Korteweg-de Vries-Burgers (KdVB) equations, i.e. scalar conservation laws with diffusive-dispersive regularization. We review the existence of traveling wave solutions for these two classes of evolution equations. For classical equations the traveling wave problem (TWP) for a local KdVB equation can be identified with the TWP for a reaction-diffusion equation. In this article we study this relationship for these two classes of evolution equations with nonlocal diffusion/dispersion. This connection is especially useful, if the TW equation is not studied directly, but the existence of a TWS is proven using one of the evolution equations instead. Finally, we present three models from fluid dynamics and discuss the TWP via its link to associated reaction-diffusion equations